{"id":194,"date":"2019-08-20T17:04:05","date_gmt":"2019-08-20T21:04:05","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/rotation-of-axes\/"},"modified":"2022-06-01T10:39:39","modified_gmt":"2022-06-01T14:39:39","slug":"rotation-of-axes","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/rotation-of-axes\/","title":{"raw":"Rotation of Axes","rendered":"Rotation of Axes"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>arning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Identify nondegenerate conic sections given their general form equations.<\/li>\n \t<li>Use rotation of axes formulas.<\/li>\n \t<li>Write equations of rotated conics in standard form.<\/li>\n \t<li>Identify conics without rotating axes.<\/li>\n<\/ul>\n<\/div>\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\/19152642\/CNX_Precalc_Figure_10_04_001.jpg\" alt=\"\" width=\"975\" height=\"650\"> <strong>Figure 1. <\/strong>The nondegenerate conic sections[\/caption]\n<p id=\"fs-id1141495\">As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a <em>cone<\/em>. The way in which we slice the cone will determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane perpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not perpendicular to the axis of symmetry. A parabola is formed by slicing the plane through the top or bottom of the double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone. See <a class=\"autogenerated-content\" href=\"#Figure_10_04_001\">(Figure)<\/a>.<span id=\"fs-id1471958\"><\/span><\/p>\n\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\/19152645\/CNX_Precalc_Figure_10_04_002n.jpg\" alt=\"\" width=\"975\" height=\"719\"> <strong>Figure 2. <\/strong>Degenerate conic sections[\/caption]\n<p id=\"fs-id2467642\">Ellipses, circles, hyperbolas, and parabolas are sometimes called the nondegenerate conic sections, in contrast to the degenerate conic sections, which are shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_002\">(Figure)<\/a>. A degenerate conic results when a plane intersects the double cone and passes through the apex. Depending on the angle of the plane, three types of degenerate conic sections are possible: a point, a line, or two intersecting lines.<span id=\"fs-id2295508\"><\/span><\/p>\n\n<div id=\"fs-id1741499\" class=\"bc-section section\">\n<h3>Identifying Nondegenerate Conics in General Form<\/h3>\n<p id=\"fs-id2726428\">In previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections. In this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below.<\/p>\n\n<div id=\"fs-id1264739\" class=\"unnumbered aligncenter\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\n<p id=\"fs-id1306881\">where[latex]\\,A,B,[\/latex] and [latex]\\,C\\,[\/latex]are not all zero. We can use the values of the coefficients to identify which type conic is represented by a given equation.<\/p>\n<p id=\"fs-id1675947\">You may notice that the general form equation has an[latex]\\,xy\\,[\/latex]term that we have not seen in any of the standard form equations. As we will discuss later, the[latex]\\,xy\\,[\/latex]term rotates the conic whenever[latex]\\text{ }B\\text{ }[\/latex]is not equal to zero.<\/p>\n\n<table id=\"Table_10_04_01\" summary=\"..\">\n<thead>\n<tr>\n<th><strong>Conic Sections<\/strong><\/th>\n<th><strong>Example<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>ellipse<\/td>\n<td>[latex]4{x}^{2}+9{y}^{2}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>circle<\/td>\n<td>[latex]4{x}^{2}+4{y}^{2}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>hyperbola<\/td>\n<td>[latex]4{x}^{2}-9{y}^{2}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>parabola<\/td>\n<td>[latex]4{x}^{2}=9y\\text{ or }4{y}^{2}=9x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>one line<\/td>\n<td>[latex]4x+9y=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>intersecting lines<\/td>\n<td>[latex]\\left(x-4\\right)\\left(y+4\\right)=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>parallel lines<\/td>\n<td>[latex]\\left(x-4\\right)\\left(x-9\\right)=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>a point<\/td>\n<td>[latex]4{x}^{2}+4{y}^{2}=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>no graph<\/td>\n<td>[latex]4{x}^{2}+4{y}^{2}=\\,-\\,1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1532180\" class=\"textbox key-takeaways\">\n<h3>General Form of Conic Sections<\/h3>\n<p id=\"fs-id1784761\">A conic section has the general form<\/p>\n\n<div>[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\n<p id=\"fs-id1264927\">where[latex]\\,A,B,[\/latex] and[latex]\\,C\\,[\/latex]are not all zero.<\/p>\n<p id=\"fs-id1400290\"><a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Table_10_04_02\">(Figure)<\/a> summarizes the different conic sections where[latex]\\,B=0,[\/latex] and[latex]\\,A\\,[\/latex]and[latex]\\,C\\,[\/latex]are nonzero real numbers. This indicates that the conic has not been rotated.<\/p>\n\n<table id=\"Table_10_04_02\" summary=\"..