{"id":196,"date":"2019-08-20T17:04:06","date_gmt":"2019-08-20T21:04:06","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/conic-sections-in-polar-coordinates\/"},"modified":"2022-06-01T10:39:39","modified_gmt":"2022-06-01T14:39:39","slug":"conic-sections-in-polar-coordinates","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/conic-sections-in-polar-coordinates\/","title":{"raw":"Conic Sections in Polar Coordinates","rendered":"Conic Sections in Polar Coordinates"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Identify a conic in polar form.<\/li>\n \t<li>Graph the polar equations of conics.<\/li>\n \t<li>De\ufb01ne conics in terms of a focus and a directrix.<\/li>\n<\/ul>\n<\/div>\n<div class=\"wp-caption-text\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152958\/CNX_Precalc_Figure_10_05_008n.jpg\" alt=\"\" width=\"975\" height=\"353\"> <strong>Figure 1. <\/strong>Planets orbiting the sun follow elliptical paths. (credit: NASA Blueshift, Flickr)[\/caption]\n\n<\/div>\n<p id=\"fs-id1258255\">Most of us are familiar with orbital motion, such as the motion of a planet around the sun or an electron around an atomic nucleus. Within the planetary system, orbits of planets, asteroids, and comets around a larger celestial body are often elliptical. Comets, however, may take on a parabolic or hyperbolic orbit instead. And, in reality, the characteristics of the planets\u2019 orbits may vary over time. Each orbit is tied to the location of the celestial body being orbited and the distance and direction of the planet or other object from that body. As a result, we tend to use polar coordinates to represent these orbits.<\/p>\n<p id=\"fs-id1482642\">In an elliptical orbit, the <span class=\"no-emphasis\">periapsis<\/span> is the point at which the two objects are closest, and the <span class=\"no-emphasis\">apoapsis<\/span> is the point at which they are farthest apart. Generally, the velocity of the orbiting body tends to increase as it approaches the periapsis and decrease as it approaches the apoapsis. Some objects reach an escape velocity, which results in an infinite orbit. These bodies exhibit either a parabolic or a hyperbolic orbit about a body; the orbiting body breaks free of the celestial body\u2019s gravitational pull and fires off into space. Each of these orbits can be modeled by a conic section in the polar coordinate system.<\/p>\n\n<div id=\"fs-id1388752\" class=\"bc-section section\">\n<h3>Identifying a Conic in Polar Form<\/h3>\n<p id=\"fs-id2264768\">Any conic may be determined by three characteristics: a single <span class=\"no-emphasis\">focus<\/span>, a fixed line called the <span class=\"no-emphasis\">directrix<\/span>, and the ratio of the distances of each to a point on the graph. Consider the <span class=\"no-emphasis\">parabola<\/span>[latex]\\,x=2+{y}^{2}\\,[\/latex]shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Figure_10_05_001\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_10_05_001\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153000\/CNX_Precalc_Figure_10_05_001.jpg\" alt=\"\" width=\"487\" height=\"316\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id2787449\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/contents\/a79194a1-878d-4c6d-a978-53670f3e1266\">The Parabola<\/a>, we learned how a parabola is defined by the focus (a fixed point) and the directrix (a fixed line). In this section, we will learn how to define any conic in the polar coordinate system in terms of a fixed point, the focus[latex]\\,P\\left(r,\\theta \\right)\\,[\/latex]at the pole, and a line, the directrix, which is perpendicular to the polar axis.<\/p>\n<p id=\"fs-id1673006\">If[latex]\\,F\\,[\/latex]is a fixed point, the focus, and[latex]\\,D\\,[\/latex]is a fixed line, the directrix, then we can let[latex]\\,e\\,[\/latex]be a fixed positive number, called the <strong>eccentricity<\/strong>, which we can define as the ratio of the distances from a point on the graph to the focus and the point on the graph to the directrix. Then the set of all points[latex]\\,P\\,[\/latex]such that[latex]\\,e=\\frac{PF}{PD}\\,[\/latex]is a conic. In other words, we can define a conic as the set of all points[latex]\\,P\\,[\/latex]with the property that the ratio of the distance from[latex]\\,P\\,[\/latex]to[latex]\\,F\\,[\/latex]to the distance from[latex]\\,P\\,[\/latex]to[latex]\\,D\\,[\/latex]is equal to the constant[latex]\\,e.[\/latex]<\/p>\n<p id=\"fs-id1540310\">For a conic with eccentricity[latex]\\,e,[\/latex]<\/p>\n\n<ul id=\"fs-id2114246\">\n \t<li>if[latex]\\,0\\le e&lt;1,[\/latex] the conic is an ellipse<\/li>\n \t<li>if[latex]\\,e=1,[\/latex] the conic is a parabola<\/li>\n \t<li>if[latex]\\,e&gt;1,[\/latex] the conic is an hyperbola<\/li>\n<\/ul>\n<p id=\"fs-id752226\">With this definition, we may now define a conic in terms of the directrix,[latex]\\,x=\u00b1p,[\/latex] the eccentricity[latex]\\,e,[\/latex] and the angle[latex]\\,\\theta .[\/latex] Thus, each conic may be written as a <strong>polar equation<\/strong>, an equation written in terms of[latex]\\,r\\,[\/latex]and[latex]\\,\\theta .[\/latex]<\/p>\n\n<div id=\"fs-id1161380\" class=\"textbox key-takeaways\">\n<h3>The Polar Equation for a Conic<\/h3>\n<p id=\"fs-id1195811\">For a conic with a focus at the origin, if the directrix is[latex]\\,x=\u00b1p,[\/latex] where[latex]\\,p\\,[\/latex]is a positive real number, and the eccentricity is a positive real number[latex]\\,e,[\/latex] the conic has a polar equation<\/p>\n\n<div id=\"fs-id2238499\" class=\"unnumbered aligncenter\">[latex]r=\\frac{ep}{1\u00b1e\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/div>\n<p id=\"fs-id1354185\">For a conic with a focus at the origin, if the directrix is[latex]\\,y=\u00b1p,[\/latex] where[latex]\\,p\\,[\/latex] is a positive real number, and the eccentricity is a positive real number[latex]\\,e,[\/latex] the conic has a polar equation<\/p>\n\n<div id=\"fs-id862339\" class=\"unnumbered aligncenter\">[latex]r=\\frac{ep}{1\u00b1e\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1331670\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id960683\"><strong>Given the polar equation for a conic, identify the type of conic, the directrix, and the eccentricity.<\/strong><\/p>\n\n<ol id=\"fs-id1814857\" type=\"1\">\n \t<li>Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form.<\/li>\n \t<li>Identify the eccentricity[latex]\\,e\\,[\/latex]as the coefficient of the trigonometric function in the denominator.<\/li>\n \t<li>Compare[latex]\\,e\\,[\/latex]with 1 to determine the shape of the conic.<\/li>\n \t<li>Determine the directrix as[latex]\\,x=p\\,[\/latex]if cosine is in the denominator and[latex]\\,y=p\\,[\/latex]if sine is in the denominator. Set[latex]\\,ep\\,[\/latex]equal to the numerator in standard form to solve for[latex]\\,x\\,[\/latex]or[latex]\\,y.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_10_05_01\" class=\"textbox examples\">\n<div id=\"fs-id2201458\">\n<div id=\"fs-id1367180\">\n<h3>Identifying a Conic Given the Polar Form<\/h3>\n<p id=\"fs-id1687490\">For each of the following equations, identify the conic with focus at the origin, the <span class=\"no-emphasis\">directrix<\/span>, and the <span class=\"no-emphasis\">eccentricity<\/span>.<\/p>\n\n<ol id=\"fs-id2430244\" type=\"a\">\n \t<li>[latex]r=\\frac{6}{3+2\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/li>\n \t<li>[latex]r=\\frac{12}{4+5\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/li>\n \t<li>[latex]r=\\frac{7}{2-2\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1329142\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1329142\"]\n<p id=\"fs-id1329142\">For each of the three conics, we will rewrite the equation in standard form. Standard form has a 1 as the constant in the denominator. Therefore, in all three parts, the first step will be to multiply the numerator and denominator by the reciprocal of the constant of the original equation,[latex]\\,\\frac{1}{c},[\/latex] where[latex]\\,c\\,[\/latex]is that constant.<\/p>\n\n<ol id=\"fs-id1822233\" type=\"a\">\n \t<li>Multiply the numerator and denominator by[latex]\\,\\frac{1}{3}.[\/latex]\n<div id=\"fs-id1227176\" class=\"unnumbered aligncenter\">[latex]r=\\frac{6}{3+2\\mathrm{sin}\\text{ }\\theta }\\cdot \\frac{\\left(\\frac{1}{3}\\right)}{\\left(\\frac{1}{3}\\right)}=\\frac{6\\left(\\frac{1}{3}\\right)}{3\\left(\\frac{1}{3}\\right)+2\\left(\\frac{1}{3}\\right)\\mathrm{sin}\\text{ }\\theta }=\\frac{2}{1+\\frac{2}{3}\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/div>\n<p id=\"fs-id1079973\">Because[latex]\\mathrm{sin}\\text{ }\\theta [\/latex] is in the denominator, the directrix is[latex]\\,y=p.\\,[\/latex]Comparing to standard form, note that[latex]\\,e=\\frac{2}{3}.[\/latex]Therefore, from the numerator,<\/p>\n\n<div id=\"fs-id1233156\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }2=ep\\hfill \\\\ \\text{ }2=\\frac{2}{3}p\\hfill \\\\ \\left(\\frac{3}{2}\\right)2=\\left(\\frac{3}{2}\\right)\\frac{2}{3}p\\hfill \\\\ \\text{ }3=p\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1925603\">Since[latex]\\,e&lt;1,[\/latex] the conic is an <span class=\"no-emphasis\">ellipse<\/span>. The eccentricity is[latex]\\,e=\\frac{2}{3}[\/latex]and the directrix is[latex]\\,y=3.[\/latex]<\/p>\n<\/li>\n \t<li>Multiply the numerator and denominator by[latex]\\,\\frac{1}{4}.[\/latex]\n<div id=\"fs-id1082549\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ r=\\frac{12}{4+5\\text{ }\\mathrm{cos}\\text{ }\\theta }\\cdot \\frac{\\left(\\frac{1}{4}\\right)}{\\left(\\frac{1}{4}\\right)}\\hfill \\end{array}\\hfill \\\\ r=\\frac{12\\left(\\frac{1}{4}\\right)}{4\\left(\\frac{1}{4}\\right)+5\\left(\\frac{1}{4}\\right)\\mathrm{cos}\\text{ }\\theta }\\hfill \\\\ r=\\frac{3}{1+\\frac{5}{4}\\text{ }\\mathrm{cos}\\text{ }\\theta }\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2339764\">Because[latex]\\text{ cos}\\,\\theta [\/latex]is in the denominator, the directrix is[latex]\\,x=p.\\,[\/latex]Comparing to standard form,[latex]\\,e=\\frac{5}{4}.\\,[\/latex]Therefore, from the numerator,<\/p>\n\n<div id=\"fs-id1923702\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }3=ep\\hfill \\\\ \\text{ }3=\\frac{5}{4}p\\hfill \\\\ \\,\\left(\\frac{4}{5}\\right)3=\\left(\\frac{4}{5}\\right)\\frac{5}{4}p\\hfill \\\\ \\text{ }\\,\\frac{12}{5}=p\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2109858\">Since[latex]\\,e&gt;1,[\/latex] the conic is a <span class=\"no-emphasis\">hyperbola<\/span>. The eccentricity is[latex]\\,e=\\frac{5}{4}\\,[\/latex]and the directrix is[latex]\\,x=\\frac{12}{5}=2.4.[\/latex]<\/p>\n<\/li>\n \t<li>Multiply the numerator and denominator by[latex]\\,\\frac{1}{2}.[\/latex]\n<div id=\"fs-id1741224\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ \\begin{array}{l}r=\\frac{7}{2-2\\text{ }\\mathrm{sin}\\text{ }\\theta }\\cdot \\frac{\\left(\\frac{1}{2}\\right)}{\\left(\\frac{1}{2}\\right)}\\hfill \\\\ r=\\frac{7\\left(\\frac{1}{2}\\right)}{2\\left(\\frac{1}{2}\\right)-2\\left(\\frac{1}{2}\\right)\\text{ }\\mathrm{sin}\\text{ }\\theta }\\hfill \\\\ r=\\frac{\\frac{7}{2}}{1-\\mathrm{sin}\\text{ }\\theta }\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1636641\">Because sine is in the denominator, the directrix is[latex]\\,y=-p.\\,[\/latex]Comparing to standard form,[latex]\\,e=1.\\,[\/latex]Therefore, from the numerator,<\/p>\n\n<div id=\"fs-id1734126\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{7}{2}=ep\\\\ \\frac{7}{2}=\\left(1\\right)p\\\\ \\frac{7}{2}=p\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1385531\">Because[latex]\\,e=1,[\/latex] the conic is a <span class=\"no-emphasis\">parabola<\/span>. The eccentricity is[latex]\\,e=1\\,[\/latex]and the directrix is[latex]\\,y=-\\frac{7}{2}=-3.5.[\/latex][\/hidden-answer]<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2892298\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_05_01\">\n<div id=\"fs-id1914310\">\n<p id=\"fs-id2135269\">Identify the conic with focus at the origin, the directrix, and the eccentricity for[latex]\\,r=\\frac{2}{3-\\mathrm{cos}\\text{ }\\theta }.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1667756\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1667756\"]\n<p id=\"fs-id1667756\">ellipse;[latex]\\,e=\\frac{1}{3};\\,x=-2[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1161787\" class=\"bc-section section\">\n<h3>Graphing the Polar Equations of Conics<\/h3>\nWhen graphing in Cartesian coordinates, each conic section has a unique equation. This is not the case when graphing in polar coordinates. We must use the eccentricity of a conic section to determine which type of curve to graph, and then determine its specific characteristics. The first step is to rewrite the conic in standard form as we have done in the previous example. In other words, we need to rewrite the equation so that the denominator begins with 1. This enables us to determine[latex]\\,e\\,[\/latex]and, therefore, the shape of the curve. The next step is to substitute values for[latex]\\,\\theta \\,[\/latex]and solve for[latex]\\,r\\,[\/latex]to plot a few key points. Setting[latex]\\,\\theta \\,[\/latex]equal to[latex]\\,0,\\frac{\\pi }{2},\\pi ,[\/latex] and[latex]\\,\\frac{3\\pi }{2}\\,[\/latex]provides the vertices so we can create a rough sketch of the graph.\n<div id=\"Example_10_05_02\" class=\"textbox examples\">\n<div id=\"fs-id2427459\">\n<div id=\"fs-id2106935\">\n<h3>Graphing a Parabola in Polar Form<\/h3>\n<p id=\"fs-id1351805\">Graph[latex]\\,r=\\frac{5}{3+3\\text{ }\\mathrm{cos}\\text{ }\\theta }.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1316220\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1316220\"]\n<p id=\"fs-id1316220\">First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 3, which is[latex]\\,\\frac{1}{3}.[\/latex]<\/p>\n\n<div id=\"fs-id1240312\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ r=\\frac{5}{3+3\\text{ }\\mathrm{cos}\\text{ }\\theta }=\\frac{5\\left(\\frac{1}{3}\\right)}{3\\left(\\frac{1}{3}\\right)+3\\left(\\frac{1}{3}\\right)\\mathrm{cos}\\text{ }\\theta }\\hfill \\end{array}\\hfill \\\\ r=\\frac{\\frac{5}{3}}{1+\\mathrm{cos}\\text{ }\\theta }\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1923255\">Because[latex]\\,e=1,[\/latex]we will graph a <span class=\"no-emphasis\">parabola<\/span> with a focus at the origin. The function has a[latex] \\mathrm{cos}\\text{ }\\theta ,[\/latex] and there is an addition sign in the denominator, so the directrix is[latex]\\,x=p.[\/latex]<\/p>\n\n<div id=\"fs-id2160621\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{5}{3}=ep\\\\ \\frac{5}{3}=\\left(1\\right)p\\\\ \\frac{5}{3}=p\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1690174\">The directrix is[latex]\\,x=\\frac{5}{3}.[\/latex]<\/p>\n<p id=\"fs-id1672859\">Plotting a few key points as in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Table_10_05_01\">(Figure)<\/a> will enable us to see the vertices. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Figure_10_05_002\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_10_05_01\" summary=\"..\">\n<thead>\n<tr>\n<th><\/th>\n<th>A<\/th>\n<th>B<\/th>\n<th>C<\/th>\n<th>D<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\theta [\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td>[latex]\\pi [\/latex]<\/td>\n<td>[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]r=\\frac{5}{3+3\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/td>\n<td>[latex]\\frac{5}{6}\\approx 0.83[\/latex]<\/td>\n<td>[latex]\\frac{5}{3}\\approx 1.67[\/latex]<\/td>\n<td>undefined<\/td>\n<td>[latex]\\frac{5}{3}\\approx 1.67[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153002\/CNX_Precalc_Figure_10_05_002.jpg\" alt=\"\" width=\"487\" height=\"376\"> <strong>Figure 3.<\/strong>[\/caption]\n\n<span id=\"fs-id1978048\"><\/span>[\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id2256443\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id2769094\">We can check our result with a graphing utility. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Figure_10_05_003\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_10_05_003\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153019\/CNX_Precalc_Figure_10_05_003.jpg\" alt=\"\" width=\"487\" height=\"376\"> <strong>Figure 4.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_10_05_03\" class=\"textbox examples\">\n<div id=\"fs-id2702007\">\n<div id=\"fs-id1908051\">\n<h3>Graphing a Hyperbola in Polar Form<\/h3>\n<p id=\"fs-id1078732\">Graph[latex]\\,r=\\frac{8}{2-3\\text{ }\\mathrm{sin}\\text{ }\\theta }.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1115003\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1115003\"]\n<p id=\"fs-id1115003\">First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 2, which is[latex]\\,\\frac{1}{2}.[\/latex]<\/p>\n\n<div id=\"fs-id2467590\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ r=\\frac{8}{2-3\\mathrm{sin}\\text{ }\\theta }=\\frac{8\\left(\\frac{1}{2}\\right)}{2\\left(\\frac{1}{2}\\right)-3\\left(\\frac{1}{2}\\right)\\mathrm{sin}\\text{ }\\theta }\\hfill \\end{array}\\hfill \\\\ r=\\frac{4}{1-\\frac{3}{2}\\text{ }\\mathrm{sin}\\text{ }\\theta }\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2153652\">Because[latex]\\,e=\\frac{3}{2},e&gt;1,[\/latex] so we will graph a <span class=\"no-emphasis\">hyperbola<\/span> with a focus at the origin. The function has a[latex]\\,\\mathrm{sin}\\text{ }\\theta \\,[\/latex]term and there is a subtraction sign in the denominator, so the directrix is[latex]\\,y=-p.[\/latex]<\/p>\n\n<div id=\"fs-id1114165\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }4=ep\\hfill \\\\ \\text{ }4=\\left(\\frac{3}{2}\\right)p\\hfill \\\\ 4\\left(\\frac{2}{3}\\right)=p\\hfill \\\\ \\text{ }\\frac{8}{3}=p\\hfill \\end{array}[\/latex]<\/div>\nThe directrix is[latex]\\,y=-\\frac{8}{3}.[\/latex]\n\nPlotting a few key points as in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Table_10_05_02\">(Figure)<\/a> will enable us to see the vertices. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Figure_10_05_004\">(Figure)<\/a>.\n<table id=\"Table_10_05_02\" summary=\"..\">\n<thead>\n<tr>\n<th><\/th>\n<th>A<\/th>\n<th>B<\/th>\n<th>C<\/th>\n<th>D<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\theta [\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td>[latex]\\pi [\/latex]<\/td>\n<td>[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]r=\\frac{8}{2-3\\mathrm{sin}\\,\\theta }[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]-8[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]\\frac{8}{5}=1.6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153023\/CNX_Precalc_Figure_10_05_004.jpg\" alt=\"\" width=\"975\" height=\"810\"> <strong>Figure 5.