\"><colgroup><col><col><\/colgroup>\n<tbody>\n<tr>\n<td><strong>ellipse<\/strong><\/td>\n<td>[latex]A{x}^{2}+C{y}^{2}+Dx+Ey+F=0,\\text{ }A\\ne C\\text{ and }AC&gt;0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>circle<\/strong><\/td>\n<td>[latex]A{x}^{2}+C{y}^{2}+Dx+Ey+F=0,\\text{ }A=C[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>hyperbola<\/strong><\/td>\n<td>[latex]A{x}^{2}-C{y}^{2}+Dx+Ey+F=0\\text{ or }-A{x}^{2}+C{y}^{2}+Dx+Ey+F=0,[\/latex]where[latex]\\,A\\,[\/latex]and[latex]\\,C\\,[\/latex]are positive<\/td>\n<\/tr>\n<tr>\n<td><strong>parabola<\/strong><\/td>\n<td>[latex]A{x}^{2}+Dx+Ey+F=0\\text{ or }C{y}^{2}+Dx+Ey+F=0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1200060\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1128527\"><strong>Given the equation of a conic, identify the type of conic.\n<\/strong><\/p>\n\n<ol id=\"fs-id1121724\" type=\"1\">\n \t<li>Rewrite the equation in the general form, [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0.[\/latex]<\/li>\n \t<li>Identify the values of[latex]\\,A\\,[\/latex]and[latex]\\,C\\,[\/latex]from the general form.\n<ol id=\"fs-id2773969\" type=\"a\">\n \t<li>If[latex]\\,A\\,[\/latex]and[latex]\\,C\\,[\/latex] are nonzero, have the same sign, and are not equal to each other, then the graph may be an ellipse.<\/li>\n \t<li>If[latex]\\,A\\,[\/latex]and[latex]\\,C\\,[\/latex]are equal and nonzero and have the same sign, then the graph may be a circle.<\/li>\n \t<li>If[latex]\\,A\\,[\/latex]and[latex]\\,C\\,[\/latex]are nonzero and have opposite signs, then the graph may be a hyperbola.<\/li>\n \t<li>If either[latex]\\,A\\,[\/latex]or[latex]\\,C\\,[\/latex] is zero, then the graph may be a parabola.<\/li>\n<\/ol>\nIf <em>B<\/em> = 0, the conic section will have a vertical and\/or horizontal axes. If <em>B<\/em> does not equal 0, as shown below, the conic section is rotated.&nbsp; Notice the phrase \u201cmay be\u201d in the definitions. That is because the equation may not represent a conic section at all, depending on the values of <em>A<\/em>, <em>B<\/em>, <em>C<\/em>, <em>D<\/em>, <em>E<\/em>, and <em>F<\/em>. For example, the degenerate case of a circle or an ellipse is a point:\n\n[latex]A{x}^{2}+B{y}^{2}=0\\text{,}[\/latex]when <em>A<\/em> and <em>B<\/em> have the same sign.\n\nThe degenerate case of a hyperbola is two intersecting straight lines:[latex]A{x}^{2}+B{y}^{2}=0\\text{,}[\/latex]when A and B have opposite signs.\n\nOn the other hand, the equation, [latex]A{x}^{2}+B{y}^{2}+1=0\\text{,}[\/latex] when A and B are positive does not represent a graph at all, since there are no real ordered pairs which satisfy it.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_10_04_01\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1926798\">\n<h3>Identifying a Conic from Its General Form<\/h3>\n<p id=\"fs-id1963747\">Identify the graph of each of the following nondegenerate conic sections.<\/p>\n\n<ol id=\"fs-id1113018\" type=\"a\">\n \t<li>[latex]4{x}^{2}-9{y}^{2}+36x+36y-125=0[\/latex]<\/li>\n \t<li>[latex]9{y}^{2}+16x+36y-10=0[\/latex]<\/li>\n \t<li>[latex]3{x}^{2}+3{y}^{2}-2x-6y-4=0[\/latex]<\/li>\n \t<li>[latex]-25{x}^{2}-4{y}^{2}+100x+16y+20=0[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1582186\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1582186\"]\n<ol id=\"fs-id1582186\" type=\"a\">\n \t<li>Rewriting the general form, we have<span id=\"eq1\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152651\/eq1_n.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id2570988\">[latex]A=4\\,[\/latex]and[latex]\\,C=-9,[\/latex] so we observe that[latex]\\,A\\,[\/latex]and[latex]\\,C\\,[\/latex]have opposite signs. The graph of this equation is a hyperbola.<\/p>\n<\/li>\n \t<li>Rewriting the general form, we have\n<span id=\"eq2\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152657\/eq2_n.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1459350\">[latex]A=0\\,[\/latex]and[latex]\\,C=9.\\,[\/latex]We can determine that the equation is a parabola, since[latex]\\,A\\,[\/latex]is zero.<\/p>\n<\/li>\n \t<li>Rewriting the general form, we have <span id=\"eq3\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152715\/eq3_n.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1154673\">[latex]A=3\\,[\/latex]and[latex]\\,C=3.\\,[\/latex]Because[latex]\\,A=C,[\/latex] the graph of this equation is a circle.<\/p>\n<\/li>\n \t<li>Rewriting the general form, we have\n<span id=\"eq4\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152734\/eq4.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id2587849\">[latex]A=-25\\,[\/latex]and[latex]\\,C=-4.\\,[\/latex]Because[latex]\\,AC&gt;0\\,[\/latex]and[latex]\\,A\\ne C,[\/latex] the graph of this equation is an ellipse.[\/hidden-answer]<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2814436\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1295024\">\n<p id=\"fs-id1342273\">Identify the graph of each of the following nondegenerate conic sections.