<\/strong>[\/caption]\n<p id=\"fs-id2427799\">[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_10_05_04\" class=\"textbox examples\">\n<div id=\"fs-id1121716\">\n<div id=\"fs-id2040203\">\n<h3>Graphing an Ellipse in Polar Form<\/h3>\n<p id=\"fs-id1983048\">Graph[latex]\\,r=\\frac{10}{5-4\\text{ }\\mathrm{cos}\\text{ }\\theta }.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2231759\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2231759\"]\n<p id=\"fs-id2231759\">First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 5, which is[latex]\\,\\frac{1}{5}.[\/latex]<\/p>\n\n<div id=\"fs-id1732373\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\begin{array}{l}r=\\frac{10}{5-4\\mathrm{cos}\\text{ }\\theta }=\\frac{10\\left(\\frac{1}{5}\\right)}{5\\left(\\frac{1}{5}\\right)-4\\left(\\frac{1}{5}\\right)\\mathrm{cos}\\text{ }\\theta }\\hfill \\\\ r=\\frac{2}{1-\\frac{4}{5}\\text{ }\\mathrm{cos}\\text{ }\\theta }\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2255518\">Because[latex]\\,e=\\frac{4}{5},e&lt;1,[\/latex] so we will graph an <span class=\"no-emphasis\">ellipse<\/span> with a <span class=\"no-emphasis\">focus<\/span> at the origin. The function has a[latex]\\,\\text{cos}\\,\\theta ,[\/latex] and there is a subtraction sign in the denominator, so the <span class=\"no-emphasis\">directrix<\/span> is[latex]\\,x=-p.[\/latex]<\/p>\n\n<div id=\"fs-id2067929\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }2=ep\\hfill \\\\ \\text{ }2=\\left(\\frac{4}{5}\\right)p\\hfill \\\\ 2\\left(\\frac{5}{4}\\right)=p\\hfill \\\\ \\text{ }\\frac{5}{2}=p\\hfill \\end{array}[\/latex]<\/div>\nThe directrix is[latex]\\,x=-\\frac{5}{2}.[\/latex]\n\nPlotting a few key points as in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Table_10_05_03\">(Figure)<\/a> will enable us to see the vertices. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Figure_10_05_006\">(Figure)<\/a>.\n<table id=\"Table_10_05_03\" summary=\"..\">\n<thead>\n<tr>\n<th><\/th>\n<th>A<\/th>\n<th>B<\/th>\n<th>C<\/th>\n<th>D<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\theta [\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td>[latex]\\pi [\/latex]<\/td>\n<td>[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]r=\\frac{10}{5-4\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]\\frac{10}{9}\\approx 1.1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153026\/CNX_Precalc_Figure_10_05_006.jpg\" alt=\"\" width=\"487\" height=\"431\"> <strong>Figure 6.<\/strong>[\/caption]\n<p id=\"fs-id2754428\"><span id=\"fs-id1551880\"><\/span>[\/hidden-answer]<\/p>\n\n<\/div>\n<div id=\"fs-id2306563\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1551888\">We can check our result using a graphing utility. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Figure_10_05_007\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_10_05_007\" class=\"small wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153033\/CNX_Precalc_Figure_10_05_007.jpg\" alt=\"\" width=\"487\" height=\"431\"> <strong>Figure 7. <\/strong>[latex]r=\\frac{10}{5-4\\text{ }\\mathrm{cos}\\text{ }\\theta }\\,[\/latex]graphed on a viewing window of[latex]\\,\\left[\u20133,12,1\\right]\\,[\/latex]by[latex]\\,\\left[\u20134,4,1\\right],\\theta \\,\\text{min =}\\,0[\/latex]and[latex]\\,\\theta \\,\\text{max =}\\,2\\pi .[\/latex][\/caption]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2255486\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_05_02\">\n<div id=\"fs-id2118885\">\n<p id=\"fs-id2118886\">Graph[latex]\\,r=\\frac{2}{4-\\mathrm{cos}\\text{ }\\theta }.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1639620\"]Show Solution[\/reveal-answer][hidden-answer a=\"1639620\"]<span id=\"fs-id1639625\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153035\/CNX_Precalc_Figure_10_05_009.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1916233\" class=\"bc-section section\">\n<h3>De\ufb01ning Conics in Terms of a Focus and a Directrix<\/h3>\n<p id=\"fs-id3026916\">So far we have been using polar equations of conics to describe and graph the curve. Now we will work in reverse; we will use information about the origin, eccentricity, and directrix to determine the polar equation.<\/p>\n\n<div id=\"fs-id3026921\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id3026927\"><strong>Given the focus, eccentricity, and directrix of a conic, determine the polar equation.\n<\/strong><\/p>\n\n<ol id=\"fs-id3026932\" type=\"1\">\n \t<li>Determine whether the directrix is horizontal or vertical. If the directrix is given in terms of[latex]\\,y,[\/latex] we use the general polar form in terms of sine. If the directrix is given in terms of[latex]\\,x,[\/latex] we use the general polar form in terms of cosine.<\/li>\n \t<li>Determine the sign in the denominator. If[latex]\\,p&lt;0,[\/latex] use subtraction. If[latex]\\,p&gt;0,[\/latex] use addition.<\/li>\n \t<li>Write the coefficient of the trigonometric function as the given eccentricity.<\/li>\n \t<li>Write the absolute value of[latex]\\,p\\,[\/latex] in the numerator, and simplify the equation.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_10_05_05\" class=\"textbox examples\">\n<div id=\"fs-id2927796\">\n<div id=\"fs-id2927798\">\n<h3>Finding the Polar Form of a Vertical Conic Given a Focus at the Origin and the Eccentricity and Directrix<\/h3>\n<p id=\"fs-id2927804\">Find the polar form of the <span class=\"no-emphasis\">conic<\/span> given a <span class=\"no-emphasis\">focus<\/span> at the origin,[latex]\\,e=3\\,[\/latex]and <span class=\"no-emphasis\">directrix<\/span>[latex]\\,y=-2.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2956031\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2956031\"]\n<p id=\"fs-id2956031\">The directrix is[latex]\\,y=-p,[\/latex] so we know the trigonometric function in the denominator is sine.<\/p>\n<p id=\"fs-id2821456\">Because[latex]\\,y=-2,\u20132&lt;0,[\/latex] so we know there is a subtraction sign in the denominator. We use the standard form of<\/p>\n\n<div id=\"fs-id2821490\" class=\"unnumbered aligncenter\">[latex]r=\\frac{ep}{1-e\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/div>\n<p id=\"fs-id2479768\">and[latex]\\,e=3\\,[\/latex]and[latex]\\,|-2|=2=p.[\/latex]<\/p>\n<p id=\"fs-id2585526\">Therefore,<\/p>\n\n<div id=\"fs-id2585529\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\begin{array}{l}r=\\frac{\\left(3\\right)\\left(2\\right)}{1-3\\text{ }\\mathrm{sin}\\text{ }\\theta }\\hfill \\\\ r=\\frac{6}{1-3\\text{ }\\mathrm{sin}\\text{ }\\theta }\\hfill \\end{array}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_10_05_06\" class=\"textbox examples\">\n<div id=\"fs-id1823159\">\n<div id=\"fs-id1823161\">\n<h3>Finding the Polar Form of a Horizontal Conic Given a Focus at the Origin and the Eccentricity and Directrix<\/h3>\n<p id=\"fs-id1823168\">Find the <span class=\"no-emphasis\">polar form of a conic<\/span> given a <span class=\"no-emphasis\">focus<\/span> at the origin,[latex]\\,e=\\frac{3}{5},[\/latex] and <span class=\"no-emphasis\">directrix<\/span>[latex]\\,x=4.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1085849\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1085849\"]\n<p id=\"fs-id1085849\">Because the directrix is[latex]\\,x=p,[\/latex]we know the function in the denominator is cosine. Because[latex]\\,x=4,4&gt;0,[\/latex]so we know there is an addition sign in the denominator. We use the standard form of<\/p>\n\n<div id=\"fs-id2735265\" class=\"unnumbered aligncenter\">[latex]r=\\frac{ep}{1+e\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/div>\n<p id=\"fs-id2735304\">and[latex]\\,e=\\frac{3}{5}\\,[\/latex]and[latex]\\,|4|=4=p.[\/latex]<\/p>\n<p id=\"fs-id1263417\">Therefore,<\/p>\n\n<div id=\"fs-id1263420\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ r=\\frac{\\left(\\frac{3}{5}\\right)\\left(4\\right)}{1+\\frac{3}{5}\\,\\mathrm{cos}\\,\\theta }\\hfill \\end{array}\\hfill \\\\ r=\\frac{\\frac{12}{5}}{1+\\frac{3}{5}\\,\\mathrm{cos}\\,\\theta }\\hfill \\\\ r=\\frac{\\frac{12}{5}}{1\\left(\\frac{5}{5}\\right)+\\frac{3}{5}\\,\\mathrm{cos}\\,\\theta }\\hfill \\\\ r=\\frac{\\frac{12}{5}}{\\frac{5}{5}+\\frac{3}{5}\\,\\mathrm{cos}\\,\\theta }\\hfill \\\\ r=\\frac{12}{5}\\cdot \\frac{5}{5+3\\,\\mathrm{cos}\\,\\theta }\\hfill \\\\ r=\\frac{12}{5+3\\,\\mathrm{cos}\\,\\theta }\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2020858\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_05_03\">\n<div id=\"fs-id2020868\">\n<p id=\"fs-id2020869\">Find the polar form of the conic given a focus at the origin,[latex]\\,e=1,[\/latex] and directrix[latex]\\,x=-1.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1121326\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1121326\"]\n<p id=\"fs-id1121326\">[latex]r=\\frac{1}{1-\\mathrm{cos}\\theta }[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_10_05_07\" class=\"textbox examples\">\n<div id=\"fs-id1121364\">\n<div id=\"fs-id1121366\">\n<h3>Converting a Conic in Polar Form to Rectangular Form<\/h3>\n<p id=\"fs-id1121372\">Convert the conic[latex]\\,r=\\frac{1}{5-5\\mathrm{sin}\\,\\theta }[\/latex]to rectangular form.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1558062\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1558062\"]\n<p id=\"fs-id1558062\">We will rearrange the formula to use the identities[latex] r=\\sqrt{{x}^{2}+{y}^{2}},x=r\\,\\mathrm{cos}\\,\\theta ,\\text{and }y=r\\,\\mathrm{sin}\\,\\theta .[\/latex]<\/p>\n\n<div id=\"fs-id2881090\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\text{ }r=\\frac{1}{5-5\\,\\mathrm{sin}\\,\\theta }\\hfill &amp; \\hfill \\\\ r\\cdot \\left(5-5\\,\\mathrm{sin}\\,\\theta \\right)=\\frac{1}{5-5\\,\\mathrm{sin}\\,\\theta }\\cdot \\left(5-5\\,\\mathrm{sin}\\,\\theta \\right)\\hfill &amp; \\text{Eliminate the fraction}.\\hfill \\\\ \\text{ }\\,5r-5r\\,\\mathrm{sin}\\,\\theta =1\\hfill &amp; \\text{Distribute}.\\hfill \\\\ \\text{ }5r=1+5r\\,\\mathrm{sin}\\,\\theta \\hfill &amp; \\text{Isolate }5r.\\hfill \\\\ \\text{ }25{r}^{2}={\\left(1+5r\\,\\mathrm{sin}\\,\\theta \\right)}^{2}\\hfill &amp; \\text{Square both sides}.\\hfill \\\\ \\text{ }25\\left({x}^{2}+{y}^{2}\\right)={\\left(1+5y\\right)}^{2}\\hfill &amp; \\text{Substitute }r=\\sqrt{{x}^{2}+{y}^{2}}\\text{ and }y=r\\,\\mathrm{sin}\\,\\theta .\\hfill \\\\ \\text{ }\\,25{x}^{2}+25{y}^{2}=1+10y+25{y}^{2}\\hfill &amp; \\text{Distribute and use FOIL}.\\hfill \\\\ \\text{ }\\,25{x}^{2}-10y=1\\hfill &amp; \\text{Rearrange terms and set equal to 1}.\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1587301\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_05_04\">\n<div id=\"fs-id1587310\">\n<p id=\"fs-id1587311\">Convert the conic[latex]\\,r=\\frac{2}{1+2\\text{ }\\mathrm{cos}\\text{ }\\theta }\\,[\/latex]to rectangular form.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2445406\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2445406\"]\n<p id=\"fs-id2445406\">[latex]4-8x+3{x}^{2}-{y}^{2}=0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2773738\" class=\"precalculus media\">\n<p id=\"fs-id2773744\">Access these online resources for additional instruction and practice with conics in polar coordinates.<\/p>\n\n<ul id=\"eip-id2676160\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/determineconic\">Polar Equations of Conic Sections<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphconic1\">Graphing Polar Equations of Conics - 1<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphconic2\">Graphing Polar Equations of Conics - 2<\/a><\/li>\n<\/ul>\n<\/div>\n<p id=\"eip-127\">Visit <a href=\"http:\/\/openstaxcollege.org\/l\/PreCalcLPC10\">this website<\/a> for additional practice questions from Learningpod.<\/p>\n\n<\/div>\n<div id=\"fs-id2773762\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id2773768\">\n \t<li>Any conic may be determined by a single focus, the corresponding eccentricity, and the directrix. We can also define a conic in terms of a fixed point, the focus[latex]\\,P\\left(r,\\theta \\right)\\,[\/latex]at the pole, and a line, the directrix, which is perpendicular to the polar axis.<\/li>\n \t<li>A conic is the set of all points[latex]\\,e=\\frac{PF}{PD},[\/latex] where eccentricity[latex]\\,e\\,[\/latex]is a positive real number. Each conic may be written in terms of its polar equation. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Example_10_05_01\">(Figure)<\/a>.<\/li>\n \t<li>The polar equations of conics can be graphed. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Example_10_05_02\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Example_10_05_03\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Example_10_05_04\">(Figure)<\/a>.<\/li>\n \t<li>Conics can be defined in terms of a focus, a directrix, and eccentricity. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Example_10_05_05\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Example_10_05_06\">(Figure)<\/a>.<\/li>\n \t<li>We can use the identities[latex]\\,r=\\sqrt{{x}^{2}+{y}^{2}},x=r\\text{ }\\mathrm{cos}\\text{ }\\theta ,[\/latex]and[latex]\\,y=r\\text{ }\\mathrm{sin}\\text{ }\\theta \\,[\/latex]to convert the equation for a conic from polar to rectangular form. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Example_10_05_07\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id2879646\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id2879649\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id2879654\">\n<div id=\"fs-id2879655\">\n<p id=\"fs-id2879656\">Explain how eccentricity determines which conic section is given.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2879662\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2879662\"]\n<p id=\"fs-id2879662\">If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2879668\">\n<div id=\"fs-id1818181\">\n<p id=\"fs-id1818182\">If a conic section is written as a polar equation, what must be true of the denominator?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1818185\">\n<div id=\"fs-id1818186\">\n<p id=\"fs-id1818187\">If a conic section is written as a polar equation, and the denominator involves[latex]\\,\\mathrm{sin}\\text{ }\\theta ,[\/latex]what conclusion can be drawn about the directrix?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1818208\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1818208\"]\n<p id=\"fs-id1818208\">The directrix will be parallel to the polar axis.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1818213\">\n<div id=\"fs-id1818214\">\n<p id=\"fs-id1818215\">If the directrix of a conic section is perpendicular to the polar axis, what do we know about the equation of the graph?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1818219\">\n<div id=\"fs-id1818220\">\n<p id=\"fs-id1818221\">What do we know about the focus\/foci of a conic section if it is written as a polar equation?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1818227\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1818227\"]\n<p id=\"fs-id1818227\">One of the foci will be located at the origin.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1818232\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1818237\">For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.<\/p>\n\n<div id=\"fs-id1818242\">\n<div id=\"fs-id1818243\">\n<p id=\"fs-id1818244\">[latex]r=\\frac{6}{1-2\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1264809\">\n<div id=\"fs-id1264810\">\n<p id=\"fs-id1264811\">[latex]r=\\frac{3}{4-4\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1264847\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1264847\"]\n<p id=\"fs-id1264847\">Parabola with[latex]\\,e=1\\,[\/latex]and directrix[latex]\\,\\frac{3}{4}\\,[\/latex]units below the pole.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2199203\">\n<div id=\"fs-id2199204\">\n<p id=\"fs-id2199205\">[latex]r=\\frac{8}{4-3\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2920062\">\n<div id=\"fs-id2920063\">\n<p id=\"fs-id2920064\">[latex]r=\\frac{5}{1+2\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2920100\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2920100\"]\n<p id=\"fs-id2920100\">Hyperbola with[latex]\\,e=2\\,[\/latex]and directrix[latex]\\,\\frac{5}{2}\\,[\/latex]units above the pole.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id2245309\">[latex]r=\\frac{16}{4+3\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2245346\">\n<div id=\"fs-id2245347\">\n<p id=\"fs-id2245348\">[latex]r=\\frac{3}{10+10\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1232410\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1232410\"]\n<p id=\"fs-id1232410\">Parabola with[latex]\\,e=1\\,[\/latex]and directrix[latex]\\,\\frac{3}{10}\\,[\/latex]units to the right of the pole.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1232457\">\n<div id=\"fs-id1232458\">\n<p id=\"fs-id1232459\">[latex]r=\\frac{2}{1-\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1403116\">\n<div id=\"fs-id1403117\">\n<p id=\"fs-id1403118\">[latex]r=\\frac{4}{7+2\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1403154\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1403154\"]\n<p id=\"fs-id1403154\">Ellipse with[latex]\\,e=\\frac{2}{7}\\,[\/latex]and directrix[latex]\\,2\\,[\/latex]units to the right of the pole.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2322949\">\n<div id=\"fs-id2322950\">[latex]r\\left(1-\\mathrm{cos}\\text{ }\\theta \\right)=3[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id2322986\">\n<div id=\"fs-id2285129\">\n<p id=\"fs-id2285130\">[latex]r\\left(3+5\\mathrm{sin}\\text{ }\\theta \\right)=11[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2285169\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2285169\"]\n<p id=\"fs-id2285169\">Hyperbola with[latex]\\,e=\\frac{5}{3}\\,[\/latex]and directrix[latex]\\,\\frac{11}{5}\\,[\/latex]units above the pole.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1999272\">\n<div id=\"fs-id1999273\">\n<p id=\"fs-id1999274\">[latex]r\\left(4-5\\mathrm{sin}\\text{ }\\theta \\right)=1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1999312\">\n<div id=\"fs-id1999313\">[latex]r\\left(7+8\\mathrm{cos}\\text{ }\\theta \\right)=7[\/latex]<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2289117\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2289117\"]\n<p id=\"fs-id2289117\">Hyperbola with[latex]\\,e=\\frac{8}{7}\\,[\/latex]and directrix[latex]\\,\\frac{7}{8}\\,[\/latex]units to the right of the pole.