<\/p>\n\n<ol id=\"fs-id1081247\" type=\"a\">\n \t<li>[latex]16{y}^{2}-{x}^{2}+x-4y-9=0[\/latex]<\/li>\n \t<li>[latex]16{x}^{2}+4{y}^{2}+16x+49y-81=0[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2335243\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2335243\"]\n<ol id=\"fs-id2335243\" type=\"a\">\n \t<li>hyperbola<\/li>\n \t<li>ellipse<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1319873\" class=\"bc-section section\">\n<h4>Finding a New Representation of the Given Equation after Rotating through a Given Angle<\/h4>\n<p id=\"fs-id1378740\">Until now, we have looked at equations of conic sections without an[latex]\\,xy\\,[\/latex]term, which aligns the graphs with the <em>x<\/em>- and <em>y<\/em>-axes. When we add an[latex]\\,xy\\,[\/latex]term, we are rotating the conic about the origin. If the <em>x<\/em>- and <em>y<\/em>-axes are rotated through an angle, say[latex]\\,\\theta ,[\/latex]then every point on the plane may be thought of as having two representations:[latex]\\,\\left(x,y\\right)\\,[\/latex]on the Cartesian plane with the original <em>x<\/em>-axis and <em>y<\/em>-axis, and[latex]\\,\\left({x}^{\\prime },{y}^{\\prime }\\right)\\,[\/latex]on the new plane defined by the new, rotated axes, called the <em>x'<\/em>-axis and <em>y'<\/em>-axis. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_003\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_10_04_003\" 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\/19152737\/CNX_Precalc_Figure_10_04_003.jpg\" alt=\"\" width=\"487\" height=\"441\"> <strong>Figure 3. <\/strong>The graph of the rotated ellipse[latex]\\,{x}^{2}+{y}^{2}\u2013xy\u201315=0[\/latex][\/caption]<\/div>\n<p id=\"fs-id1352584\">We will find the relationships between[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]on the Cartesian plane with[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]on the new rotated plane. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_004\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_10_04_004\" 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\/19152741\/CNX_Precalc_Figure_10_04_004.jpg\" alt=\"\" width=\"487\" height=\"366\"> <strong>Figure 4. <\/strong>The Cartesian plane with x- and y-axes and the resulting x\u2032\u2212 and y\u2032\u2212axes formed by a rotation by an angle[latex]\\text{ }\\theta .[\/latex][\/caption]<\/div>\n<p id=\"fs-id1333360\">The original coordinate <em>x<\/em>- and <em>y<\/em>-axes have unit vectors[latex]\\,i\\,[\/latex]and[latex]\\,j\\,.[\/latex]The rotated coordinate axes have unit vectors[latex]\\,{i}^{\\prime }\\,[\/latex]and[latex]\\,{j}^{\\prime }.[\/latex]The angle[latex]\\,\\theta \\,[\/latex]is known as the angle of rotation. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_005\">(Figure)<\/a>. We may write the new unit vectors in terms of the original ones.<\/p>\n\n<div id=\"fs-id2753493\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{i}^{\\prime }=\\mathrm{cos}\\text{ }\\theta i+\\mathrm{sin}\\text{ }\\theta j\\hfill \\\\ {j}^{\\prime }=-\\mathrm{sin}\\text{ }\\theta i+\\mathrm{cos}\\text{ }\\theta j\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"Figure_10_04_005\" 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\/19152747\/CNX_Precalc_Figure_10_04_005.jpg\" alt=\"\" width=\"487\" height=\"364\"> <strong>Figure 5. <\/strong>Relationship between the old and new coordinate planes.[\/caption]\n\n<\/div>\n<p id=\"fs-id2129430\">Consider a vector<strong>[latex]\\,u\\,[\/latex]<\/strong>in the new coordinate plane. It may be represented in terms of its coordinate axes.<\/p>\n\n<div id=\"fs-id1260569\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime }\\hfill &amp; \\hfill \\\\ u={x}^{\\prime }\\left(i\\text{ }\\mathrm{cos}\\text{ }\\theta +j\\text{ }\\mathrm{sin}\\text{ }\\theta \\right)+{y}^{\\prime }\\left(-i\\text{ }\\mathrm{sin}\\text{ }\\theta +j\\text{ }\\mathrm{cos}\\text{ }\\theta \\right)\\hfill &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{Substitute}.\\hfill \\\\ u=ix\\text{'}\\text{ }\\mathrm{cos}\\text{ }\\theta +jx\\text{'}\\text{ }\\mathrm{sin}\\text{ }\\theta -iy\\text{'}\\text{ }\\mathrm{sin}\\text{ }\\theta +jy\\text{'}\\text{ }\\mathrm{cos}\\text{ }\\theta \\hfill &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{Distribute}.\\hfill \\\\ u=ix\\text{'}\\text{ }\\mathrm{cos}\\text{ }\\theta -iy\\text{'}\\text{ }\\mathrm{sin}\\text{ }\\theta +jx\\text{'}\\text{ }\\mathrm{sin}\\text{ }\\theta +jy\\text{'}\\text{ }\\mathrm{cos}\\text{ }\\theta \\hfill &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{Apply commutative property}.\\hfill \\\\ u=\\left(x\\text{'}\\text{ }\\mathrm{cos}\\text{ }\\theta -y\\text{'}\\text{ }\\mathrm{sin}\\text{ }\\theta \\right)i+\\left(x\\text{'}\\text{ }\\mathrm{sin}\\text{ }\\theta +y\\text{'}\\text{ }\\mathrm{cos}\\text{ }\\theta \\right)j\\hfill &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{Factor by grouping}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1319681\">Because[latex]\\,u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime },[\/latex] we have representations of[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]in terms of the new coordinate system.<\/p>\n\n<div id=\"fs-id1339818\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\\\ \\text{and}\\\\ y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1741266\" class=\"textbox key-takeaways\">\n<h3>Equations of Rotation<\/h3>\n<p id=\"fs-id1385513\">If a point[latex]\\,\\left(x,y\\right)\\,[\/latex]on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle[latex]\\,\\theta \\,[\/latex]from the positive <em>x<\/em>-axis, then the coordinates of the point with respect to the new axes are[latex]\\,\\left({x}^{\\prime },{y}^{\\prime }\\right).\\,[\/latex]We can use the following equations of rotation to define the relationship between[latex]\\,\\left(x,y\\right)\\,[\/latex]and[latex]\\,\\left({x}^{\\prime },{y}^{\\prime }\\right):[\/latex]<\/p>\n\n<div id=\"Equation_10_04_02\">[latex]x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta [\/latex]<\/div>\n<p id=\"fs-id2281028\">and<\/p>\n\n<div id=\"Equation_10_04_03\">[latex]y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta [\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1355297\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1560707\"><strong>Given the equation of a conic, find a new representation after rotating through an angle.\n<\/strong><\/p>\n\n<ol id=\"fs-id1146233\" type=\"1\">\n \t<li>Find[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]where[latex]\\,x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta .[\/latex]<\/li>\n \t<li>Substitute the expression for[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]into in the given equation, then simplify.<\/li>\n \t<li>Write the equations with[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]in standard form.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_10_04_02\" class=\"textbox examples\">\n<div id=\"fs-id1357778\">\n<div id=\"fs-id2354314\">\n<h3>Finding a New Representation of an Equation after Rotating through a Given Angle<\/h3>\n<p id=\"fs-id1541593\">Find a new representation of the equation[latex]\\,2{x}^{2}-xy+2{y}^{2}-30=0\\,[\/latex]after rotating through an angle of[latex]\\,\\theta =45\u00b0.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1503906\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1503906\"]\n<p id=\"fs-id1503906\">Find[latex]\\,x\\,[\/latex]and[latex]\\,y,[\/latex]where[latex]\\,x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta .[\/latex]<\/p>\n<p id=\"fs-id1290570\">Because[latex]\\,\\theta =45\u00b0,[\/latex]<\/p>\n\n<div id=\"fs-id1989238\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ x={x}^{\\prime }\\mathrm{cos}\\left(45\u00b0\\right)-{y}^{\\prime }\\mathrm{sin}\\left(45\u00b0\\right)\\hfill \\\\ x={x}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)-{y}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)\\hfill \\\\ x=\\frac{{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{2}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2113047\">and<\/p>\n\n<div id=\"fs-id2091380\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\\\ \\begin{array}{l}y={x}^{\\prime }\\mathrm{sin}\\left(45\u00b0\\right)+{y}^{\\prime }\\mathrm{cos}\\left(45\u00b0\\right)\\hfill \\\\ y={x}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)+{y}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)\\hfill \\\\ y=\\frac{{x}^{\\prime }+{y}^{\\prime }}{\\sqrt{2}}\\hfill \\end{array}\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1674091\">Substitute[latex]\\,x={x}^{\\prime }\\mathrm{cos}\\theta -{y}^{\\prime }\\mathrm{sin}\\theta \\,[\/latex]and[latex]\\,y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta \\,[\/latex]into[latex]\\,2{x}^{2}-xy+2{y}^{2}-30=0.[\/latex]<\/p>\n\n<div id=\"fs-id1375279\" class=\"unnumbered aligncenter\">[latex]2{\\left(\\frac{{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{2}}\\right)}^{2}-\\left(\\frac{{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{2}}\\right)\\left(\\frac{{x}^{\\prime }+{y}^{\\prime }}{\\sqrt{2}}\\right)+2{\\left(\\frac{{x}^{\\prime }+{y}^{\\prime }}{\\sqrt{2}}\\right)}^{2}-30=0[\/latex]<\/div>\n<p id=\"fs-id1901790\">Simplify.<\/p>\n\n<div id=\"fs-id1901793\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\overline{)2}\\frac{\\left({x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }-{y}^{\\prime }\\right)}{\\overline{)2}}-\\frac{\\left({x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }+{y}^{\\prime }\\right)}{2}+\\overline{)2}\\frac{\\left({x}^{\\prime }+{y}^{\\prime }\\right)\\left({x}^{\\prime }+{y}^{\\prime }\\right)}{\\overline{)2}}-30=0\\hfill &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{FOIL method}\\hfill \\\\ \\text{ }{x}^{\\prime }{}^{2}{\\overline{)-2{x}^{\\prime }y}}^{\\prime }+{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}+{x}^{\\prime }{}^{2}\\overline{)+2{x}^{\\prime }{y}^{\\prime }}+{y}^{\\prime }{}^{2}-30=0\\hfill &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{Combine like terms}.\\hfill \\\\ \\text{ }2{x}^{\\prime }{}^{2}+2{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}=30\\hfill &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{Combine like terms}.