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2289168\">For the following exercises, convert the polar equation of a conic section to a rectangular equation.<\/p>\n\n<div id=\"fs-id2289171\">\n<div>\n<p id=\"fs-id2106793\">[latex]r=\\frac{4}{1+3\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2106827\">\n<div id=\"fs-id2106828\">\n<p id=\"fs-id2106829\">[latex]r=\\frac{2}{5-3\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2106865\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2106865\"]\n<p id=\"fs-id2106865\">[latex]25{x}^{2}+16{y}^{2}-12y-4=0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2430897\">\n<div id=\"fs-id2430898\">\n<p id=\"fs-id2430899\">[latex]r=\\frac{8}{3-2\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2261851\">\n<div id=\"fs-id2261852\">\n<p id=\"fs-id2261853\">[latex]r=\\frac{3}{2+5\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2261889\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2261889\"]\n<p id=\"fs-id2261889\">[latex]21{x}^{2}-4{y}^{2}-30x+9=0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1468839\">\n<div id=\"fs-id1468840\">\n<p id=\"fs-id1468841\">[latex]r=\\frac{4}{2+2\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1468875\">\n<div id=\"fs-id1468876\">\n<p id=\"fs-id1468877\">[latex]r=\\frac{3}{8-8\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1468914\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1468914\"]\n<p id=\"fs-id1468914\">[latex]64{y}^{2}=48x+9[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2601785\">\n<div id=\"fs-id2601786\">\n<p id=\"fs-id2601787\">[latex]r=\\frac{2}{6+7\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2601821\">\n<div id=\"fs-id2601822\">\n<p id=\"fs-id2601823\">[latex]r=\\frac{5}{5-11\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1233482\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1233482\"]\n<p id=\"fs-id1233482\">[latex]96{y}^{2}-25{x}^{2}+110y+25=0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1233533\">\n<div id=\"fs-id1233534\">\n<p id=\"fs-id1233535\">[latex]r\\left(5+2\\text{ }\\mathrm{cos}\\text{ }\\theta \\right)=6[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1924146\">\n<div id=\"fs-id1924148\">\n<p id=\"fs-id1924149\">[latex]r\\left(2-\\mathrm{cos}\\text{ }\\theta \\right)=1[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"645551\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"645551\"][latex]3{x}^{2}+4{y}^{2}-2x-1=0[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1649438\">\n<div id=\"fs-id1649439\">\n<p id=\"fs-id1649440\">[latex]r\\left(2.5-2.5\\text{ }\\mathrm{sin}\\text{ }\\theta \\right)=5[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1649477\">\n<div id=\"fs-id1649478\">\n<p id=\"fs-id1649479\">[latex]r=\\frac{6\\mathrm{sec}\\text{ }\\theta }{-2+3\\text{ }\\mathrm{sec}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2439850\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2439850\"]\n<p id=\"fs-id2439850\">[latex]5{x}^{2}+9{y}^{2}-24x-36=0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2439901\">\n<div id=\"fs-id2439902\">\n<p id=\"fs-id2439903\">[latex]r=\\frac{6\\mathrm{csc}\\text{ }\\theta }{3+2\\text{ }\\mathrm{csc}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1978601\">For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.<\/p>\n\n<div id=\"fs-id1978606\">\n<div id=\"fs-id1978607\">\n<p id=\"fs-id1978608\">[latex]r=\\frac{5}{2+\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1978640\"]Show Solution[\/reveal-answer][hidden-answer a=\"1978640\"]<span id=\"fs-id1978645\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153037\/CNX_Precalc_Figure_10_05_201.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1978656\">\n<div>\n<p id=\"fs-id1978658\">[latex]r=\\frac{2}{3+3\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1872482\">\n<div id=\"fs-id1872483\">\n<p id=\"fs-id1872484\">[latex]r=\\frac{10}{5-4\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1872521\"]Show Solution[\/reveal-answer][hidden-answer a=\"1872521\"]<span id=\"fs-id1872527\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153047\/CNX_Precalc_Figure_10_05_203.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1872537\">\n<div id=\"fs-id1872538\">\n<p id=\"fs-id1872539\">[latex]r=\\frac{3}{1+2\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1479120\">\n<div id=\"fs-id1479121\">\n<p id=\"fs-id1479122\">[latex]r=\\frac{8}{4-5\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1479156\"]Show Solution[\/reveal-answer][hidden-answer a=\"1479156\"]<span id=\"fs-id1479161\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153050\/CNX_Precalc_Figure_10_05_205.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1479172\">\n<div id=\"fs-id1479173\">\n<p id=\"fs-id1479174\">[latex]r=\\frac{3}{4-4\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1479208\">\n<div id=\"fs-id1479209\">\n<p id=\"fs-id1479210\">[latex]r=\\frac{2}{1-\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1703892\"]Show Solution[\/reveal-answer][hidden-answer a=\"1703892\"]<span id=\"fs-id1703897\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153052\/CNX_Precalc_Figure_10_05_207.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1703908\">\n<div id=\"fs-id1703909\">\n<p id=\"fs-id1703910\">[latex]r=\\frac{6}{3+2\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1703944\">\n<div id=\"fs-id1703945\">\n<p id=\"fs-id1703946\">[latex]r\\left(1+\\mathrm{cos}\\text{ }\\theta \\right)=5[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"2089733\"]Show Solution[\/reveal-answer][hidden-answer a=\"2089733\"]<span id=\"fs-id2089738\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153102\/CNX_Precalc_Figure_10_05_209.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id2089749\">\n<div id=\"fs-id2089750\">\n<p id=\"fs-id2089751\">[latex]r\\left(3-4\\mathrm{sin}\\text{ }\\theta \\right)=9[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2089788\">\n<div id=\"fs-id2089789\">\n<p id=\"fs-id2089790\">[latex]r\\left(3-2\\mathrm{sin}\\text{ }\\theta \\right)=6[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"2089828\"]Show Solution[\/reveal-answer][hidden-answer a=\"2089828\"]<span id=\"fs-id2089833\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153112\/CNX_Precalc_Figure_10_05_211.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id2253204\">\n<div id=\"fs-id2253206\">\n<p id=\"fs-id2253207\">[latex]r\\left(6-4\\mathrm{cos}\\text{ }\\theta \\right)=5[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2253244\">For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.<\/p>\n\n<div id=\"fs-id2253248\">\n<div id=\"fs-id2253249\">\n<p id=\"fs-id2253250\">Directrix:[latex]x=4;\\,e=\\frac{1}{5}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2253287\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2253287\"]\n<p id=\"fs-id2253287\">[latex]r=\\frac{4}{5+\\mathrm{cos}\\theta }[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2222207\">\n<div id=\"fs-id2222208\">\n<p id=\"fs-id2222209\">Directrix:[latex]x=-4;\\,e=5[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2222239\">\n<div id=\"fs-id2222240\">\n<p id=\"fs-id2222241\">Directrix:[latex]y=2;\\,e=2[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2222264\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2222264\"]\n<p id=\"fs-id2222264\">[latex]r=\\frac{4}{1+2\\mathrm{sin}\\theta }[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2222301\">\n<div id=\"fs-id2222302\">\n<p id=\"fs-id2222303\">Directrix: [latex]y=-2;\\,e=\\frac{1}{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1929509\">\n<p id=\"fs-id1929510\">Directrix:[latex]x=1;\\,e=1[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1929537\"]Show Solution[\/reveal-answer][hidden-answer a=\"1929537\"]\n[latex]r=\\frac{1}{1+\\mathrm{cos}\\theta }[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1929573\">\n<div id=\"fs-id1929574\">\n<p id=\"fs-id1929575\">Directrix:[latex]x=-1;\\,e=1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1783904\">\n<div id=\"fs-id1783905\">\n<p id=\"fs-id1783906\">Directrix: [latex]x=-\\frac{1}{4};\\,e=\\frac{7}{2}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1783952\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1783952\"]\n<p id=\"fs-id1783952\">[latex]r=\\frac{7}{8-28\\mathrm{cos}\\theta }[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1783989\">\n<div id=\"fs-id1783990\">\n<p id=\"fs-id1783991\">Directrix:[latex]y=\\frac{2}{5};\\,e=\\frac{7}{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2900253\">\n<div id=\"fs-id2900254\">\n<p id=\"fs-id2900255\">Directrix: [latex]y=4;\\,e=\\frac{3}{2}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2900291\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2900291\"]\n<p id=\"fs-id2900291\">[latex]r=\\frac{12}{2+3\\mathrm{sin}\\theta }[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2900329\">\n<div id=\"fs-id2900330\">\n<p id=\"fs-id2900332\">Directrix:[latex]x=-2;\\,e=\\frac{8}{3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1529994\">\n<div id=\"fs-id1529996\">\n<p id=\"fs-id1529997\">Directrix:[latex]x=-5;\\,e=\\frac{3}{4}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1530035\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1530035\"]\n<p id=\"fs-id1530035\">[latex]r=\\frac{15}{4-3\\mathrm{cos}\\theta }[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1530073\">\n<div>\n<p id=\"fs-id1530076\">Directrix:[latex]y=2;\\,e=2.5[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2893428\">\n<div id=\"fs-id2893429\">\n<p id=\"fs-id2893430\">Directrix:[latex]x=-3;\\,e=\\frac{1}{3}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2893468\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2893468\"]\n<p id=\"fs-id2893468\">[latex]r=\\frac{3}{3-3\\mathrm{cos}\\theta }[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2893505\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id2893510\">Recall from <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/rotation-of-axes\/\">Rotation of Axes<\/a> that equations of conics with an[latex]\\,xy\\,[\/latex]term have rotated graphs. For the following exercises, express each equation in polar form with[latex]\\,r\\,[\/latex]as a function of[latex]\\,\\theta .[\/latex]<\/p>\n\n<div id=\"fs-id2000559\">\n<div id=\"fs-id2000560\">\n<p id=\"fs-id2000562\">[latex]xy=2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2000580\">\n<div id=\"fs-id2000581\">\n<p id=\"fs-id2000582\">[latex]{x}^{2}+xy+{y}^{2}=4[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2000626\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2000626\"]\n<p id=\"fs-id2000626\">[latex]r=\u00b1\\frac{2}{\\sqrt{1+\\mathrm{sin}\\theta \\mathrm{cos}\\theta }}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2191166\">\n<div id=\"fs-id2191167\">\n<p id=\"fs-id2191168\">[latex]2{x}^{2}+4xy+2{y}^{2}=9[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id2191217\">\n<p id=\"fs-id2191218\">[latex]16{x}^{2}+24xy+9{y}^{2}=4[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1237498\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1237498\"]\n<p id=\"fs-id1237498\">[latex]r=\u00b1\\frac{2}{4\\mathrm{cos}\\theta +3\\mathrm{sin}\\theta }[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1237539\">\n<div id=\"fs-id1237540\">\n<p id=\"fs-id1237541\">[latex]2xy+y=1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1237568\" class=\"review-exercises textbox exercises\">\n<h3>Chapter Review Exercises<\/h3>\n<div id=\"fs-id1237574\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/contents\/99d38770-49c7-49d3-b567-88f393ffb4fe\">The Ellipse<\/a><\/h4>\n<p id=\"fs-id1237580\">For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.<\/p>\n\n<div id=\"fs-id1237584\">\n<div id=\"fs-id1237585\">\n<p id=\"fs-id1237586\">[latex]\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{64}=1[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1493914\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1493914\"]\n<p id=\"fs-id1493914\">[latex]\\frac{{x}^{2}}{{5}^{2}}+\\frac{{y}^{2}}{{8}^{2}}=1;\\,[\/latex]center:[latex]\\,\\left(0,0\\right);\\,[\/latex]vertices:[latex]\\,\\left(5,0\\right),\\left(-5,0\\right),\\left(0,8\\right),\\left(0,-8\\right);\\,[\/latex]foci:[latex]\\,\\left(0,\\sqrt{39}\\right),\\left(0,-\\sqrt{39}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1860702\">\n<div id=\"fs-id1860703\">\n<p id=\"fs-id1860704\">[latex]\\frac{{\\left(x-2\\right)}^{2}}{100}+\\frac{{\\left(y+3\\right)}^{2}}{36}=1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2146165\">\n<div id=\"fs-id2146166\">\n<p id=\"fs-id2146167\">[latex]9{x}^{2}+{y}^{2}+54x-4y+76=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2146224\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2146224\"]\n<p id=\"fs-id2146224\">[latex]\\frac{{\\left(x+3\\right)}^{2}}{{1}^{2}}+\\frac{{\\left(y-2\\right)}^{2}}{{3}^{2}}=1\\,\\,\\left(-3,2\\right);\\,\\,\\left(-2,2\\right),\\left(-4,2\\right),\\left(-3,5\\right),\\left(-3,-1\\right);\\,\\,\\left(-3,2+2\\sqrt{2}\\right),\\left(-3,2-2\\sqrt{2}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2613644\">\n<div id=\"fs-id2613646\">\n<p id=\"fs-id2613647\">[latex]9{x}^{2}+36{y}^{2}-36x+72y+36=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2613703\">For the following exercises, graph the ellipse, noting center, vertices, and foci.<\/p>\n\n<div id=\"fs-id2613706\">\n<div id=\"fs-id2613707\">\n<p id=\"fs-id2613708\">[latex]\\frac{{x}^{2}}{36}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2888444\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2888444\"]\n<p id=\"fs-id2888444\">center:[latex]\\,\\left(0,0\\right);\\,[\/latex]vertices:[latex]\\,\\left(6,0\\right),\\left(-6,0\\right),\\left(0,3\\right),\\left(0,-3\\right);\\,[\/latex]foci:[latex]\\,\\left(3\\sqrt{3},0\\right),\\left(-3\\sqrt{3},0\\right)[\/latex]<span id=\"fs-id2196864\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153114\/CNX_Precalc_Figure_10_05_223.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2196874\">\n<div id=\"fs-id2196876\">\n<p id=\"fs-id2196877\">[latex]\\frac{{\\left(x-4\\right)}^{2}}{25}+\\frac{{\\left(y+3\\right)}^{2}}{49}=1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1548201\">\n<div id=\"fs-id1548202\">\n<p id=\"fs-id1548203\">[latex]4{x}^{2}+{y}^{2}+16x+4y-44=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1737673\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1737673\"]\n<p id=\"fs-id1737673\">center:[latex]\\,\\left(-2,-2\\right);\\,[\/latex]vertices:[latex]\\,\\left(2,-2\\right),\\left(-6,-2\\right),\\left(-2,6\\right),\\left(-2,-10\\right);\\,[\/latex]foci:[latex]\\,\\left(-2,-2+4\\sqrt{3},\\right),\\left(-2,-2-4\\sqrt{3}\\right)[\/latex]<\/p>\n\n<div><\/div>\n<span id=\"fs-id1363974\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153117\/CNX_Precalc_Figure_10_05_225.jpg\" alt=\"\">[\/hidden-answer]<\/span>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1363984\">\n<div id=\"fs-id1363985\">\n<p id=\"fs-id1363986\">[latex]\\,2{x}^{2}+3{y}^{2}-20x+12y+38=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id3032766\">For the following exercises, use the given information to find the equation for the ellipse.<\/p>\n\n<div id=\"fs-id3032771\">\n<div id=\"fs-id3032772\">\n<p id=\"fs-id3032773\">Center at [latex]\\,\\left(0,0\\right),[\/latex]focus at[latex]\\,\\left(3,0\\right),[\/latex]vertex at[latex]\\,\\left(-5,0\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2122257\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2122257\"]\n<p id=\"fs-id2122257\">[latex]\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{16}=1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2122319\">\n<div id=\"fs-id2122320\">\n<p id=\"fs-id2122321\">Center at[latex]\\,\\left(2,-2\\right),[\/latex]vertex at[latex]\\,\\left(7,-2\\right),[\/latex]focus at[latex]\\,\\left(4,-2\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2329260\">\n<div id=\"fs-id2329261\">\n<p id=\"fs-id2329262\">A whispering gallery is to be constructed such that the foci are located 35 feet from the center. If the length of the gallery is to be 100 feet, what should the height of the ceiling be?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2329269\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2329269\"]\n<p id=\"fs-id2329269\">Approximately 35.71 feet<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2329274\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/contents\/23e28179-8eb8-4790-a332-861509a3d9fb\">The Hyperbola<\/a><\/h4>\n<p id=\"fs-id2329278\">For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci.<\/p>\n\n<div id=\"fs-id2329283\">\n<div id=\"fs-id2329284\">\n<p id=\"fs-id2329285\">[latex]\\frac{{x}^{2}}{81}-\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2329343\">\n<div id=\"fs-id2329344\">\n<p id=\"fs-id2329345\">[latex]\\frac{{\\left(y+1\\right)}^{2}}{16}-\\frac{{\\left(x-4\\right)}^{2}}{36}=1[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1408198\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1408198\"]\n<p id=\"fs-id1408198\">[latex]\\frac{{\\left(y+1\\right)}^{2}}{{4}^{2}}-\\frac{{\\left(x-4\\right)}^{2}}{{6}^{2}}=1;\\,[\/latex]center:[latex]\\,\\left(4,-1\\right);\\,[\/latex]vertices:[latex]\\,\\left(4,3\\right),\\left(4,-5\\right);\\,[\/latex]foci:[latex]\\,\\left(4,-1+2\\sqrt{13}\\right),\\left(4,-1-2\\sqrt{13}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1550436\">\n<div id=\"fs-id1550438\">\n<p id=\"fs-id1550439\">[latex]9{y}^{2}-4{x}^{2}+54y-16x+29=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2875786\">\n<div id=\"fs-id2875787\">\n<p id=\"fs-id2875788\">[latex]3{x}^{2}-{y}^{2}-12x-6y-9=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2875845\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2875845\"]\n<p id=\"fs-id2875845\">[latex]\\frac{{\\left(x-2\\right)}^{2}}{{2}^{2}}-\\frac{{\\left(y+3\\right)}^{2}}{{\\left(2\\sqrt{3}\\right)}^{2}}=1;\\,[\/latex]center:[latex]\\,\\left(2,-3\\right);\\,[\/latex]vertices:[latex]\\,\\left(4,-3\\right),\\left(0,-3\\right);\\,[\/latex]foci:[latex]\\,\\left(6,-3\\right),\\left(-2,-3\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1899768\">For the following exercises, graph the hyperbola, labeling vertices and foci.