\\hfill \\\\ \\text{ }2\\left(2{x}^{\\prime }{}^{2}+2{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}\\right)=2\\left(30\\right)\\hfill &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{Multiply both sides by 2}.\\hfill \\\\ \\text{ }4{x}^{\\prime }{}^{2}+4{y}^{\\prime }{}^{2}-\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)=60\\hfill &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{Simplify}.\\hfill \\\\ \\text{ }4{x}^{\\prime }{}^{2}+4{y}^{\\prime }{}^{2}-{x}^{\\prime }{}^{2}+{y}^{\\prime }{}^{2}=60\\hfill &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{Distribute}.\\hfill \\\\ \\text{ }\\frac{3{x}^{\\prime }{}^{2}}{60}+\\frac{5{y}^{\\prime }{}^{2}}{60}=\\frac{60}{60}\\hfill &amp; \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{Set equal to 1}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2230049\">Write the equations with[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]in the standard form.<\/p>\n\n<div id=\"fs-id2465469\" class=\"unnumbered aligncenter\">[latex]\\frac{{{x}^{\\prime }}^{2}}{20}+\\frac{{{y}^{\\prime }}^{2}}{12}=1[\/latex]<\/div>\nThis equation is an ellipse. <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_006\">(Figure)<\/a> shows the graph.\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\/19152753\/CNX_Precalc_Figure_10_04_006.jpg\" alt=\"\" width=\"487\" height=\"441\"> <strong>Figure 6.<\/strong>[\/caption]\n<p id=\"fs-id1923420\">[\/hidden-answer]<span id=\"fs-id1194819\"><\/span><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1194834\" class=\"bc-section section\">\n<h3>Writing Equations of Rotated Conics in Standard Form<\/h3>\n<p id=\"fs-id1194839\">Now that we can find the standard form of a conic when we are given an angle of rotation, we will learn how to transform the equation of a conic given in the form[latex]\\,A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0\\,[\/latex]into standard form by rotating the axes. To do so, we will rewrite the general form as an equation in the[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]coordinate system without the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]term, by rotating the axes by a measure of[latex]\\,\\theta \\,[\/latex]that satisfies<\/p>\n\n<div id=\"Equation_10_04_04\">[latex]\\mathrm{cot}\\left(2\\theta \\right)=\\frac{A-C}{B}[\/latex]<\/div>\n<p id=\"fs-id1186178\">We have learned already that any conic may be represented by the second degree equation<\/p>\n\n<div id=\"fs-id1186181\" class=\"unnumbered aligncenter\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\n<p id=\"fs-id2702021\">where[latex]\\,A,B,[\/latex]and[latex]\\,C\\,[\/latex]are not all zero. However, if[latex]\\,B\\ne 0,[\/latex] then we have an[latex]\\,xy\\,[\/latex]term that prevents us from rewriting the equation in standard form. To eliminate it, we can rotate the axes by an acute angle[latex]\\,\\theta \\,[\/latex]where[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)=\\frac{A-C}{B}.[\/latex]<\/p>\n\n<ul id=\"fs-id2132367\">\n \t<li>If[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)&gt;0,[\/latex] then[latex]\\,2\\theta \\,[\/latex] is in the first quadrant, and[latex]\\,\\theta \\,[\/latex] is between[latex]\\,\\left(0\u00b0,45\u00b0\\right).[\/latex]<\/li>\n \t<li>If[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)&lt;0,[\/latex] then[latex]\\,2\\theta \\,[\/latex] is in the second quadrant, and[latex]\\,\\theta \\,[\/latex] is between[latex]\\,\\left(45\u00b0,90\u00b0\\right).[\/latex]<\/li>\n \t<li>If[latex]\\,A=C,[\/latex] then[latex]\\,\\theta =45\u00b0.[\/latex]<\/li>\n<\/ul>\n<div id=\"fs-id1503870\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1503876\"><strong>Given an equation for a conic in the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]system, rewrite the equation without the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]term in terms of[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime },[\/latex]where the[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]axes are rotations of the standard axes by[latex]\\,\\theta \\,[\/latex]degrees.<\/strong><\/p>\n\n<ol id=\"fs-id2274393\" type=\"1\">\n \t<li>Find[latex]\\,\\mathrm{cot}\\left(2\\theta \\right).[\/latex]<\/li>\n \t<li>Find[latex]\\,\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,\\mathrm{cos}\\text{ }\\theta .[\/latex]<\/li>\n \t<li>Substitute[latex]\\,\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,\\mathrm{cos}\\text{ }\\theta \\,[\/latex]into[latex]\\,x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta .[\/latex]<\/li>\n \t<li>Substitute the expression for[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]into in the given equation, and then simplify.<\/li>\n \t<li>Write the equations with[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]in the standard form with respect to the rotated axes.