<\/p>\n\n<div id=\"fs-id1899771\">\n<div id=\"fs-id1899772\">\n<p id=\"fs-id1899774\">[latex]\\frac{{x}^{2}}{9}-\\frac{{y}^{2}}{16}=1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2130346\">\n<div id=\"fs-id2130347\">\n<p id=\"fs-id2130348\">[latex]\\frac{{\\left(y-1\\right)}^{2}}{49}-\\frac{{\\left(x+1\\right)}^{2}}{4}=1[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"2130442\"]Show Solution[\/reveal-answer][hidden-answer a=\"2130442\"]\n\n<img src=\"https:\/\/oasis.geneseo.edu\/trig-pics\/CNX_Precalc_Figure_10_05_228.jpg#fixme\" alt=\"\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2130458\">\n<div id=\"fs-id2130459\">\n<p id=\"fs-id2130460\">[latex]{x}^{2}-4{y}^{2}+6x+32y-91=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2732697\">\n<div id=\"fs-id2732698\">\n<p id=\"fs-id2732699\">[latex]2{y}^{2}-{x}^{2}-12y-6=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"2732747\"]Show Solution[\/reveal-answer][hidden-answer a=\"2732747\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153619\/CNX_Precalc_Figure_10_05_230.jpg\" alt=\"\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2732763\">For the following exercises, find the equation of the hyperbola.<\/p>\n\n<div id=\"fs-id2732766\">\n<div id=\"fs-id2732767\">\n<p id=\"fs-id2732768\">Center at[latex]\\,\\left(0,0\\right),[\/latex]vertex at[latex]\\,\\left(0,4\\right),[\/latex]focus at[latex]\\,\\left(0,-6\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1245770\">\n<div id=\"fs-id1245771\">\n<p id=\"fs-id1245772\">Foci at[latex]\\,\\left(3,7\\right)\\,[\/latex]and[latex]\\,\\left(7,7\\right),[\/latex]vertex at[latex]\\,\\left(6,7\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1931177\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1931177\"]\n<p id=\"fs-id1931177\">[latex]\\frac{{\\left(x-5\\right)}^{2}}{1}-\\frac{{\\left(y-7\\right)}^{2}}{3}=1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1931269\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/contents\/a79194a1-878d-4c6d-a978-53670f3e1266\">The Parabola<\/a><\/h4>\n<p id=\"fs-id1931274\">For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.<\/p>\n\n<div id=\"fs-id1931279\">\n<div id=\"fs-id1931280\">\n<p id=\"fs-id1931281\">[latex]{y}^{2}=12x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2614655\">\n<div id=\"fs-id2614656\">\n<p id=\"fs-id2614657\">[latex]{\\left(x+2\\right)}^{2}=\\frac{1}{2}\\left(y-1\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2614726\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2614726\"]\n<p id=\"fs-id2614726\">[latex]{\\left(x+2\\right)}^{2}=\\frac{1}{2}\\left(y-1\\right);\\,[\/latex]vertex:[latex]\\,\\left(-2,1\\right);\\,[\/latex]focus:[latex]\\,\\left(-2,\\frac{9}{8}\\right);\\,[\/latex]directrix:[latex]\\,y=\\frac{7}{8}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2955405\">\n<div id=\"fs-id2955406\">\n<p id=\"fs-id2955407\">[latex]{y}^{2}-6y-6x-3=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2613227\">\n<div id=\"fs-id2613228\">\n<p id=\"fs-id2613229\">[latex]{x}^{2}+10x-y+23=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2613269\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2613269\"]\n<p id=\"fs-id2613269\">[latex]{\\left(x+5\\right)}^{2}=\\left(y+2\\right);\\,[\/latex]vertex:[latex]\\,\\left(-5,-2\\right);\\,[\/latex]focus:[latex]\\,\\left(-5,-\\frac{7}{4}\\right);\\,[\/latex]directrix:[latex]\\,y=-\\frac{9}{4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2186040\">For the following exercises, graph the parabola, labeling vertex, focus, and directrix.<\/p>\n\n<div id=\"fs-id2186043\">\n<div id=\"fs-id2186044\">\n<p id=\"fs-id2186045\">[latex]{x}^{2}+4y=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2186075\">\n<div id=\"fs-id2186076\">\n<p id=\"fs-id2186077\">[latex]{\\left(y-1\\right)}^{2}=\\frac{1}{2}\\left(x+3\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"2131471\"]Show Solution[\/reveal-answer][hidden-answer a=\"2131471\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153621\/CNX_Precalc_Figure_10_05_232.jpg\" alt=\"\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2131487\">\n<div id=\"fs-id2131488\">\n<p id=\"fs-id2131489\">[latex]{x}^{2}-8x-10y+46=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2131530\">\n<div id=\"fs-id2131531\">\n<p id=\"fs-id2131532\">[latex]2{y}^{2}+12y+6x+15=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"2131574\"]Show Solution[\/reveal-answer][hidden-answer a=\"2131574\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153623\/CNX_Precalc_Figure_10_05_234.jpg\" alt=\"\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2450076\">For the following exercises, write the equation of the parabola using the given information.<\/p>\n\n<div id=\"fs-id2450079\">\n<div id=\"fs-id2450080\">\n<p id=\"fs-id2450081\">Focus at [latex]\\,\\left(-4,0\\right);\\,[\/latex]directrix is[latex]\\,x=4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2450134\">\n<div id=\"fs-id2450135\">\n<p id=\"fs-id2450136\">Focus at[latex]\\,\\left(2,\\frac{9}{8}\\right);\\,[\/latex]directrix is[latex]\\,y=\\frac{7}{8}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2450204\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2450204\"]\n<p id=\"fs-id2450204\">[latex]{\\left(x-2\\right)}^{2}=\\left(\\frac{1}{2}\\right)\\left(y-1\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1790044\">\n<div id=\"fs-id1790045\">\n<p id=\"fs-id1790046\">A cable TV receiving dish is the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the focus, if the dish is 5 feet across at its opening and 1.5 feet deep.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1790052\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/contents\/35cc5135-47da-41e2-8041-90f4e34c5210\">Rotation of Axes<\/a><\/h4>\n<p id=\"fs-id1790058\">For the following exercises, determine which of the conic sections is represented.<\/p>\n\n<div id=\"fs-id1790061\">\n<div id=\"fs-id1790062\">\n<p id=\"fs-id1790063\">[latex]16{x}^{2}+24xy+9{y}^{2}+24x-60y-60=0[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"237600\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"237600\"][latex]{B}^{2}-4AC=0,[\/latex]\nparabola[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id3063430\">\n<div id=\"fs-id3063431\">\n<p id=\"fs-id3063432\">[latex]4{x}^{2}+14xy+5{y}^{2}+18x-6y+30=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id3063497\">\n<div id=\"fs-id3063498\">\n<p id=\"fs-id3063499\">[latex]4{x}^{2}+xy+2{y}^{2}+8x-26y+9=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2157336\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2157336\"]\n<p id=\"fs-id2157336\">[latex]{B}^{2}-4AC=-31&lt;0,[\/latex]\nellipse[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2157377\">For the following exercises, determine the angle[latex]\\,\\theta \\,[\/latex]that will eliminate the[latex]\\,xy\\,[\/latex]term, and write the corresponding equation without the[latex]\\,xy\\,[\/latex]term.<\/p>\n\n<div id=\"fs-id2157426\">\n<div id=\"fs-id2157428\">\n<p id=\"fs-id2157429\">[latex]{x}^{2}+4xy-2{y}^{2}-6=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2887192\">\n<div id=\"fs-id2887193\">\n<p id=\"fs-id2887194\">[latex]{x}^{2}-xy+{y}^{2}-6=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2887242\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2887242\"]\n<p id=\"fs-id2887242\">[latex]\\theta ={45}^{\\circ },{{x}^{\\prime }}^{2}+3{{y}^{\\prime }}^{2}-12=0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2887317\">For the following exercises, graph the equation relative to the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]system in which the equation has no[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]term.<\/p>\n\n<div id=\"fs-id1871198\">\n<div id=\"fs-id1871199\">\n<p id=\"fs-id1871200\">[latex]9{x}^{2}-24xy+16{y}^{2}-80x-60y+100=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1871264\">\n<div id=\"fs-id1871265\">\n<p id=\"fs-id1871266\">[latex]{x}^{2}-xy+{y}^{2}-2=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2886326\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2886326\"]\n<p id=\"fs-id2886326\">[latex]\\theta ={45}^{\\circ }[\/latex]<\/p>\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153625\/CNX_Precalc_Figure_10_05_236.jpg\" alt=\"\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2886365\">\n<div id=\"fs-id2886366\">\n<p id=\"fs-id2886367\">[latex]6{x}^{2}+24xy-{y}^{2}-12x+26y+11=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2886431\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/contents\/17672e6c-13e2-4f93-82f9-ec2ce1759a57\">Conic Sections in Polar Coordinates<\/a><\/h4>\n<p id=\"fs-id2886436\">For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix.<\/p>\n\n<div id=\"fs-id2886441\">\n<div id=\"fs-id2886442\">\n<p id=\"fs-id2886443\">[latex]r=\\frac{10}{1-5\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1267271\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1267271\"]\n<p id=\"fs-id1267271\">Hyperbola with[latex]\\,e=5\\,[\/latex]and directrix[latex]\\,2\\,[\/latex]units to the left of the pole.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1267293\">\n<div id=\"fs-id1267294\">\n<p id=\"fs-id1267295\">[latex]r=\\frac{6}{3+2\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1267329\">\n<div id=\"fs-id1267330\">\n<p id=\"fs-id1267332\">[latex]r=\\frac{1}{4+3\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1267368\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1267368\"]\n<p id=\"fs-id1267368\">Ellipse with[latex]\\,e=\\frac{3}{4}\\,[\/latex]and directrix[latex]\\,\\frac{1}{3}\\,[\/latex]unit above the pole.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1740821\">\n<div id=\"fs-id1740822\">\n<p id=\"fs-id1740823\">[latex]r=\\frac{3}{5-5\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1740857\">For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci.<\/p>\n\n<div id=\"fs-id1740863\">\n<div id=\"fs-id1740864\">[latex]r=\\frac{3}{1-\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1740897\"]Show Solution[\/reveal-answer][hidden-answer a=\"1740897\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153632\/CNX_Precalc_Figure_10_05_238.jpg\" alt=\"\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1740913\">\n<div id=\"fs-id1740914\">\n<p id=\"fs-id1740915\">[latex]r=\\frac{8}{4+3\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1740949\">\n<div id=\"fs-id1740950\">\n<p id=\"fs-id1740951\">[latex]r=\\frac{10}{4+5\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1727593\"]Show Solution[\/reveal-answer][hidden-answer a=\"1727593\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153634\/CNX_Precalc_Figure_10_05_240.jpg\" alt=\"\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1727609\">\n<div id=\"fs-id1727610\">\n<p id=\"fs-id1727611\">[latex]r=\\frac{9}{3-6\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1727645\">For the following exercises, given information about the graph of a conic with focus at the origin, find the equation in polar form.<\/p>\n\n<div id=\"fs-id1727649\">\n<div id=\"fs-id1727650\">\n<p id=\"fs-id1727651\">Directrix is[latex]\\,x=3\\,[\/latex]and eccentricity[latex]\\,e=1[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1727689\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1727689\"]\n<p id=\"fs-id1727689\">[latex]r=\\frac{3}{1+\\mathrm{cos}\\text{ } \\theta }[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id3028880\">\n<div id=\"fs-id3028882\">\n<p id=\"fs-id3028883\">Directrix is[latex]\\,y=-2\\,[\/latex]and eccentricity[latex]\\,e=4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id3028922\" class=\"practice-test\">\n<h3>Practice Test<\/h3>\n<p id=\"fs-id3028928\">For the following exercises, write the equation in standard form and state the center, vertices, and foci.<\/p>\n\n<div id=\"fs-id3028933\">\n<div id=\"fs-id3028934\">\n<p id=\"fs-id3028935\">[latex]\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{4}=1[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id3028992\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id3028992\"]\n<p id=\"fs-id3028992\">[latex]\\frac{{x}^{2}}{{3}^{2}}+\\frac{{y}^{2}}{{2}^{2}}=1;\\,[\/latex]center:[latex]\\,\\left(0,0\\right);\\,[\/latex]vertices:[latex]\\,\\left(3,0\\right),\\left(\u20133,0\\right),\\left(0,2\\right),\\left(0,-2\\right);\\,[\/latex]foci:[latex]\\left(\\sqrt{5},0\\right),\\left(-\\sqrt{5},0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2237123\">\n<div id=\"fs-id2237124\">\n<p id=\"fs-id2237125\">[latex]9{y}^{2}+16{x}^{2}-36y+32x-92=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2237181\">For the following exercises, sketch the graph, identifying the center, vertices, and foci.<\/p>\n\n<div id=\"fs-id2237184\">\n<div id=\"fs-id2237186\">\n<p id=\"fs-id2237187\">[latex]\\frac{{\\left(x-3\\right)}^{2}}{64}+\\frac{{\\left(y-2\\right)}^{2}}{36}=1[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2869597\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2869597\"]\n<p id=\"fs-id2869597\">center:[latex]\\,\\left(3,2\\right);\\,[\/latex]vertices:[latex]\\,\\left(11,2\\right),\\left(-5,2\\right),\\left(3,8\\right),\\left(3,-4\\right);\\,[\/latex]foci:[latex]\\,\\left(3+2\\sqrt{7},2\\right),\\left(3-2\\sqrt{7},2\\right)[\/latex]<\/p>\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153644\/CNX_Precalc_Figure_10_05_242.jpg\" alt=\"\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2234975\">\n<div id=\"fs-id2234976\">\n<p id=\"fs-id2234978\">[latex]2{x}^{2}+{y}^{2}+8x-6y-7=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2584310\">\n<div id=\"fs-id2584311\">\n<p id=\"fs-id2584312\">Write the standard form equation of an ellipse with a center at[latex]\\,\\left(1,2\\right),[\/latex]vertex at[latex]\\,\\left(7,2\\right),[\/latex]and focus at[latex]\\,\\left(4,2\\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2584401\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2584401\"]\n<p id=\"fs-id2584401\">[latex]\\frac{{\\left(x-1\\right)}^{2}}{36}+\\frac{{\\left(y-2\\right)}^{2}}{27}=1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2602092\">\n<div id=\"fs-id2602093\">\n<p id=\"fs-id2602094\">A whispering gallery is to be constructed with a length of 150 feet. If the foci are to be located 20 feet away from the wall, how high should the ceiling be?<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2602099\">For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes.<\/p>\n\n<div id=\"fs-id2602103\">\n<div id=\"fs-id2602104\">\n<p id=\"fs-id2602105\">[latex]\\frac{{x}^{2}}{49}-\\frac{{y}^{2}}{81}=1[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2602168\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2602168\"]\n<p id=\"fs-id2602168\">[latex]\\frac{{x}^{2}}{{7}^{2}}-\\frac{{y}^{2}}{{9}^{2}}=1;\\,[\/latex]center:[latex]\\,\\left(0,0\\right);\\,[\/latex]vertices[latex]\\,\\left(7,0\\right),\\left(-7,0\\right);\\,[\/latex]foci:[latex]\\,\\left(\\sqrt{130},0\\right),\\left(-\\sqrt{130},0\\right);\\,[\/latex]asymptotes:[latex]\\,y=\u00b1\\frac{9}{7}x[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2888159\">\n<div id=\"fs-id2888160\">\n<p id=\"fs-id2888162\">[latex]16{y}^{2}-9{x}^{2}+128y+112=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2888212\">For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.<\/p>\n\n<div id=\"fs-id2888216\">\n<div id=\"fs-id2888217\">\n<p id=\"fs-id2888218\">[latex]\\frac{{\\left(x-3\\right)}^{2}}{25}-\\frac{{\\left(y+3\\right)}^{2}}{1}=1[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2686597\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2686597\"]\n<p id=\"fs-id2686597\">center:[latex]\\,\\left(3,-3\\right);\\,[\/latex]vertices:[latex]\\,\\left(8,-3\\right),\\left(-2,-3\\right);[\/latex]foci:[latex]\\,\\left(3+\\sqrt{26},-3\\right),\\left(3-\\sqrt{26},-3\\right);\\,[\/latex]asymptotes:[latex]\\,y=\u00b1\\frac{1}{5}\\left(x-3\\right)-3[\/latex]<\/p>\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153651\/CNX_Precalc_Figure_10_05_244.jpg\" alt=\"\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2575313\">\n<div id=\"fs-id2575314\">\n<p id=\"fs-id2575315\">[latex]{y}^{2}-{x}^{2}+4y-4x-18=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2575367\">\n<div id=\"fs-id2575368\">\n<p id=\"fs-id2575369\">Write the standard form equation of a hyperbola with foci at[latex]\\,\\left(1,0\\right)\\,[\/latex]and[latex]\\,\\left(1,6\\right),[\/latex]and a vertex at[latex]\\,\\left(1,2\\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1187406\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1187406\"]\n<p id=\"fs-id1187406\">[latex]\\frac{{\\left(y-3\\right)}^{2}}{1}-\\frac{{\\left(x-1\\right)}^{2}}{8}=1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1187498\">For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.<\/p>\n\n<div id=\"fs-id1187503\">\n<div id=\"fs-id1187504\">\n<p id=\"fs-id1187505\">[latex]{y}^{2}+10x=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1187535\">\n<div id=\"fs-id2106507\">\n<p id=\"fs-id2106508\">[latex]3{x}^{2}-12x-y+11=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2106551\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2106551\"]\n<p id=\"fs-id2106551\">[latex]{\\left(x-2\\right)}^{2}=\\frac{1}{3}\\left(y+1\\right);\\,[\/latex]vertex:[latex]\\,\\left(2,-1\\right);\\,[\/latex]focus:[latex]\\,\\left(2,-\\frac{11}{12}\\right);\\,[\/latex]directrix:[latex]\\,y=-\\frac{13}{12}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2067554\">For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.<\/p>\n\n<div id=\"fs-id2067558\">\n<div id=\"fs-id2067559\">\n<p id=\"fs-id2067560\">[latex]{\\left(x-1\\right)}^{2}=-4\\left(y+3\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2067620\">\n<div id=\"fs-id2067622\">\n<p id=\"fs-id2067623\">[latex]{y}^{2}+8x-8y+40=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"2067663\"]Show Solution[\/reveal-answer][hidden-answer a=\"2067663\"]\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153654\/CNX_Precalc_Figure_10_05_247.jpg\" alt=\"\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2067679\">\n<div id=\"fs-id2067680\">\n<p id=\"fs-id2067681\">Write the equation of a parabola with a focus at[latex]\\,\\left(2,3\\right)\\,[\/latex]and directrix[latex]\\,y=-1.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2814254\">\n<div id=\"fs-id2814256\">\n<p id=\"fs-id2814257\">A searchlight is shaped like a paraboloid of revolution. If the light source is located 1.5 feet from the base along the axis of symmetry, and the depth of the searchlight is 3 feet, what should the width of the opening be?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2814264\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2814264\"]\n<p id=\"fs-id2814264\">Approximately[latex]\\,8.49\\,[\/latex]feet<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2814269\">For the following exercises, determine which conic section is represented by the given equation, and then determine the angle[latex]\\,\\theta \\,[\/latex]that will eliminate the[latex]\\,xy\\,[\/latex]term.