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_10_04_03\" class=\"textbox examples\">\n<div id=\"fs-id1414093\">\n<div id=\"fs-id1414095\">\n<h3>Rewriting an Equation with respect to the <em>x\u2032<\/em> and <em>y\u2032<\/em> axes without the <em>x\u2032y\u2032<\/em> Term<\/h3>\n<p id=\"fs-id1259372\">Rewrite the equation[latex]\\,8{x}^{2}-12xy+17{y}^{2}=20\\,[\/latex]in the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]system without an[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]term.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1741772\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1741772\"]\n<p id=\"fs-id1741772\">First, we find[latex]\\,\\mathrm{cot}\\left(2\\theta \\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_007\">(Figure)<\/a>.<\/p>\n\n<div id=\"fs-id1403863\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}8{x}^{2}-12xy+17{y}^{2}=20\u21d2A=8,\\,B=-12\\,\\text{and}\\,C=17\\hfill \\\\ \\text{ }\\,\\,\\,\\mathrm{cot}\\left(2\\theta \\right)=\\frac{A-C}{B}=\\frac{8-17}{-12}\\hfill \\\\ \\text{ }\\,\\,\\,\\mathrm{cot}\\left(2\\theta \\right)=\\frac{-9}{-12}=\\frac{3}{4}\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"Figure_10_04_007\" 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\/19152755\/CNX_Precalc_Figure_10_04_007.jpg\" alt=\"\" width=\"487\" height=\"328\"> <strong>Figure 7.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1901664\" class=\"unnumbered aligncenter\">[latex]\\mathrm{cot}\\left(2\\theta \\right)=\\frac{3}{4}=\\frac{\\text{adjacent}}{\\text{opposite}}[\/latex]<\/div>\n<p id=\"fs-id2206549\">So the hypotenuse is<\/p>\n\n<div id=\"fs-id2206552\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{r}\\hfill {3}^{2}+{4}^{2}={h}^{2}\\\\ \\hfill 9+16={h}^{2}\\\\ \\hfill 25={h}^{2}\\\\ \\hfill h=5\\,\\,\\,\\end{array}[\/latex]<\/div>\n<p id=\"fs-id2175665\">Next, we find[latex]\\,\\mathrm{sin}\\text{ }\\theta [\/latex] and [latex]\\mathrm{cos}\\text{ }\\theta .[\/latex]<\/p>\n\n<div id=\"fs-id1316548\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ \\mathrm{sin}\\text{ }\\theta =\\sqrt{\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}}=\\sqrt{\\frac{1-\\frac{3}{5}}{2}}=\\sqrt{\\frac{\\frac{5}{5}-\\frac{3}{5}}{2}}=\\sqrt{\\frac{5-3}{5}\\cdot \\frac{1}{2}}=\\sqrt{\\frac{2}{10}}=\\sqrt{\\frac{1}{5}}\\hfill \\end{array}\\hfill \\\\ \\mathrm{sin}\\text{ }\\theta =\\frac{1}{\\sqrt{5}}\\hfill \\\\ \\mathrm{cos}\\text{ }\\theta =\\sqrt{\\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}}=\\sqrt{\\frac{1+\\frac{3}{5}}{2}}=\\sqrt{\\frac{\\frac{5}{5}+\\frac{3}{5}}{2}}=\\sqrt{\\frac{5+3}{5}\\cdot \\frac{1}{2}}=\\sqrt{\\frac{8}{10}}=\\sqrt{\\frac{4}{5}}\\hfill \\\\ \\mathrm{cos}\\text{ }\\theta =\\frac{2}{\\sqrt{5}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1888188\">Substitute the values of[latex]\\,\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,\\mathrm{cos}\\text{ }\\theta \\,[\/latex]into[latex]\\,x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta .[\/latex]<\/p>\n\n<div id=\"fs-id1587444\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\begin{array}{l}x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\hfill \\\\ x={x}^{\\prime }\\left(\\frac{2}{\\sqrt{5}}\\right)-{y}^{\\prime }\\left(\\frac{1}{\\sqrt{5}}\\right)\\hfill \\\\ x=\\frac{2{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{5}}\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2691278\">and<\/p>\n\n<div id=\"fs-id2691281\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta \\hfill \\end{array}\\hfill \\\\ y={x}^{\\prime }\\left(\\frac{1}{\\sqrt{5}}\\right)+{y}^{\\prime }\\left(\\frac{2}{\\sqrt{5}}\\right)\\hfill \\\\ y=\\frac{{x}^{\\prime }+2{y}^{\\prime }}{\\sqrt{5}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2290144\">Substitute the expressions for[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]into in the given equation, and then simplify.<\/p>\n\n<div id=\"fs-id2030209\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }8{\\left(\\frac{2{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{5}}\\right)}^{2}-12\\left(\\frac{2{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{5}}\\right)\\left(\\frac{{x}^{\\prime }+2{y}^{\\prime }}{\\sqrt{5}}\\right)+17{\\left(\\frac{{x}^{\\prime }+2{y}^{\\prime }}{\\sqrt{5}}\\right)}^{2}=20\\text{ }\\hfill \\\\ \\text{ }8\\left(\\frac{\\left(2{x}^{\\prime }-{y}^{\\prime }\\right)\\left(2{x}^{\\prime }-{y}^{\\prime }\\right)}{5}\\right)-12\\left(\\frac{\\left(2{x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }+2{y}^{\\prime }\\right)}{5}\\right)+17\\left(\\frac{\\left({x}^{\\prime }+2{y}^{\\prime }\\right)\\left({x}^{\\prime }+2{y}^{\\prime }\\right)}{5}\\right)=20\\text{ }\\hfill \\\\ \\text{ }8\\left(4{x}^{\\prime }{}^{2}-4{x}^{\\prime }{y}^{\\prime }+{y}^{\\prime }{}^{2}\\right)-12\\left(2{x}^{\\prime }{}^{2}+3{x}^{\\prime }{y}^{\\prime }-2{y}^{\\prime }{}^{2}\\right)+17\\left({x}^{\\prime }{}^{2}+4{x}^{\\prime }{y}^{\\prime }+4{y}^{\\prime }{}^{2}\\right)=100\\hfill \\\\ 32{x}^{\\prime }{}^{2}-32{x}^{\\prime }{y}^{\\prime }+8{y}^{\\prime }{}^{2}-24{x}^{\\prime }{}^{2}-36{x}^{\\prime }{y}^{\\prime }+24{y}^{\\prime }{}^{2}+17{x}^{\\prime }{}^{2}+68{x}^{\\prime }{y}^{\\prime }+68{y}^{\\prime }{}^{2}=100\\hfill \\\\ \\text{ }25{x}^{\\prime }{}^{2}+100{y}^{\\prime }{}^{2}=100\\text{ }\\hfill \\\\ \\text{ }\\frac{25}{100}{x}^{\\prime }{}^{2}+\\frac{100}{100}{y}^{\\prime }{}^{2}=\\frac{100}{100} \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1805836\">Write the equations with[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]in the standard form with respect to the new coordinate system.