<\/p>\n\n<div id=\"fs-id2814303\">\n<div id=\"fs-id2814304\">\n<p id=\"fs-id2814305\">[latex]3{x}^{2}-2xy+3{y}^{2}=4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2814353\">\n<div id=\"fs-id2814354\">\n<p id=\"fs-id2814355\">[latex]{x}^{2}+4xy+4{y}^{2}+6x-8y=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2246942\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2246942\"]\n<p id=\"fs-id2246942\">parabola;[latex]\\,\\theta \\approx {63.4}^{\\circ }[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2246971\">For the following exercises, rewrite in the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]system without the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]term, and graph the rotated graph.<\/p>\n\n<div>\n<div id=\"fs-id2247038\">\n<p id=\"fs-id2247039\">[latex]11{x}^{2}+10\\sqrt{3}xy+{y}^{2}=4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2081302\">\n<div id=\"fs-id2081303\">\n<p id=\"fs-id2081304\">[latex]16{x}^{2}+24xy+9{y}^{2}-125x=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2081360\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2081360\"]\n<p id=\"fs-id2081360\">[latex]{{x}^{\\prime }}^{2}-4{x}^{\\prime }+3{y}^{\\prime }=0[\/latex]<\/p>\n<span id=\"fs-id2081421\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153657\/CNX_Precalc_Figure_10_05_249.jpg\" alt=\"\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2081432\">For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity.<\/p>\n\n<div id=\"fs-id2081436\">\n<div id=\"fs-id2081437\">\n<p id=\"fs-id2081438\">[latex]r=\\frac{3}{2-\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2081470\">\n<div id=\"fs-id2081471\">\n<p id=\"fs-id2081472\">[latex]r=\\frac{5}{4+6\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2883838\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2883838\"]\n<p id=\"fs-id2883838\">Hyperbola with[latex]\\,e=\\frac{3}{2},\\,[\/latex]and directrix[latex]\\,\\frac{5}{6}\\,[\/latex]units to the right of the pole.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2883891\">For the following exercises, graph the given conic section. If it is a parabola, label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci.<\/p>\n\n<div id=\"fs-id2883896\">\n<div id=\"fs-id2883897\">\n<p id=\"fs-id2883898\">[latex]r=\\frac{12}{4-8\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2883936\">\n<div id=\"fs-id2883937\">\n<p id=\"fs-id2883938\">[latex]r=\\frac{2}{4+4\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"2883972\"]Show Solution[\/reveal-answer][hidden-answer a=\"2883972\"]<span id=\"fs-id2883977\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153714\/CNX_Precalc_Figure_10_05_251n.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id2883988\">\n<div id=\"fs-id2883989\">\n<p id=\"fs-id2883990\">Find a polar equation of the conic with focus at the origin, eccentricity of[latex]\\,e=2,[\/latex]and directrix:[latex]\\,x=3.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id2172786\">\n \t<dt>eccentricity<\/dt>\n \t<dd id=\"fs-id2172791\">the ratio of the distances from a point[latex]\\,P\\,[\/latex]on the graph to the focus[latex]\\,F\\,[\/latex]and to the directrix[latex]\\,D\\,[\/latex]represented by[latex]\\,e=\\frac{PF}{PD},[\/latex]where[latex]\\,e\\,[\/latex]is a positive real number<\/dd>\n<\/dl>\n<dl id=\"fs-id2172885\">\n \t<dt>polar equation<\/dt>\n \t<dd id=\"fs-id1271646\">an equation of a curve in polar coordinates[latex]\\,r\\,[\/latex]and[latex]\\,\\theta [\/latex]<\/dd>\n<\/dl>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Identify a conic in polar form.<\/li>\n<li>Graph the polar equations of conics.<\/li>\n<li>De\ufb01ne conics in terms of a focus and a directrix.<\/li>\n<\/ul>\n<\/div>\n<div class=\"wp-caption-text\">\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152958\/CNX_Precalc_Figure_10_05_008n.jpg\" alt=\"\" width=\"975\" height=\"353\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1. <\/strong>Planets orbiting the sun follow elliptical paths. (credit: NASA Blueshift, Flickr)<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1258255\">Most of us are familiar with orbital motion, such as the motion of a planet around the sun or an electron around an atomic nucleus. Within the planetary system, orbits of planets, asteroids, and comets around a larger celestial body are often elliptical. Comets, however, may take on a parabolic or hyperbolic orbit instead. And, in reality, the characteristics of the planets\u2019 orbits may vary over time. Each orbit is tied to the location of the celestial body being orbited and the distance and direction of the planet or other object from that body. As a result, we tend to use polar coordinates to represent these orbits.<\/p>\n<p id=\"fs-id1482642\">In an elliptical orbit, the <span class=\"no-emphasis\">periapsis<\/span> is the point at which the two objects are closest, and the <span class=\"no-emphasis\">apoapsis<\/span> is the point at which they are farthest apart. Generally, the velocity of the orbiting body tends to increase as it approaches the periapsis and decrease as it approaches the apoapsis. Some objects reach an escape velocity, which results in an infinite orbit. These bodies exhibit either a parabolic or a hyperbolic orbit about a body; the orbiting body breaks free of the celestial body\u2019s gravitational pull and fires off into space. Each of these orbits can be modeled by a conic section in the polar coordinate system.<\/p>\n<div id=\"fs-id1388752\" class=\"bc-section section\">\n<h3>Identifying a Conic in Polar Form<\/h3>\n<p id=\"fs-id2264768\">Any conic may be determined by three characteristics: a single <span class=\"no-emphasis\">focus<\/span>, a fixed line called the <span class=\"no-emphasis\">directrix<\/span>, and the ratio of the distances of each to a point on the graph. Consider the <span class=\"no-emphasis\">parabola<\/span>[latex]\\,x=2+{y}^{2}\\,[\/latex]shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Figure_10_05_001\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_10_05_001\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153000\/CNX_Precalc_Figure_10_05_001.jpg\" alt=\"\" width=\"487\" height=\"316\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id2787449\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/contents\/a79194a1-878d-4c6d-a978-53670f3e1266\">The Parabola<\/a>, we learned how a parabola is defined by the focus (a fixed point) and the directrix (a fixed line). In this section, we will learn how to define any conic in the polar coordinate system in terms of a fixed point, the focus[latex]\\,P\\left(r,\\theta \\right)\\,[\/latex]at the pole, and a line, the directrix, which is perpendicular to the polar axis.<\/p>\n<p id=\"fs-id1673006\">If[latex]\\,F\\,[\/latex]is a fixed point, the focus, and[latex]\\,D\\,[\/latex]is a fixed line, the directrix, then we can let[latex]\\,e\\,[\/latex]be a fixed positive number, called the <strong>eccentricity<\/strong>, which we can define as the ratio of the distances from a point on the graph to the focus and the point on the graph to the directrix. Then the set of all points[latex]\\,P\\,[\/latex]such that[latex]\\,e=\\frac{PF}{PD}\\,[\/latex]is a conic. In other words, we can define a conic as the set of all points[latex]\\,P\\,[\/latex]with the property that the ratio of the distance from[latex]\\,P\\,[\/latex]to[latex]\\,F\\,[\/latex]to the distance from[latex]\\,P\\,[\/latex]to[latex]\\,D\\,[\/latex]is equal to the constant[latex]\\,e.[\/latex]<\/p>\n<p id=\"fs-id1540310\">For a conic with eccentricity[latex]\\,e,[\/latex]<\/p>\n<ul id=\"fs-id2114246\">\n<li>if[latex]\\,0\\le e<1,[\/latex] the conic is an ellipse<\/li>\n<li>if[latex]\\,e=1,[\/latex] the conic is a parabola<\/li>\n<li>if[latex]\\,e>1,[\/latex] the conic is an hyperbola<\/li>\n<\/ul>\n<p id=\"fs-id752226\">With this definition, we may now define a conic in terms of the directrix,[latex]\\,x=\u00b1p,[\/latex] the eccentricity[latex]\\,e,[\/latex] and the angle[latex]\\,\\theta .[\/latex] Thus, each conic may be written as a <strong>polar equation<\/strong>, an equation written in terms of[latex]\\,r\\,[\/latex]and[latex]\\,\\theta .[\/latex]<\/p>\n<div id=\"fs-id1161380\" class=\"textbox key-takeaways\">\n<h3>The Polar Equation for a Conic<\/h3>\n<p id=\"fs-id1195811\">For a conic with a focus at the origin, if the directrix is[latex]\\,x=\u00b1p,[\/latex] where[latex]\\,p\\,[\/latex]is a positive real number, and the eccentricity is a positive real number[latex]\\,e,[\/latex] the conic has a polar equation<\/p>\n<div id=\"fs-id2238499\" class=\"unnumbered aligncenter\">[latex]r=\\frac{ep}{1\u00b1e\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/div>\n<p id=\"fs-id1354185\">For a conic with a focus at the origin, if the directrix is[latex]\\,y=\u00b1p,[\/latex] where[latex]\\,p\\,[\/latex] is a positive real number, and the eccentricity is a positive real number[latex]\\,e,[\/latex] the conic has a polar equation<\/p>\n<div id=\"fs-id862339\" class=\"unnumbered aligncenter\">[latex]r=\\frac{ep}{1\u00b1e\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1331670\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id960683\"><strong>Given the polar equation for a conic, identify the type of conic, the directrix, and the eccentricity.<\/strong><\/p>\n<ol id=\"fs-id1814857\" type=\"1\">\n<li>Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form.<\/li>\n<li>Identify the eccentricity[latex]\\,e\\,[\/latex]as the coefficient of the trigonometric function in the denominator.<\/li>\n<li>Compare[latex]\\,e\\,[\/latex]with 1 to determine the shape of the conic.<\/li>\n<li>Determine the directrix as[latex]\\,x=p\\,[\/latex]if cosine is in the denominator and[latex]\\,y=p\\,[\/latex]if sine is in the denominator. Set[latex]\\,ep\\,[\/latex]equal to the numerator in standard form to solve for[latex]\\,x\\,[\/latex]or[latex]\\,y.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_10_05_01\" class=\"textbox examples\">\n<div id=\"fs-id2201458\">\n<div id=\"fs-id1367180\">\n<h3>Identifying a Conic Given the Polar Form<\/h3>\n<p id=\"fs-id1687490\">For each of the following equations, identify the conic with focus at the origin, the <span class=\"no-emphasis\">directrix<\/span>, and the <span class=\"no-emphasis\">eccentricity<\/span>.<\/p>\n<ol id=\"fs-id2430244\" type=\"a\">\n<li>[latex]r=\\frac{6}{3+2\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/li>\n<li>[latex]r=\\frac{12}{4+5\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/li>\n<li>[latex]r=\\frac{7}{2-2\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1329142\">For each of the three conics, we will rewrite the equation in standard form. Standard form has a 1 as the constant in the denominator. Therefore, in all three parts, the first step will be to multiply the numerator and denominator by the reciprocal of the constant of the original equation,[latex]\\,\\frac{1}{c},[\/latex] where[latex]\\,c\\,[\/latex]is that constant.<\/p>\n<ol id=\"fs-id1822233\" type=\"a\">\n<li>Multiply the numerator and denominator by[latex]\\,\\frac{1}{3}.[\/latex]\n<div id=\"fs-id1227176\" class=\"unnumbered aligncenter\">[latex]r=\\frac{6}{3+2\\mathrm{sin}\\text{ }\\theta }\\cdot \\frac{\\left(\\frac{1}{3}\\right)}{\\left(\\frac{1}{3}\\right)}=\\frac{6\\left(\\frac{1}{3}\\right)}{3\\left(\\frac{1}{3}\\right)+2\\left(\\frac{1}{3}\\right)\\mathrm{sin}\\text{ }\\theta }=\\frac{2}{1+\\frac{2}{3}\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/div>\n<p id=\"fs-id1079973\">Because[latex]\\mathrm{sin}\\text{ }\\theta[\/latex] is in the denominator, the directrix is[latex]\\,y=p.\\,[\/latex]Comparing to standard form, note that[latex]\\,e=\\frac{2}{3}.[\/latex]Therefore, from the numerator,<\/p>\n<div id=\"fs-id1233156\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }2=ep\\hfill \\\\ \\text{ }2=\\frac{2}{3}p\\hfill \\\\ \\left(\\frac{3}{2}\\right)2=\\left(\\frac{3}{2}\\right)\\frac{2}{3}p\\hfill \\\\ \\text{ }3=p\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1925603\">Since[latex]\\,e<1,[\/latex] the conic is an <span class=\"no-emphasis\">ellipse<\/span>. The eccentricity is[latex]\\,e=\\frac{2}{3}[\/latex]and the directrix is[latex]\\,y=3.[\/latex]<\/p>\n<\/li>\n<li>Multiply the numerator and denominator by[latex]\\,\\frac{1}{4}.[\/latex]\n<div id=\"fs-id1082549\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ r=\\frac{12}{4+5\\text{ }\\mathrm{cos}\\text{ }\\theta }\\cdot \\frac{\\left(\\frac{1}{4}\\right)}{\\left(\\frac{1}{4}\\right)}\\hfill \\end{array}\\hfill \\\\ r=\\frac{12\\left(\\frac{1}{4}\\right)}{4\\left(\\frac{1}{4}\\right)+5\\left(\\frac{1}{4}\\right)\\mathrm{cos}\\text{ }\\theta }\\hfill \\\\ r=\\frac{3}{1+\\frac{5}{4}\\text{ }\\mathrm{cos}\\text{ }\\theta }\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2339764\">Because[latex]\\text{ cos}\\,\\theta[\/latex]is in the denominator, the directrix is[latex]\\,x=p.\\,[\/latex]Comparing to standard form,[latex]\\,e=\\frac{5}{4}.\\,[\/latex]Therefore, from the numerator,<\/p>\n<div id=\"fs-id1923702\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }3=ep\\hfill \\\\ \\text{ }3=\\frac{5}{4}p\\hfill \\\\ \\,\\left(\\frac{4}{5}\\right)3=\\left(\\frac{4}{5}\\right)\\frac{5}{4}p\\hfill \\\\ \\text{ }\\,\\frac{12}{5}=p\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2109858\">Since[latex]\\,e>1,[\/latex] the conic is a <span class=\"no-emphasis\">hyperbola<\/span>. The eccentricity is[latex]\\,e=\\frac{5}{4}\\,[\/latex]and the directrix is[latex]\\,x=\\frac{12}{5}=2.4.[\/latex]<\/p>\n<\/li>\n<li>Multiply the numerator and denominator by[latex]\\,\\frac{1}{2}.[\/latex]\n<div id=\"fs-id1741224\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ \\begin{array}{l}r=\\frac{7}{2-2\\text{ }\\mathrm{sin}\\text{ }\\theta }\\cdot \\frac{\\left(\\frac{1}{2}\\right)}{\\left(\\frac{1}{2}\\right)}\\hfill \\\\ r=\\frac{7\\left(\\frac{1}{2}\\right)}{2\\left(\\frac{1}{2}\\right)-2\\left(\\frac{1}{2}\\right)\\text{ }\\mathrm{sin}\\text{ }\\theta }\\hfill \\\\ r=\\frac{\\frac{7}{2}}{1-\\mathrm{sin}\\text{ }\\theta }\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1636641\">Because sine is in the denominator, the directrix is[latex]\\,y=-p.\\,[\/latex]Comparing to standard form,[latex]\\,e=1.\\,[\/latex]Therefore, from the numerator,<\/p>\n<div id=\"fs-id1734126\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{7}{2}=ep\\\\ \\frac{7}{2}=\\left(1\\right)p\\\\ \\frac{7}{2}=p\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1385531\">Because[latex]\\,e=1,[\/latex] the conic is a <span class=\"no-emphasis\">parabola<\/span>. The eccentricity is[latex]\\,e=1\\,[\/latex]and the directrix is[latex]\\,y=-\\frac{7}{2}=-3.5.[\/latex]<\/details>\n<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2892298\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_05_01\">\n<div id=\"fs-id1914310\">\n<p id=\"fs-id2135269\">Identify the conic with focus at the origin, the directrix, and the eccentricity for[latex]\\,r=\\frac{2}{3-\\mathrm{cos}\\text{ }\\theta }.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1667756\">ellipse;[latex]\\,e=\\frac{1}{3};\\,x=-2[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1161787\" class=\"bc-section section\">\n<h3>Graphing the Polar Equations of Conics<\/h3>\n<p>When graphing in Cartesian coordinates, each conic section has a unique equation. This is not the case when graphing in polar coordinates. We must use the eccentricity of a conic section to determine which type of curve to graph, and then determine its specific characteristics. The first step is to rewrite the conic in standard form as we have done in the previous example. In other words, we need to rewrite the equation so that the denominator begins with 1. This enables us to determine[latex]\\,e\\,[\/latex]and, therefore, the shape of the curve. The next step is to substitute values for[latex]\\,\\theta \\,[\/latex]and solve for[latex]\\,r\\,[\/latex]to plot a few key points. Setting[latex]\\,\\theta \\,[\/latex]equal to[latex]\\,0,\\frac{\\pi }{2},\\pi ,[\/latex] and[latex]\\,\\frac{3\\pi }{2}\\,[\/latex]provides the vertices so we can create a rough sketch of the graph.<\/p>\n<div id=\"Example_10_05_02\" class=\"textbox examples\">\n<div id=\"fs-id2427459\">\n<div id=\"fs-id2106935\">\n<h3>Graphing a Parabola in Polar Form<\/h3>\n<p id=\"fs-id1351805\">Graph[latex]\\,r=\\frac{5}{3+3\\text{ }\\mathrm{cos}\\text{ }\\theta }.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1316220\">First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 3, which is[latex]\\,\\frac{1}{3}.[\/latex]<\/p>\n<div id=\"fs-id1240312\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ r=\\frac{5}{3+3\\text{ }\\mathrm{cos}\\text{ }\\theta }=\\frac{5\\left(\\frac{1}{3}\\right)}{3\\left(\\frac{1}{3}\\right)+3\\left(\\frac{1}{3}\\right)\\mathrm{cos}\\text{ }\\theta }\\hfill \\end{array}\\hfill \\\\ r=\\frac{\\frac{5}{3}}{1+\\mathrm{cos}\\text{ }\\theta }\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1923255\">Because[latex]\\,e=1,[\/latex]we will graph a <span class=\"no-emphasis\">parabola<\/span> with a focus at the origin. The function has a[latex]\\mathrm{cos}\\text{ }\\theta ,[\/latex] and there is an addition sign in the denominator, so the directrix is[latex]\\,x=p.[\/latex]<\/p>\n<div id=\"fs-id2160621\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{5}{3}=ep\\\\ \\frac{5}{3}=\\left(1\\right)p\\\\ \\frac{5}{3}=p\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1690174\">The directrix is[latex]\\,x=\\frac{5}{3}.[\/latex]<\/p>\n<p id=\"fs-id1672859\">Plotting a few key points as in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Table_10_05_01\">(Figure)<\/a> will enable us to see the vertices. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Figure_10_05_002\">(Figure)<\/a>.<\/p>\n<table id=\"Table_10_05_01\" summary=\"..\">\n<thead>\n<tr>\n<th><\/th>\n<th>A<\/th>\n<th>B<\/th>\n<th>C<\/th>\n<th>D<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\theta[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td>[latex]\\pi[\/latex]<\/td>\n<td>[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]r=\\frac{5}{3+3\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/td>\n<td>[latex]\\frac{5}{6}\\approx 0.83[\/latex]<\/td>\n<td>[latex]\\frac{5}{3}\\approx 1.67[\/latex]<\/td>\n<td>undefined<\/td>\n<td>[latex]\\frac{5}{3}\\approx 1.67[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153002\/CNX_Precalc_Figure_10_05_002.jpg\" alt=\"\" width=\"487\" height=\"376\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<p><span id=\"fs-id1978048\"><\/span><\/details>\n<\/div>\n<div id=\"fs-id2256443\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id2769094\">We can check our result with a graphing utility. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Figure_10_05_003\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_10_05_003\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153019\/CNX_Precalc_Figure_10_05_003.jpg\" alt=\"\" width=\"487\" height=\"376\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_10_05_03\" class=\"textbox examples\">\n<div id=\"fs-id2702007\">\n<div id=\"fs-id1908051\">\n<h3>Graphing a Hyperbola in Polar Form<\/h3>\n<p id=\"fs-id1078732\">Graph[latex]\\,r=\\frac{8}{2-3\\text{ }\\mathrm{sin}\\text{ }\\theta }.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1115003\">First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 2, which is[latex]\\,\\frac{1}{2}.[\/latex]<\/p>\n<div id=\"fs-id2467590\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ r=\\frac{8}{2-3\\mathrm{sin}\\text{ }\\theta }=\\frac{8\\left(\\frac{1}{2}\\right)}{2\\left(\\frac{1}{2}\\right)-3\\left(\\frac{1}{2}\\right)\\mathrm{sin}\\text{ }\\theta }\\hfill \\end{array}\\hfill \\\\ r=\\frac{4}{1-\\frac{3}{2}\\text{ }\\mathrm{sin}\\text{ }\\theta }\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2153652\">Because[latex]\\,e=\\frac{3}{2},e>1,[\/latex] so we will graph a <span class=\"no-emphasis\">hyperbola<\/span> with a focus at the origin. The function has a[latex]\\,\\mathrm{sin}\\text{ }\\theta \\,[\/latex]term and there is a subtraction sign in the denominator, so the directrix is[latex]\\,y=-p.[\/latex]<\/p>\n<div id=\"fs-id1114165\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }4=ep\\hfill \\\\ \\text{ }4=\\left(\\frac{3}{2}\\right)p\\hfill \\\\ 4\\left(\\frac{2}{3}\\right)=p\\hfill \\\\ \\text{ }\\frac{8}{3}=p\\hfill \\end{array}[\/latex]<\/div>\n<p>The directrix is[latex]\\,y=-\\frac{8}{3}.[\/latex]<\/p>\n<p>Plotting a few key points as in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Table_10_05_02\">(Figure)<\/a> will enable us to see the vertices. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Figure_10_05_004\">(Figure)<\/a>.<\/p>\n<table id=\"Table_10_05_02\" summary=\"..\">\n<thead>\n<tr>\n<th><\/th>\n<th>A<\/th>\n<th>B<\/th>\n<th>C<\/th>\n<th>D<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\theta[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td>[latex]\\pi[\/latex]<\/td>\n<td>[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]r=\\frac{8}{2-3\\mathrm{sin}\\,\\theta }[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]-8[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]\\frac{8}{5}=1.6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153023\/CNX_Precalc_Figure_10_05_004.jpg\" alt=\"\" width=\"975\" height=\"810\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id2427799\"><\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_10_05_04\" class=\"textbox examples\">\n<div id=\"fs-id1121716\">\n<div id=\"fs-id2040203\">\n<h3>Graphing an Ellipse in Polar Form<\/h3>\n<p id=\"fs-id1983048\">Graph[latex]\\,r=\\frac{10}{5-4\\text{ }\\mathrm{cos}\\text{ }\\theta }.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2231759\">First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 5, which is[latex]\\,\\frac{1}{5}.[\/latex]<\/p>\n<div id=\"fs-id1732373\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\begin{array}{l}r=\\frac{10}{5-4\\mathrm{cos}\\text{ }\\theta }=\\frac{10\\left(\\frac{1}{5}\\right)}{5\\left(\\frac{1}{5}\\right)-4\\left(\\frac{1}{5}\\right)\\mathrm{cos}\\text{ }\\theta }\\hfill \\\\ r=\\frac{2}{1-\\frac{4}{5}\\text{ }\\mathrm{cos}\\text{ }\\theta }\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2255518\">Because[latex]\\,e=\\frac{4}{5},e<1,[\/latex] so we will graph an <span class=\"no-emphasis\">ellipse<\/span> with a <span class=\"no-emphasis\">focus<\/span> at the origin. The function has a[latex]\\,\\text{cos}\\,\\theta ,[\/latex] and there is a subtraction sign in the denominator, so the <span class=\"no-emphasis\">directrix<\/span> is[latex]\\,x=-p.[\/latex]<\/p>\n<div id=\"fs-id2067929\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }2=ep\\hfill \\\\ \\text{ }2=\\left(\\frac{4}{5}\\right)p\\hfill \\\\ 2\\left(\\frac{5}{4}\\right)=p\\hfill \\\\ \\text{ }\\frac{5}{2}=p\\hfill \\end{array}[\/latex]<\/div>\n<p>The directrix is[latex]\\,x=-\\frac{5}{2}.[\/latex]<\/p>\n<p>Plotting a few key points as in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Table_10_05_03\">(Figure)<\/a> will enable us to see the vertices. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Figure_10_05_006\">(Figure)<\/a>.<\/p>\n<table id=\"Table_10_05_03\" summary=\"..\">\n<thead>\n<tr>\n<th><\/th>\n<th>A<\/th>\n<th>B<\/th>\n<th>C<\/th>\n<th>D<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\theta[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td>[latex]\\pi[\/latex]<\/td>\n<td>[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]r=\\frac{10}{5-4\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]\\frac{10}{9}\\approx 1.1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153026\/CNX_Precalc_Figure_10_05_006.jpg\" alt=\"\" width=\"487\" height=\"431\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id2754428\"><span id=\"fs-id1551880\"><\/span><\/details>\n<\/p>\n<\/div>\n<div id=\"fs-id2306563\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1551888\">We can check our result using a graphing utility. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Figure_10_05_007\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_10_05_007\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153033\/CNX_Precalc_Figure_10_05_007.jpg\" alt=\"\" width=\"487\" height=\"431\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 7. <\/strong>[latex]r=\\frac{10}{5-4\\text{ }\\mathrm{cos}\\text{ }\\theta }\\,[\/latex]graphed on a viewing window of[latex]\\,\\left[\u20133,12,1\\right]\\,[\/latex]by[latex]\\,\\left[\u20134,4,1\\right],\\theta \\,\\text{min =}\\,0[\/latex]and[latex]\\,\\theta \\,\\text{max =}\\,2\\pi .[\/latex]<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2255486\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_05_02\">\n<div id=\"fs-id2118885\">\n<p id=\"fs-id2118886\">Graph[latex]\\,r=\\frac{2}{4-\\mathrm{cos}\\text{ }\\theta }.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1639625\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153035\/CNX_Precalc_Figure_10_05_009.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1916233\" class=\"bc-section section\">\n<h3>De\ufb01ning Conics in Terms of a Focus and a Directrix<\/h3>\n<p id=\"fs-id3026916\">So far we have been using polar equations of conics to describe and graph the curve. Now we will work in reverse; we will use information about the origin, eccentricity, and directrix to determine the polar equation.<\/p>\n<div id=\"fs-id3026921\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id3026927\"><strong>Given the focus, eccentricity, and directrix of a conic, determine the polar equation.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id3026932\" type=\"1\">\n<li>Determine whether the directrix is horizontal or vertical. If the directrix is given in terms of[latex]\\,y,[\/latex] we use the general polar form in terms of sine. If the directrix is given in terms of[latex]\\,x,[\/latex] we use the general polar form in terms of cosine.<\/li>\n<li>Determine the sign in the denominator. If[latex]\\,p<0,[\/latex] use subtraction. If[latex]\\,p>0,[\/latex] use addition.<\/li>\n<li>Write the coefficient of the trigonometric function as the given eccentricity.<\/li>\n<li>Write the absolute value of[latex]\\,p\\,[\/latex] in the numerator, and simplify the equation.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_10_05_05\" class=\"textbox examples\">\n<div id=\"fs-id2927796\">\n<div id=\"fs-id2927798\">\n<h3>Finding the Polar Form of a Vertical Conic Given a Focus at the Origin and the Eccentricity and Directrix<\/h3>\n<p id=\"fs-id2927804\">Find the polar form of the <span class=\"no-emphasis\">conic<\/span> given a <span class=\"no-emphasis\">focus<\/span> at the origin,[latex]\\,e=3\\,[\/latex]and <span class=\"no-emphasis\">directrix<\/span>[latex]\\,y=-2.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2956031\">The directrix is[latex]\\,y=-p,[\/latex] so we know the trigonometric function in the denominator is sine.<\/p>\n<p id=\"fs-id2821456\">Because[latex]\\,y=-2,\u20132<0,[\/latex] so we know there is a subtraction sign in the denominator. We use the standard form of<\/p>\n<div id=\"fs-id2821490\" class=\"unnumbered aligncenter\">[latex]r=\\frac{ep}{1-e\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/div>\n<p id=\"fs-id2479768\">and[latex]\\,e=3\\,[\/latex]and[latex]\\,|-2|=2=p.[\/latex]<\/p>\n<p id=\"fs-id2585526\">Therefore,<\/p>\n<div id=\"fs-id2585529\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\begin{array}{l}r=\\frac{\\left(3\\right)\\left(2\\right)}{1-3\\text{ }\\mathrm{sin}\\text{ }\\theta }\\hfill \\\\ r=\\frac{6}{1-3\\text{ }\\mathrm{sin}\\text{ }\\theta }\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_10_05_06\" class=\"textbox examples\">\n<div id=\"fs-id1823159\">\n<div id=\"fs-id1823161\">\n<h3>Finding the Polar Form of a Horizontal Conic Given a Focus at the Origin and the Eccentricity and Directrix<\/h3>\n<p id=\"fs-id1823168\">Find the <span class=\"no-emphasis\">polar form of a conic<\/span> given a <span class=\"no-emphasis\">focus<\/span> at the origin,[latex]\\,e=\\frac{3}{5},[\/latex] and <span class=\"no-emphasis\">directrix<\/span>[latex]\\,x=4.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1085849\">Because the directrix is[latex]\\,x=p,[\/latex]we know the function in the denominator is cosine. Because[latex]\\,x=4,4>0,[\/latex]so we know there is an addition sign in the denominator. We use the standard form of<\/p>\n<div id=\"fs-id2735265\" class=\"unnumbered aligncenter\">[latex]r=\\frac{ep}{1+e\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/div>\n<p id=\"fs-id2735304\">and[latex]\\,e=\\frac{3}{5}\\,[\/latex]and[latex]\\,|4|=4=p.[\/latex]<\/p>\n<p id=\"fs-id1263417\">Therefore,<\/p>\n<div id=\"fs-id1263420\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ r=\\frac{\\left(\\frac{3}{5}\\right)\\left(4\\right)}{1+\\frac{3}{5}\\,\\mathrm{cos}\\,\\theta }\\hfill \\end{array}\\hfill \\\\ r=\\frac{\\frac{12}{5}}{1+\\frac{3}{5}\\,\\mathrm{cos}\\,\\theta }\\hfill \\\\ r=\\frac{\\frac{12}{5}}{1\\left(\\frac{5}{5}\\right)+\\frac{3}{5}\\,\\mathrm{cos}\\,\\theta }\\hfill \\\\ r=\\frac{\\frac{12}{5}}{\\frac{5}{5}+\\frac{3}{5}\\,\\mathrm{cos}\\,\\theta }\\hfill \\\\ r=\\frac{12}{5}\\cdot \\frac{5}{5+3\\,\\mathrm{cos}\\,\\theta }\\hfill \\\\ r=\\frac{12}{5+3\\,\\mathrm{cos}\\,\\theta }\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2020858\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_05_03\">\n<div id=\"fs-id2020868\">\n<p id=\"fs-id2020869\">Find the polar form of the conic given a focus at the origin,[latex]\\,e=1,[\/latex] and directrix[latex]\\,x=-1.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1121326\">[latex]r=\\frac{1}{1-\\mathrm{cos}\\theta }[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_10_05_07\" class=\"textbox examples\">\n<div id=\"fs-id1121364\">\n<div id=\"fs-id1121366\">\n<h3>Converting a Conic in Polar Form to Rectangular Form<\/h3>\n<p id=\"fs-id1121372\">Convert the conic[latex]\\,r=\\frac{1}{5-5\\mathrm{sin}\\,\\theta }[\/latex]to rectangular form.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1558062\">We will rearrange the formula to use the identities[latex]r=\\sqrt{{x}^{2}+{y}^{2}},x=r\\,\\mathrm{cos}\\,\\theta ,\\text{and }y=r\\,\\mathrm{sin}\\,\\theta .[\/latex]<\/p>\n<div id=\"fs-id2881090\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\text{ }r=\\frac{1}{5-5\\,\\mathrm{sin}\\,\\theta }\\hfill & \\hfill \\\\ r\\cdot \\left(5-5\\,\\mathrm{sin}\\,\\theta \\right)=\\frac{1}{5-5\\,\\mathrm{sin}\\,\\theta }\\cdot \\left(5-5\\,\\mathrm{sin}\\,\\theta \\right)\\hfill & \\text{Eliminate the fraction}.\\hfill \\\\ \\text{ }\\,5r-5r\\,\\mathrm{sin}\\,\\theta =1\\hfill & \\text{Distribute}.\\hfill \\\\ \\text{ }5r=1+5r\\,\\mathrm{sin}\\,\\theta \\hfill & \\text{Isolate }5r.\\hfill \\\\ \\text{ }25{r}^{2}={\\left(1+5r\\,\\mathrm{sin}\\,\\theta \\right)}^{2}\\hfill & \\text{Square both sides}.\\hfill \\\\ \\text{ }25\\left({x}^{2}+{y}^{2}\\right)={\\left(1+5y\\right)}^{2}\\hfill & \\text{Substitute }r=\\sqrt{{x}^{2}+{y}^{2}}\\text{ and }y=r\\,\\mathrm{sin}\\,\\theta .\\hfill \\\\ \\text{ }\\,25{x}^{2}+25{y}^{2}=1+10y+25{y}^{2}\\hfill & \\text{Distribute and use FOIL}.\\hfill \\\\ \\text{ }\\,25{x}^{2}-10y=1\\hfill & \\text{Rearrange terms and set equal to 1}.\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1587301\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_05_04\">\n<div id=\"fs-id1587310\">\n<p id=\"fs-id1587311\">Convert the conic[latex]\\,r=\\frac{2}{1+2\\text{ }\\mathrm{cos}\\text{ }\\theta }\\,[\/latex]to rectangular form.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2445406\">[latex]4-8x+3{x}^{2}-{y}^{2}=0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2773738\" class=\"precalculus media\">\n<p id=\"fs-id2773744\">Access these online resources for additional instruction and practice with conics in polar coordinates.<\/p>\n<ul id=\"eip-id2676160\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/determineconic\">Polar Equations of Conic Sections<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphconic1\">Graphing Polar Equations of Conics &#8211; 1<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphconic2\">Graphing Polar Equations of Conics &#8211; 2<\/a><\/li>\n<\/ul>\n<\/div>\n<p id=\"eip-127\">Visit <a href=\"http:\/\/openstaxcollege.org\/l\/PreCalcLPC10\">this website<\/a> for additional practice questions from Learningpod.<\/p>\n<\/div>\n<div id=\"fs-id2773762\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id2773768\">\n<li>Any conic may be determined by a single focus, the corresponding eccentricity, and the directrix. We can also define a conic in terms of a fixed point, the focus[latex]\\,P\\left(r,\\theta \\right)\\,[\/latex]at the pole, and a line, the directrix, which is perpendicular to the polar axis.<\/li>\n<li>A conic is the set of all points[latex]\\,e=\\frac{PF}{PD},[\/latex] where eccentricity[latex]\\,e\\,[\/latex]is a positive real number. Each conic may be written in terms of its polar equation. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Example_10_05_01\">(Figure)<\/a>.<\/li>\n<li>The polar equations of conics can be graphed. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Example_10_05_02\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Example_10_05_03\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Example_10_05_04\">(Figure)<\/a>.<\/li>\n<li>Conics can be defined in terms of a focus, a directrix, and eccentricity. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Example_10_05_05\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Example_10_05_06\">(Figure)<\/a>.<\/li>\n<li>We can use the identities[latex]\\,r=\\sqrt{{x}^{2}+{y}^{2}},x=r\\text{ }\\mathrm{cos}\\text{ }\\theta ,[\/latex]and[latex]\\,y=r\\text{ }\\mathrm{sin}\\text{ }\\theta \\,[\/latex]to convert the equation for a conic from polar to rectangular form. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3425&amp;action=edit#Example_10_05_07\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id2879646\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id2879649\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id2879654\">\n<div id=\"fs-id2879655\">\n<p id=\"fs-id2879656\">Explain how eccentricity determines which conic section is given.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2879662\">If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2879668\">\n<div id=\"fs-id1818181\">\n<p id=\"fs-id1818182\">If a conic section is written as a polar equation, what must be true of the denominator?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1818185\">\n<div id=\"fs-id1818186\">\n<p id=\"fs-id1818187\">If a conic section is written as a polar equation, and the denominator involves[latex]\\,\\mathrm{sin}\\text{ }\\theta ,[\/latex]what conclusion can be drawn about the directrix?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1818208\">The directrix will be parallel to the polar axis.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1818213\">\n<div id=\"fs-id1818214\">\n<p id=\"fs-id1818215\">If the directrix of a conic section is perpendicular to the polar axis, what do we know about the equation of the graph?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1818219\">\n<div id=\"fs-id1818220\">\n<p id=\"fs-id1818221\">What do we know about the focus\/foci of a conic section if it is written as a polar equation?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1818227\">One of the foci will be located at the origin.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1818232\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1818237\">For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.<\/p>\n<div id=\"fs-id1818242\">\n<div id=\"fs-id1818243\">\n<p id=\"fs-id1818244\">[latex]r=\\frac{6}{1-2\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1264809\">\n<div id=\"fs-id1264810\">\n<p id=\"fs-id1264811\">[latex]r=\\frac{3}{4-4\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1264847\">Parabola with[latex]\\,e=1\\,[\/latex]and directrix[latex]\\,\\frac{3}{4}\\,[\/latex]units below the pole.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2199203\">\n<div id=\"fs-id2199204\">\n<p id=\"fs-id2199205\">[latex]r=\\frac{8}{4-3\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2920062\">\n<div id=\"fs-id2920063\">\n<p id=\"fs-id2920064\">[latex]r=\\frac{5}{1+2\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2920100\">Hyperbola with[latex]\\,e=2\\,[\/latex]and directrix[latex]\\,\\frac{5}{2}\\,[\/latex]units above the pole.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id2245309\">[latex]r=\\frac{16}{4+3\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2245346\">\n<div id=\"fs-id2245347\">\n<p id=\"fs-id2245348\">[latex]r=\\frac{3}{10+10\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1232410\">Parabola with[latex]\\,e=1\\,[\/latex]and directrix[latex]\\,\\frac{3}{10}\\,[\/latex]units to the right of the pole.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1232457\">\n<div id=\"fs-id1232458\">\n<p id=\"fs-id1232459\">[latex]r=\\frac{2}{1-\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1403116\">\n<div id=\"fs-id1403117\">\n<p id=\"fs-id1403118\">[latex]r=\\frac{4}{7+2\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1403154\">Ellipse with[latex]\\,e=\\frac{2}{7}\\,[\/latex]and directrix[latex]\\,2\\,[\/latex]units to the right of the pole.