<\/p>\n\n<div id=\"fs-id2044859\" class=\"unnumbered aligncenter\">[latex]\\frac{{{x}^{\\prime }}^{2}}{4}+\\frac{{{y}^{\\prime }}^{2}}{1}=1[\/latex]<\/div>\n<a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_008\">(Figure)<\/a> shows the graph of the ellipse.\n\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\/19152809\/CNX_Precalc_Figure_10_04_008.jpg\" alt=\"\" width=\"487\" height=\"217\"> <strong>Figure 8.<\/strong>[\/caption]\n<p id=\"fs-id1419500\"><span id=\"fs-id1419508\"><\/span>[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2008343\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id2008352\">\n<p id=\"fs-id2008353\">Rewrite the[latex]\\,13{x}^{2}-6\\sqrt{3}xy+7{y}^{2}=16\\,[\/latex]in the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]system without the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]term.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2579717\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2579717\"]\n<p id=\"fs-id2579717\">[latex]\\frac{{{x}^{\\prime }}^{2}}{4}+\\frac{{{y}^{\\prime }}^{2}}{1}=1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_10_04_04\" class=\"textbox examples\">\n<div id=\"fs-id2753537\">\n<div id=\"fs-id2050743\">\n<h3>Graphing an Equation That Has No <em>x\u2032y\u2032<\/em> Terms<\/h3>\n<p id=\"fs-id2050753\">Graph the following equation relative to the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]system:<\/p>\n\n<div id=\"fs-id2067868\" class=\"unnumbered aligncenter\">[latex]{x}^{2}+12xy-4{y}^{2}=30[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1770490\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1770490\"]\n<p id=\"fs-id1770490\">First, we find[latex]\\,\\mathrm{cot}\\left(2\\theta \\right).[\/latex]<\/p>\n\n<div id=\"fs-id2102392\" class=\"unnumbered aligncenter\">[latex]{x}^{2}+12xy-4{y}^{2}=20\u21d2A=1,\\text{ }B=12,\\text{and }C=-4[\/latex]<\/div>\n<div id=\"fs-id1649246\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{cot}\\left(2\\theta \\right)=\\frac{A-C}{B}\\hfill \\\\ \\mathrm{cot}\\left(2\\theta \\right)=\\frac{1-\\left(-4\\right)}{12}\\hfill \\\\ \\mathrm{cot}\\left(2\\theta \\right)=\\frac{5}{12}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2349896\">Because[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)=\\frac{5}{12},[\/latex] we can draw a reference triangle as in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_009\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_10_04_009\" 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\/19152811\/CNX_Precalc_Figure_10_04_009.jpg\" alt=\"\" width=\"487\" height=\"591\"> <strong>Figure 9.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id2637723\" class=\"unnumbered aligncenter\">[latex]\\mathrm{cot}\\left(2\\theta \\right)=\\frac{5}{12}=\\frac{\\text{adjacent}}{\\text{opposite}}[\/latex]<\/div>\n<p id=\"fs-id2309119\">Thus, the hypotenuse is<\/p>\n\n<div id=\"fs-id2309122\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{r}\\hfill {5}^{2}+{12}^{2}={h}^{2}\\\\ \\hfill 25+144={h}^{2}\\\\ \\hfill 169={h}^{2}\\\\ \\hfill h=13\\end{array}[\/latex]<\/div>\n<p id=\"fs-id2292950\">Next, we find[latex]\\,\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,\\mathrm{cos}\\text{ }\\theta .\\,[\/latex]We will use half-angle identities.<\/p>\n\n<div id=\"fs-id2594568\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ \\mathrm{sin}\\text{ }\\theta =\\sqrt{\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}}=\\sqrt{\\frac{1-\\frac{5}{13}}{2}}=\\sqrt{\\frac{\\frac{13}{13}-\\frac{5}{13}}{2}}=\\sqrt{\\frac{8}{13}\\cdot \\frac{1}{2}}=\\frac{2}{\\sqrt{13}}\\hfill \\end{array}\\hfill \\\\ \\mathrm{cos}\\text{ }\\theta =\\sqrt{\\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}}=\\sqrt{\\frac{1+\\frac{5}{13}}{2}}=\\sqrt{\\frac{\\frac{13}{13}+\\frac{5}{13}}{2}}=\\sqrt{\\frac{18}{13}\\cdot \\frac{1}{2}}=\\frac{3}{\\sqrt{13}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1405820\">Now we find[latex]\\,x\\,[\/latex]and[latex]\\,y\\text{.\\hspace{0.17em}}[\/latex]<\/p>\n\n<div id=\"fs-id1936549\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\hfill \\\\ x={x}^{\\prime }\\left(\\frac{3}{\\sqrt{13}}\\right)-{y}^{\\prime }\\left(\\frac{2}{\\sqrt{13}}\\right)\\hfill \\\\ x=\\frac{3{x}^{\\prime }-2{y}^{\\prime }}{\\sqrt{13}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1576589\">and<\/p>\n\n<div id=\"fs-id1576592\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta \\hfill \\\\ y={x}^{\\prime }\\left(\\frac{2}{\\sqrt{13}}\\right)+{y}^{\\prime }\\left(\\frac{3}{\\sqrt{13}}\\right)\\hfill \\\\ y=\\frac{2{x}^{\\prime }+3{y}^{\\prime }}{\\sqrt{13}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2290414\">Now we substitute[latex]\\,x=\\frac{3{x}^{\\prime }-2{y}^{\\prime }}{\\sqrt{13}}\\,[\/latex]and[latex]\\,y=\\frac{2{x}^{\\prime }+3{y}^{\\prime }}{\\sqrt{13}}\\,[\/latex]into[latex]\\,{x}^{2}+12xy-4{y}^{2}=30.