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2322949\">\n<div id=\"fs-id2322950\">[latex]r\\left(1-\\mathrm{cos}\\text{ }\\theta \\right)=3[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id2322986\">\n<div id=\"fs-id2285129\">\n<p id=\"fs-id2285130\">[latex]r\\left(3+5\\mathrm{sin}\\text{ }\\theta \\right)=11[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2285169\">Hyperbola with[latex]\\,e=\\frac{5}{3}\\,[\/latex]and directrix[latex]\\,\\frac{11}{5}\\,[\/latex]units above the pole.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1999272\">\n<div id=\"fs-id1999273\">\n<p id=\"fs-id1999274\">[latex]r\\left(4-5\\mathrm{sin}\\text{ }\\theta \\right)=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1999312\">\n<div id=\"fs-id1999313\">[latex]r\\left(7+8\\mathrm{cos}\\text{ }\\theta \\right)=7[\/latex]<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2289117\">Hyperbola with[latex]\\,e=\\frac{8}{7}\\,[\/latex]and directrix[latex]\\,\\frac{7}{8}\\,[\/latex]units to the right of the pole.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2289168\">For the following exercises, convert the polar equation of a conic section to a rectangular equation.<\/p>\n<div id=\"fs-id2289171\">\n<div>\n<p id=\"fs-id2106793\">[latex]r=\\frac{4}{1+3\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2106827\">\n<div id=\"fs-id2106828\">\n<p id=\"fs-id2106829\">[latex]r=\\frac{2}{5-3\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2106865\">[latex]25{x}^{2}+16{y}^{2}-12y-4=0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2430897\">\n<div id=\"fs-id2430898\">\n<p id=\"fs-id2430899\">[latex]r=\\frac{8}{3-2\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2261851\">\n<div id=\"fs-id2261852\">\n<p id=\"fs-id2261853\">[latex]r=\\frac{3}{2+5\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2261889\">[latex]21{x}^{2}-4{y}^{2}-30x+9=0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1468839\">\n<div id=\"fs-id1468840\">\n<p id=\"fs-id1468841\">[latex]r=\\frac{4}{2+2\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1468875\">\n<div id=\"fs-id1468876\">\n<p id=\"fs-id1468877\">[latex]r=\\frac{3}{8-8\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1468914\">[latex]64{y}^{2}=48x+9[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2601785\">\n<div id=\"fs-id2601786\">\n<p id=\"fs-id2601787\">[latex]r=\\frac{2}{6+7\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2601821\">\n<div id=\"fs-id2601822\">\n<p id=\"fs-id2601823\">[latex]r=\\frac{5}{5-11\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1233482\">[latex]96{y}^{2}-25{x}^{2}+110y+25=0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1233533\">\n<div id=\"fs-id1233534\">\n<p id=\"fs-id1233535\">[latex]r\\left(5+2\\text{ }\\mathrm{cos}\\text{ }\\theta \\right)=6[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1924146\">\n<div id=\"fs-id1924148\">\n<p id=\"fs-id1924149\">[latex]r\\left(2-\\mathrm{cos}\\text{ }\\theta \\right)=1[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]3{x}^{2}+4{y}^{2}-2x-1=0[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1649438\">\n<div id=\"fs-id1649439\">\n<p id=\"fs-id1649440\">[latex]r\\left(2.5-2.5\\text{ }\\mathrm{sin}\\text{ }\\theta \\right)=5[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1649477\">\n<div id=\"fs-id1649478\">\n<p id=\"fs-id1649479\">[latex]r=\\frac{6\\mathrm{sec}\\text{ }\\theta }{-2+3\\text{ }\\mathrm{sec}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2439850\">[latex]5{x}^{2}+9{y}^{2}-24x-36=0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2439901\">\n<div id=\"fs-id2439902\">\n<p id=\"fs-id2439903\">[latex]r=\\frac{6\\mathrm{csc}\\text{ }\\theta }{3+2\\text{ }\\mathrm{csc}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1978601\">For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.<\/p>\n<div id=\"fs-id1978606\">\n<div id=\"fs-id1978607\">\n<p id=\"fs-id1978608\">[latex]r=\\frac{5}{2+\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1978645\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153037\/CNX_Precalc_Figure_10_05_201.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1978656\">\n<div>\n<p id=\"fs-id1978658\">[latex]r=\\frac{2}{3+3\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1872482\">\n<div id=\"fs-id1872483\">\n<p id=\"fs-id1872484\">[latex]r=\\frac{10}{5-4\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1872527\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153047\/CNX_Precalc_Figure_10_05_203.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1872537\">\n<div id=\"fs-id1872538\">\n<p id=\"fs-id1872539\">[latex]r=\\frac{3}{1+2\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1479120\">\n<div id=\"fs-id1479121\">\n<p id=\"fs-id1479122\">[latex]r=\\frac{8}{4-5\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1479161\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153050\/CNX_Precalc_Figure_10_05_205.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1479172\">\n<div id=\"fs-id1479173\">\n<p id=\"fs-id1479174\">[latex]r=\\frac{3}{4-4\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1479208\">\n<div id=\"fs-id1479209\">\n<p id=\"fs-id1479210\">[latex]r=\\frac{2}{1-\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1703897\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153052\/CNX_Precalc_Figure_10_05_207.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1703908\">\n<div id=\"fs-id1703909\">\n<p id=\"fs-id1703910\">[latex]r=\\frac{6}{3+2\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1703944\">\n<div id=\"fs-id1703945\">\n<p id=\"fs-id1703946\">[latex]r\\left(1+\\mathrm{cos}\\text{ }\\theta \\right)=5[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id2089738\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153102\/CNX_Precalc_Figure_10_05_209.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2089749\">\n<div id=\"fs-id2089750\">\n<p id=\"fs-id2089751\">[latex]r\\left(3-4\\mathrm{sin}\\text{ }\\theta \\right)=9[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2089788\">\n<div id=\"fs-id2089789\">\n<p id=\"fs-id2089790\">[latex]r\\left(3-2\\mathrm{sin}\\text{ }\\theta \\right)=6[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id2089833\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153112\/CNX_Precalc_Figure_10_05_211.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2253204\">\n<div id=\"fs-id2253206\">\n<p id=\"fs-id2253207\">[latex]r\\left(6-4\\mathrm{cos}\\text{ }\\theta \\right)=5[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2253244\">For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.<\/p>\n<div id=\"fs-id2253248\">\n<div id=\"fs-id2253249\">\n<p id=\"fs-id2253250\">Directrix:[latex]x=4;\\,e=\\frac{1}{5}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2253287\">[latex]r=\\frac{4}{5+\\mathrm{cos}\\theta }[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2222207\">\n<div id=\"fs-id2222208\">\n<p id=\"fs-id2222209\">Directrix:[latex]x=-4;\\,e=5[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2222239\">\n<div id=\"fs-id2222240\">\n<p id=\"fs-id2222241\">Directrix:[latex]y=2;\\,e=2[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2222264\">[latex]r=\\frac{4}{1+2\\mathrm{sin}\\theta }[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2222301\">\n<div id=\"fs-id2222302\">\n<p id=\"fs-id2222303\">Directrix: [latex]y=-2;\\,e=\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1929509\">\n<p id=\"fs-id1929510\">Directrix:[latex]x=1;\\,e=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]r=\\frac{1}{1+\\mathrm{cos}\\theta }[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1929573\">\n<div id=\"fs-id1929574\">\n<p id=\"fs-id1929575\">Directrix:[latex]x=-1;\\,e=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1783904\">\n<div id=\"fs-id1783905\">\n<p id=\"fs-id1783906\">Directrix: [latex]x=-\\frac{1}{4};\\,e=\\frac{7}{2}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1783952\">[latex]r=\\frac{7}{8-28\\mathrm{cos}\\theta }[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1783989\">\n<div id=\"fs-id1783990\">\n<p id=\"fs-id1783991\">Directrix:[latex]y=\\frac{2}{5};\\,e=\\frac{7}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2900253\">\n<div id=\"fs-id2900254\">\n<p id=\"fs-id2900255\">Directrix: [latex]y=4;\\,e=\\frac{3}{2}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2900291\">[latex]r=\\frac{12}{2+3\\mathrm{sin}\\theta }[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2900329\">\n<div id=\"fs-id2900330\">\n<p id=\"fs-id2900332\">Directrix:[latex]x=-2;\\,e=\\frac{8}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1529994\">\n<div id=\"fs-id1529996\">\n<p id=\"fs-id1529997\">Directrix:[latex]x=-5;\\,e=\\frac{3}{4}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1530035\">[latex]r=\\frac{15}{4-3\\mathrm{cos}\\theta }[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1530073\">\n<div>\n<p id=\"fs-id1530076\">Directrix:[latex]y=2;\\,e=2.5[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2893428\">\n<div id=\"fs-id2893429\">\n<p id=\"fs-id2893430\">Directrix:[latex]x=-3;\\,e=\\frac{1}{3}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2893468\">[latex]r=\\frac{3}{3-3\\mathrm{cos}\\theta }[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2893505\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id2893510\">Recall from <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/rotation-of-axes\/\">Rotation of Axes<\/a> that equations of conics with an[latex]\\,xy\\,[\/latex]term have rotated graphs. For the following exercises, express each equation in polar form with[latex]\\,r\\,[\/latex]as a function of[latex]\\,\\theta .[\/latex]<\/p>\n<div id=\"fs-id2000559\">\n<div id=\"fs-id2000560\">\n<p id=\"fs-id2000562\">[latex]xy=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2000580\">\n<div id=\"fs-id2000581\">\n<p id=\"fs-id2000582\">[latex]{x}^{2}+xy+{y}^{2}=4[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2000626\">[latex]r=\u00b1\\frac{2}{\\sqrt{1+\\mathrm{sin}\\theta \\mathrm{cos}\\theta }}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2191166\">\n<div id=\"fs-id2191167\">\n<p id=\"fs-id2191168\">[latex]2{x}^{2}+4xy+2{y}^{2}=9[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id2191217\">\n<p id=\"fs-id2191218\">[latex]16{x}^{2}+24xy+9{y}^{2}=4[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1237498\">[latex]r=\u00b1\\frac{2}{4\\mathrm{cos}\\theta +3\\mathrm{sin}\\theta }[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1237539\">\n<div id=\"fs-id1237540\">\n<p id=\"fs-id1237541\">[latex]2xy+y=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1237568\" class=\"review-exercises textbox exercises\">\n<h3>Chapter Review Exercises<\/h3>\n<div id=\"fs-id1237574\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/contents\/99d38770-49c7-49d3-b567-88f393ffb4fe\">The Ellipse<\/a><\/h4>\n<p id=\"fs-id1237580\">For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.<\/p>\n<div id=\"fs-id1237584\">\n<div id=\"fs-id1237585\">\n<p id=\"fs-id1237586\">[latex]\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{64}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1493914\">[latex]\\frac{{x}^{2}}{{5}^{2}}+\\frac{{y}^{2}}{{8}^{2}}=1;\\,[\/latex]center:[latex]\\,\\left(0,0\\right);\\,[\/latex]vertices:[latex]\\,\\left(5,0\\right),\\left(-5,0\\right),\\left(0,8\\right),\\left(0,-8\\right);\\,[\/latex]foci:[latex]\\,\\left(0,\\sqrt{39}\\right),\\left(0,-\\sqrt{39}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1860702\">\n<div id=\"fs-id1860703\">\n<p id=\"fs-id1860704\">[latex]\\frac{{\\left(x-2\\right)}^{2}}{100}+\\frac{{\\left(y+3\\right)}^{2}}{36}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2146165\">\n<div id=\"fs-id2146166\">\n<p id=\"fs-id2146167\">[latex]9{x}^{2}+{y}^{2}+54x-4y+76=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2146224\">[latex]\\frac{{\\left(x+3\\right)}^{2}}{{1}^{2}}+\\frac{{\\left(y-2\\right)}^{2}}{{3}^{2}}=1\\,\\,\\left(-3,2\\right);\\,\\,\\left(-2,2\\right),\\left(-4,2\\right),\\left(-3,5\\right),\\left(-3,-1\\right);\\,\\,\\left(-3,2+2\\sqrt{2}\\right),\\left(-3,2-2\\sqrt{2}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2613644\">\n<div id=\"fs-id2613646\">\n<p id=\"fs-id2613647\">[latex]9{x}^{2}+36{y}^{2}-36x+72y+36=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2613703\">For the following exercises, graph the ellipse, noting center, vertices, and foci.<\/p>\n<div id=\"fs-id2613706\">\n<div id=\"fs-id2613707\">\n<p id=\"fs-id2613708\">[latex]\\frac{{x}^{2}}{36}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2888444\">center:[latex]\\,\\left(0,0\\right);\\,[\/latex]vertices:[latex]\\,\\left(6,0\\right),\\left(-6,0\\right),\\left(0,3\\right),\\left(0,-3\\right);\\,[\/latex]foci:[latex]\\,\\left(3\\sqrt{3},0\\right),\\left(-3\\sqrt{3},0\\right)[\/latex]<span id=\"fs-id2196864\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153114\/CNX_Precalc_Figure_10_05_223.jpg\" alt=\"\" \/><\/span><\/details>\n<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2196874\">\n<div id=\"fs-id2196876\">\n<p id=\"fs-id2196877\">[latex]\\frac{{\\left(x-4\\right)}^{2}}{25}+\\frac{{\\left(y+3\\right)}^{2}}{49}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1548201\">\n<div id=\"fs-id1548202\">\n<p id=\"fs-id1548203\">[latex]4{x}^{2}+{y}^{2}+16x+4y-44=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1737673\">center:[latex]\\,\\left(-2,-2\\right);\\,[\/latex]vertices:[latex]\\,\\left(2,-2\\right),\\left(-6,-2\\right),\\left(-2,6\\right),\\left(-2,-10\\right);\\,[\/latex]foci:[latex]\\,\\left(-2,-2+4\\sqrt{3},\\right),\\left(-2,-2-4\\sqrt{3}\\right)[\/latex]<\/p>\n<div><\/div>\n<p><span id=\"fs-id1363974\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153117\/CNX_Precalc_Figure_10_05_225.jpg\" alt=\"\" \/><\/details>\n<p><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1363984\">\n<div id=\"fs-id1363985\">\n<p id=\"fs-id1363986\">[latex]\\,2{x}^{2}+3{y}^{2}-20x+12y+38=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id3032766\">For the following exercises, use the given information to find the equation for the ellipse.<\/p>\n<div id=\"fs-id3032771\">\n<div id=\"fs-id3032772\">\n<p id=\"fs-id3032773\">Center at [latex]\\,\\left(0,0\\right),[\/latex]focus at[latex]\\,\\left(3,0\\right),[\/latex]vertex at[latex]\\,\\left(-5,0\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2122257\">[latex]\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{16}=1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2122319\">\n<div id=\"fs-id2122320\">\n<p id=\"fs-id2122321\">Center at[latex]\\,\\left(2,-2\\right),[\/latex]vertex at[latex]\\,\\left(7,-2\\right),[\/latex]focus at[latex]\\,\\left(4,-2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2329260\">\n<div id=\"fs-id2329261\">\n<p id=\"fs-id2329262\">A whispering gallery is to be constructed such that the foci are located 35 feet from the center. If the length of the gallery is to be 100 feet, what should the height of the ceiling be?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2329269\">Approximately 35.71 feet<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2329274\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/contents\/23e28179-8eb8-4790-a332-861509a3d9fb\">The Hyperbola<\/a><\/h4>\n<p id=\"fs-id2329278\">For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci.<\/p>\n<div id=\"fs-id2329283\">\n<div id=\"fs-id2329284\">\n<p id=\"fs-id2329285\">[latex]\\frac{{x}^{2}}{81}-\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2329343\">\n<div id=\"fs-id2329344\">\n<p id=\"fs-id2329345\">[latex]\\frac{{\\left(y+1\\right)}^{2}}{16}-\\frac{{\\left(x-4\\right)}^{2}}{36}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1408198\">[latex]\\frac{{\\left(y+1\\right)}^{2}}{{4}^{2}}-\\frac{{\\left(x-4\\right)}^{2}}{{6}^{2}}=1;\\,[\/latex]center:[latex]\\,\\left(4,-1\\right);\\,[\/latex]vertices:[latex]\\,\\left(4,3\\right),\\left(4,-5\\right);\\,[\/latex]foci:[latex]\\,\\left(4,-1+2\\sqrt{13}\\right),\\left(4,-1-2\\sqrt{13}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1550436\">\n<div id=\"fs-id1550438\">\n<p id=\"fs-id1550439\">[latex]9{y}^{2}-4{x}^{2}+54y-16x+29=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2875786\">\n<div id=\"fs-id2875787\">\n<p id=\"fs-id2875788\">[latex]3{x}^{2}-{y}^{2}-12x-6y-9=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2875845\">[latex]\\frac{{\\left(x-2\\right)}^{2}}{{2}^{2}}-\\frac{{\\left(y+3\\right)}^{2}}{{\\left(2\\sqrt{3}\\right)}^{2}}=1;\\,[\/latex]center:[latex]\\,\\left(2,-3\\right);\\,[\/latex]vertices:[latex]\\,\\left(4,-3\\right),\\left(0,-3\\right);\\,[\/latex]foci:[latex]\\,\\left(6,-3\\right),\\left(-2,-3\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1899768\">For the following exercises, graph the hyperbola, labeling vertices and foci.<\/p>\n<div id=\"fs-id1899771\">\n<div id=\"fs-id1899772\">\n<p id=\"fs-id1899774\">[latex]\\frac{{x}^{2}}{9}-\\frac{{y}^{2}}{16}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2130346\">\n<div id=\"fs-id2130347\">\n<p id=\"fs-id2130348\">[latex]\\frac{{\\left(y-1\\right)}^{2}}{49}-\\frac{{\\left(x+1\\right)}^{2}}{4}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/oasis.geneseo.edu\/trig-pics\/CNX_Precalc_Figure_10_05_228.jpg#fixme\" alt=\"\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2130458\">\n<div id=\"fs-id2130459\">\n<p id=\"fs-id2130460\">[latex]{x}^{2}-4{y}^{2}+6x+32y-91=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2732697\">\n<div id=\"fs-id2732698\">\n<p id=\"fs-id2732699\">[latex]2{y}^{2}-{x}^{2}-12y-6=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153619\/CNX_Precalc_Figure_10_05_230.jpg\" alt=\"\" \/><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2732763\">For the following exercises, find the equation of the hyperbola.<\/p>\n<div id=\"fs-id2732766\">\n<div id=\"fs-id2732767\">\n<p id=\"fs-id2732768\">Center at[latex]\\,\\left(0,0\\right),[\/latex]vertex at[latex]\\,\\left(0,4\\right),[\/latex]focus at[latex]\\,\\left(0,-6\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1245770\">\n<div id=\"fs-id1245771\">\n<p id=\"fs-id1245772\">Foci at[latex]\\,\\left(3,7\\right)\\,[\/latex]and[latex]\\,\\left(7,7\\right),[\/latex]vertex at[latex]\\,\\left(6,7\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1931177\">[latex]\\frac{{\\left(x-5\\right)}^{2}}{1}-\\frac{{\\left(y-7\\right)}^{2}}{3}=1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1931269\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/contents\/a79194a1-878d-4c6d-a978-53670f3e1266\">The Parabola<\/a><\/h4>\n<p id=\"fs-id1931274\">For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.<\/p>\n<div id=\"fs-id1931279\">\n<div id=\"fs-id1931280\">\n<p id=\"fs-id1931281\">[latex]{y}^{2}=12x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2614655\">\n<div id=\"fs-id2614656\">\n<p id=\"fs-id2614657\">[latex]{\\left(x+2\\right)}^{2}=\\frac{1}{2}\\left(y-1\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2614726\">[latex]{\\left(x+2\\right)}^{2}=\\frac{1}{2}\\left(y-1\\right);\\,[\/latex]vertex:[latex]\\,\\left(-2,1\\right);\\,[\/latex]focus:[latex]\\,\\left(-2,\\frac{9}{8}\\right);\\,[\/latex]directrix:[latex]\\,y=\\frac{7}{8}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2955405\">\n<div id=\"fs-id2955406\">\n<p id=\"fs-id2955407\">[latex]{y}^{2}-6y-6x-3=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2613227\">\n<div id=\"fs-id2613228\">\n<p id=\"fs-id2613229\">[latex]{x}^{2}+10x-y+23=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2613269\">[latex]{\\left(x+5\\right)}^{2}=\\left(y+2\\right);\\,[\/latex]vertex:[latex]\\,\\left(-5,-2\\right);\\,[\/latex]focus:[latex]\\,\\left(-5,-\\frac{7}{4}\\right);\\,[\/latex]directrix:[latex]\\,y=-\\frac{9}{4}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2186040\">For the following exercises, graph the parabola, labeling vertex, focus, and directrix.<\/p>\n<div id=\"fs-id2186043\">\n<div id=\"fs-id2186044\">\n<p id=\"fs-id2186045\">[latex]{x}^{2}+4y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2186075\">\n<div id=\"fs-id2186076\">\n<p id=\"fs-id2186077\">[latex]{\\left(y-1\\right)}^{2}=\\frac{1}{2}\\left(x+3\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153621\/CNX_Precalc_Figure_10_05_232.jpg\" alt=\"\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2131487\">\n<div id=\"fs-id2131488\">\n<p id=\"fs-id2131489\">[latex]{x}^{2}-8x-10y+46=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2131530\">\n<div id=\"fs-id2131531\">\n<p id=\"fs-id2131532\">[latex]2{y}^{2}+12y+6x+15=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153623\/CNX_Precalc_Figure_10_05_234.jpg\" alt=\"\" \/><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2450076\">For the following exercises, write the equation of the parabola using the given information.<\/p>\n<div id=\"fs-id2450079\">\n<div id=\"fs-id2450080\">\n<p id=\"fs-id2450081\">Focus at [latex]\\,\\left(-4,0\\right);\\,[\/latex]directrix is[latex]\\,x=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2450134\">\n<div id=\"fs-id2450135\">\n<p id=\"fs-id2450136\">Focus at[latex]\\,\\left(2,\\frac{9}{8}\\right);\\,[\/latex]directrix is[latex]\\,y=\\frac{7}{8}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2450204\">[latex]{\\left(x-2\\right)}^{2}=\\left(\\frac{1}{2}\\right)\\left(y-1\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1790044\">\n<div id=\"fs-id1790045\">\n<p id=\"fs-id1790046\">A cable TV receiving dish is the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the focus, if the dish is 5 feet across at its opening and 1.5 feet deep.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1790052\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/contents\/35cc5135-47da-41e2-8041-90f4e34c5210\">Rotation of Axes<\/a><\/h4>\n<p id=\"fs-id1790058\">For the following exercises, determine which of the conic sections is represented.<\/p>\n<div id=\"fs-id1790061\">\n<div id=\"fs-id1790062\">\n<p id=\"fs-id1790063\">[latex]16{x}^{2}+24xy+9{y}^{2}+24x-60y-60=0[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]{B}^{2}-4AC=0,[\/latex]<br \/>\nparabola<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id3063430\">\n<div id=\"fs-id3063431\">\n<p id=\"fs-id3063432\">[latex]4{x}^{2}+14xy+5{y}^{2}+18x-6y+30=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id3063497\">\n<div id=\"fs-id3063498\">\n<p id=\"fs-id3063499\">[latex]4{x}^{2}+xy+2{y}^{2}+8x-26y+9=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2157336\">[latex]{B}^{2}-4AC=-31<0,[\/latex]\nellipse<\/details>\n<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2157377\">For the following exercises, determine the angle[latex]\\,\\theta \\,[\/latex]that will eliminate the[latex]\\,xy\\,[\/latex]term, and write the corresponding equation without the[latex]\\,xy\\,[\/latex]term.<\/p>\n<div id=\"fs-id2157426\">\n<div id=\"fs-id2157428\">\n<p id=\"fs-id2157429\">[latex]{x}^{2}+4xy-2{y}^{2}-6=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2887192\">\n<div id=\"fs-id2887193\">\n<p id=\"fs-id2887194\">[latex]{x}^{2}-xy+{y}^{2}-6=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2887242\">[latex]\\theta ={45}^{\\circ },{{x}^{\\prime }}^{2}+3{{y}^{\\prime }}^{2}-12=0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2887317\">For the following exercises, graph the equation relative to the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]system in which the equation has no[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]term.<\/p>\n<div id=\"fs-id1871198\">\n<div id=\"fs-id1871199\">\n<p id=\"fs-id1871200\">[latex]9{x}^{2}-24xy+16{y}^{2}-80x-60y+100=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1871264\">\n<div id=\"fs-id1871265\">\n<p id=\"fs-id1871266\">[latex]{x}^{2}-xy+{y}^{2}-2=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2886326\">[latex]\\theta ={45}^{\\circ }[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153625\/CNX_Precalc_Figure_10_05_236.jpg\" alt=\"\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2886365\">\n<div id=\"fs-id2886366\">\n<p id=\"fs-id2886367\">[latex]6{x}^{2}+24xy-{y}^{2}-12x+26y+11=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2886431\" class=\"bc-section section\">\n<h4><a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/contents\/17672e6c-13e2-4f93-82f9-ec2ce1759a57\">Conic Sections in Polar Coordinates<\/a><\/h4>\n<p id=\"fs-id2886436\">For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix.<\/p>\n<div id=\"fs-id2886441\">\n<div id=\"fs-id2886442\">\n<p id=\"fs-id2886443\">[latex]r=\\frac{10}{1-5\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1267271\">Hyperbola with[latex]\\,e=5\\,[\/latex]and directrix[latex]\\,2\\,[\/latex]units to the left of the pole.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1267293\">\n<div id=\"fs-id1267294\">\n<p id=\"fs-id1267295\">[latex]r=\\frac{6}{3+2\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1267329\">\n<div id=\"fs-id1267330\">\n<p id=\"fs-id1267332\">[latex]r=\\frac{1}{4+3\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1267368\">Ellipse with[latex]\\,e=\\frac{3}{4}\\,[\/latex]and directrix[latex]\\,\\frac{1}{3}\\,[\/latex]unit above the pole.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1740821\">\n<div id=\"fs-id1740822\">\n<p id=\"fs-id1740823\">[latex]r=\\frac{3}{5-5\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1740857\">For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci.<\/p>\n<div id=\"fs-id1740863\">\n<div id=\"fs-id1740864\">[latex]r=\\frac{3}{1-\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153632\/CNX_Precalc_Figure_10_05_238.jpg\" alt=\"\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1740913\">\n<div id=\"fs-id1740914\">\n<p id=\"fs-id1740915\">[latex]r=\\frac{8}{4+3\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1740949\">\n<div id=\"fs-id1740950\">\n<p id=\"fs-id1740951\">[latex]r=\\frac{10}{4+5\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153634\/CNX_Precalc_Figure_10_05_240.jpg\" alt=\"\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1727609\">\n<div id=\"fs-id1727610\">\n<p id=\"fs-id1727611\">[latex]r=\\frac{9}{3-6\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1727645\">For the following exercises, given information about the graph of a conic with focus at the origin, find the equation in polar form.<\/p>\n<div id=\"fs-id1727649\">\n<div id=\"fs-id1727650\">\n<p id=\"fs-id1727651\">Directrix is[latex]\\,x=3\\,[\/latex]and eccentricity[latex]\\,e=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1727689\">[latex]r=\\frac{3}{1+\\mathrm{cos}\\text{ } \\theta }[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id3028880\">\n<div id=\"fs-id3028882\">\n<p id=\"fs-id3028883\">Directrix is[latex]\\,y=-2\\,[\/latex]and eccentricity[latex]\\,e=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id3028922\" class=\"practice-test\">\n<h3>Practice Test<\/h3>\n<p id=\"fs-id3028928\">For the following exercises, write the equation in standard form and state the center, vertices, and foci.<\/p>\n<div id=\"fs-id3028933\">\n<div id=\"fs-id3028934\">\n<p id=\"fs-id3028935\">[latex]\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{4}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id3028992\">[latex]\\frac{{x}^{2}}{{3}^{2}}+\\frac{{y}^{2}}{{2}^{2}}=1;\\,[\/latex]center:[latex]\\,\\left(0,0\\right);\\,[\/latex]vertices:[latex]\\,\\left(3,0\\right),\\left(\u20133,0\\right),\\left(0,2\\right),\\left(0,-2\\right);\\,[\/latex]foci:[latex]\\left(\\sqrt{5},0\\right),\\left(-\\sqrt{5},0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2237123\">\n<div id=\"fs-id2237124\">\n<p id=\"fs-id2237125\">[latex]9{y}^{2}+16{x}^{2}-36y+32x-92=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2237181\">For the following exercises, sketch the graph, identifying the center, vertices, and foci.<\/p>\n<div id=\"fs-id2237184\">\n<div id=\"fs-id2237186\">\n<p id=\"fs-id2237187\">[latex]\\frac{{\\left(x-3\\right)}^{2}}{64}+\\frac{{\\left(y-2\\right)}^{2}}{36}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2869597\">center:[latex]\\,\\left(3,2\\right);\\,[\/latex]vertices:[latex]\\,\\left(11,2\\right),\\left(-5,2\\right),\\left(3,8\\right),\\left(3,-4\\right);\\,[\/latex]foci:[latex]\\,\\left(3+2\\sqrt{7},2\\right),\\left(3-2\\sqrt{7},2\\right)[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153644\/CNX_Precalc_Figure_10_05_242.jpg\" alt=\"\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2234975\">\n<div id=\"fs-id2234976\">\n<p id=\"fs-id2234978\">[latex]2{x}^{2}+{y}^{2}+8x-6y-7=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2584310\">\n<div id=\"fs-id2584311\">\n<p id=\"fs-id2584312\">Write the standard form equation of an ellipse with a center at[latex]\\,\\left(1,2\\right),[\/latex]vertex at[latex]\\,\\left(7,2\\right),[\/latex]and focus at[latex]\\,\\left(4,2\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2584401\">[latex]\\frac{{\\left(x-1\\right)}^{2}}{36}+\\frac{{\\left(y-2\\right)}^{2}}{27}=1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2602092\">\n<div id=\"fs-id2602093\">\n<p id=\"fs-id2602094\">A whispering gallery is to be constructed with a length of 150 feet. If the foci are to be located 20 feet away from the wall, how high should the ceiling be?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2602099\">For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes.<\/p>\n<div id=\"fs-id2602103\">\n<div id=\"fs-id2602104\">\n<p id=\"fs-id2602105\">[latex]\\frac{{x}^{2}}{49}-\\frac{{y}^{2}}{81}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2602168\">[latex]\\frac{{x}^{2}}{{7}^{2}}-\\frac{{y}^{2}}{{9}^{2}}=1;\\,[\/latex]center:[latex]\\,\\left(0,0\\right);\\,[\/latex]vertices[latex]\\,\\left(7,0\\right),\\left(-7,0\\right);\\,[\/latex]foci:[latex]\\,\\left(\\sqrt{130},0\\right),\\left(-\\sqrt{130},0\\right);\\,[\/latex]asymptotes:[latex]\\,y=\u00b1\\frac{9}{7}x[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2888159\">\n<div id=\"fs-id2888160\">\n<p id=\"fs-id2888162\">[latex]16{y}^{2}-9{x}^{2}+128y+112=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2888212\">For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.<\/p>\n<div id=\"fs-id2888216\">\n<div id=\"fs-id2888217\">\n<p id=\"fs-id2888218\">[latex]\\frac{{\\left(x-3\\right)}^{2}}{25}-\\frac{{\\left(y+3\\right)}^{2}}{1}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2686597\">center:[latex]\\,\\left(3,-3\\right);\\,[\/latex]vertices:[latex]\\,\\left(8,-3\\right),\\left(-2,-3\\right);[\/latex]foci:[latex]\\,\\left(3+\\sqrt{26},-3\\right),\\left(3-\\sqrt{26},-3\\right);\\,[\/latex]asymptotes:[latex]\\,y=\u00b1\\frac{1}{5}\\left(x-3\\right)-3[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153651\/CNX_Precalc_Figure_10_05_244.jpg\" alt=\"\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2575313\">\n<div id=\"fs-id2575314\">\n<p id=\"fs-id2575315\">[latex]{y}^{2}-{x}^{2}+4y-4x-18=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2575367\">\n<div id=\"fs-id2575368\">\n<p id=\"fs-id2575369\">Write the standard form equation of a hyperbola with foci at[latex]\\,\\left(1,0\\right)\\,[\/latex]and[latex]\\,\\left(1,6\\right),[\/latex]and a vertex at[latex]\\,\\left(1,2\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1187406\">[latex]\\frac{{\\left(y-3\\right)}^{2}}{1}-\\frac{{\\left(x-1\\right)}^{2}}{8}=1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1187498\">For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.<\/p>\n<div id=\"fs-id1187503\">\n<div id=\"fs-id1187504\">\n<p id=\"fs-id1187505\">[latex]{y}^{2}+10x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1187535\">\n<div id=\"fs-id2106507\">\n<p id=\"fs-id2106508\">[latex]3{x}^{2}-12x-y+11=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2106551\">[latex]{\\left(x-2\\right)}^{2}=\\frac{1}{3}\\left(y+1\\right);\\,[\/latex]vertex:[latex]\\,\\left(2,-1\\right);\\,[\/latex]focus:[latex]\\,\\left(2,-\\frac{11}{12}\\right);\\,[\/latex]directrix:[latex]\\,y=-\\frac{13}{12}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2067554\">For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.<\/p>\n<div id=\"fs-id2067558\">\n<div id=\"fs-id2067559\">\n<p id=\"fs-id2067560\">[latex]{\\left(x-1\\right)}^{2}=-4\\left(y+3\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2067620\">\n<div id=\"fs-id2067622\">\n<p id=\"fs-id2067623\">[latex]{y}^{2}+8x-8y+40=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153654\/CNX_Precalc_Figure_10_05_247.jpg\" alt=\"\" \/><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2067679\">\n<div id=\"fs-id2067680\">\n<p id=\"fs-id2067681\">Write the equation of a parabola with a focus at[latex]\\,\\left(2,3\\right)\\,[\/latex]and directrix[latex]\\,y=-1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2814254\">\n<div id=\"fs-id2814256\">\n<p id=\"fs-id2814257\">A searchlight is shaped like a paraboloid of revolution. If the light source is located 1.5 feet from the base along the axis of symmetry, and the depth of the searchlight is 3 feet, what should the width of the opening be?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2814264\">Approximately[latex]\\,8.49\\,[\/latex]feet<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2814269\">For the following exercises, determine which conic section is represented by the given equation, and then determine the angle[latex]\\,\\theta \\,[\/latex]that will eliminate the[latex]\\,xy\\,[\/latex]term.<\/p>\n<div id=\"fs-id2814303\">\n<div id=\"fs-id2814304\">\n<p id=\"fs-id2814305\">[latex]3{x}^{2}-2xy+3{y}^{2}=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2814353\">\n<div id=\"fs-id2814354\">\n<p id=\"fs-id2814355\">[latex]{x}^{2}+4xy+4{y}^{2}+6x-8y=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2246942\">parabola;[latex]\\,\\theta \\approx {63.4}^{\\circ }[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2246971\">For the following exercises, rewrite in the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]system without the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]term, and graph the rotated graph.<\/p>\n<div>\n<div id=\"fs-id2247038\">\n<p id=\"fs-id2247039\">[latex]11{x}^{2}+10\\sqrt{3}xy+{y}^{2}=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2081302\">\n<div id=\"fs-id2081303\">\n<p id=\"fs-id2081304\">[latex]16{x}^{2}+24xy+9{y}^{2}-125x=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2081360\">[latex]{{x}^{\\prime }}^{2}-4{x}^{\\prime }+3{y}^{\\prime }=0[\/latex]<\/p>\n<p><span id=\"fs-id2081421\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153657\/CNX_Precalc_Figure_10_05_249.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2081432\">For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity.<\/p>\n<div id=\"fs-id2081436\">\n<div id=\"fs-id2081437\">\n<p id=\"fs-id2081438\">[latex]r=\\frac{3}{2-\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2081470\">\n<div id=\"fs-id2081471\">\n<p id=\"fs-id2081472\">[latex]r=\\frac{5}{4+6\\text{ }\\mathrm{cos}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2883838\">Hyperbola with[latex]\\,e=\\frac{3}{2},\\,[\/latex]and directrix[latex]\\,\\frac{5}{6}\\,[\/latex]units to the right of the pole.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2883891\">For the following exercises, graph the given conic section. If it is a parabola, label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci.<\/p>\n<div id=\"fs-id2883896\">\n<div id=\"fs-id2883897\">\n<p id=\"fs-id2883898\">[latex]r=\\frac{12}{4-8\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2883936\">\n<div id=\"fs-id2883937\">\n<p id=\"fs-id2883938\">[latex]r=\\frac{2}{4+4\\text{ }\\mathrm{sin}\\text{ }\\theta }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id2883977\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19153714\/CNX_Precalc_Figure_10_05_251n.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2883988\">\n<div id=\"fs-id2883989\">\n<p id=\"fs-id2883990\">Find a polar equation of the conic with focus at the origin, eccentricity of[latex]\\,e=2,[\/latex]and directrix:[latex]\\,x=3.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id2172786\">\n<dt>eccentricity<\/dt>\n<dd id=\"fs-id2172791\">the ratio of the distances from a point[latex]\\,P\\,[\/latex]on the graph to the focus[latex]\\,F\\,[\/latex]and to the directrix[latex]\\,D\\,[\/latex]represented by[latex]\\,e=\\frac{PF}{PD},[\/latex]where[latex]\\,e\\,[\/latex]is a positive real number<\/dd>\n<\/dl>\n<dl id=\"fs-id2172885\">\n<dt>polar equation<\/dt>\n<dd id=\"fs-id1271646\">an equation of a curve in polar coordinates[latex]\\,r\\,[\/latex]and[latex]\\,\\theta[\/latex]<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":291,"menu_order":6,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-196","chapter","type-chapter","status-publish","hentry"],"part":185,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/196","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/196\/revisions"}],"predecessor-version":[{"id":197,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/196\/revisions\/197"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/185"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/196\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=196"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=196"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=196"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=196"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}