[\/latex]<\/p>\n\n<div id=\"fs-id2067955\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{llll}\\text{ }{\\left(\\frac{3{x}^{\\prime }-2{y}^{\\prime }}{\\sqrt{13}}\\right)}^{2}+12\\left(\\frac{3{x}^{\\prime }-2{y}^{\\prime }}{\\sqrt{13}}\\right)\\left(\\frac{2{x}^{\\prime }+3{y}^{\\prime }}{\\sqrt{13}}\\right)-4{\\left(\\frac{2{x}^{\\prime }+3{y}^{\\prime }}{\\sqrt{13}}\\right)}^{2}=30\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill \\\\ \\text{ }\\,\\,\\left(\\frac{1}{13}\\right)\\left[{\\left(3{x}^{\\prime }-2{y}^{\\prime }\\right)}^{2}+12\\left(3{x}^{\\prime }-2{y}^{\\prime }\\right)\\left(2{x}^{\\prime }+3{y}^{\\prime }\\right)-4{\\left(2{x}^{\\prime }+3{y}^{\\prime }\\right)}^{2}\\right]=30 \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Factor}.\\hfill \\\\ \\left(\\frac{1}{13}\\right)\\left[9{x}^{\\prime }{}^{2}-12{x}^{\\prime }{y}^{\\prime }+4{y}^{\\prime }{}^{2}+12\\left(6{x}^{\\prime }{}^{2}+5{x}^{\\prime }{y}^{\\prime }-6{y}^{\\prime }{}^{2}\\right)-4\\left(4{x}^{\\prime }{}^{2}+12{x}^{\\prime }{y}^{\\prime }+9{y}^{\\prime }{}^{2}\\right)\\right]=30\\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Multiply}.\\hfill \\\\ \\text{ }\\left(\\frac{1}{13}\\right)\\left[9{x}^{\\prime }{}^{2}-12{x}^{\\prime }{y}^{\\prime }+4{y}^{\\prime }{}^{2}+72{x}^{\\prime }{}^{2}+60{x}^{\\prime }{y}^{\\prime }-72{y}^{\\prime }{}^{2}-16{x}^{\\prime }{}^{2}-48{x}^{\\prime }{y}^{\\prime }-36{y}^{\\prime }{}^{2}\\right]=30\\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Distribute}.\\hfill \\\\ \\text{ }\\,\\text{ }\\left(\\frac{1}{13}\\right)\\left[65{x}^{\\prime }{}^{2}-104{y}^{\\prime }{}^{2}\\right]=30\\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ \\text{ }65{x}^{\\prime }{}^{2}-104{y}^{\\prime }{}^{2}=390\\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Multiply}.\\text{ }\\hfill \\\\ \\text{ }\\frac{{x}^{\\prime }{}^{2}}{6}-\\frac{4{y}^{\\prime }{}^{2}}{15}=1 \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Divide by 390}.\\hfill \\end{array}[\/latex]<\/div>\n<a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_010\">(Figure)<\/a> shows the graph of the hyperbola[latex]\\,\\frac{{{x}^{\\prime }}^{2}}{6}-\\frac{4{{y}^{\\prime }}^{2}}{15}=1.\\text{ }[\/latex]\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\/19152818\/CNX_Precalc_Figure_10_04_010.jpg\" alt=\"\" width=\"487\" height=\"441\"> <strong>Figure 10.<\/strong>[\/caption]\n<p id=\"fs-id1500365\">[\/hidden-answer]<span id=\"fs-id2011757\"><\/span><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2011771\" class=\"bc-section section\">\n<h3>Identifying Conics without Rotating Axes<\/h3>\n<p id=\"fs-id2822590\">Now we have come full circle. How do we identify the type of conic described by an equation? What happens when the axes are rotated? Recall, the general form of a conic is<\/p>\n\n<div id=\"fs-id2822595\" class=\"unnumbered aligncenter\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\n<p id=\"fs-id1966856\">If we apply the rotation formulas to this equation we get the form<\/p>\n\n<div id=\"fs-id1966860\" class=\"unnumbered aligncenter\">[latex]{A}^{\\prime }{{x}^{\\prime }}^{2}+{B}^{\\prime }{x}^{\\prime }{y}^{\\prime }+{C}^{\\prime }{{y}^{\\prime }}^{2}+{D}^{\\prime }{x}^{\\prime }+{E}^{\\prime }{y}^{\\prime }+{F}^{\\prime }=0[\/latex]<\/div>\n<p id=\"fs-id1587522\">It may be shown that[latex]\\,{B}^{2}-4AC={{B}^{\\prime }}^{2}-4{A}^{\\prime }{C}^{\\prime }.\\,[\/latex]The expression does not vary after rotation, so we call the expression invariant<strong>.<\/strong> The discriminant,[latex]\\,{B}^{2}-4AC,[\/latex] is invariant and remains unchanged after rotation. Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section.<\/p>\n\n<div id=\"fs-id1971164\" class=\"textbox key-takeaways\">\n<h3>Using the Discriminant to Identify a Conic<\/h3>\n<p id=\"fs-id1971173\">If the equation[latex]\\,A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0\\,[\/latex]is transformed by rotating axes into the equation[latex]\\,{A}^{\\prime }{{x}^{\\prime }}^{2}+{B}^{\\prime }{x}^{\\prime }{y}^{\\prime }+{C}^{\\prime }{{y}^{\\prime }}^{2}+{D}^{\\prime }{x}^{\\prime }+{E}^{\\prime }{y}^{\\prime }+{F}^{\\prime }=0,[\/latex] then[latex]\\,{B}^{2}-4AC={{B}^{\\prime }}^{2}-4{A}^{\\prime }{C}^{\\prime }.[\/latex]<\/p>\n<p id=\"fs-id1527850\">The equation[latex]\\,A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0\\,[\/latex]is an ellipse, a parabola, or a hyperbola, or a degenerate case of one of these.<\/p>\n<p id=\"fs-id1246926\">If the discriminant,[latex]\\,{B}^{2}-4AC,[\/latex]is<\/p>\n\n<ul id=\"fs-id1246959\">\n \t<li>[latex]&lt;0,[\/latex] the conic section is an ellipse<\/li>\n \t<li>[latex]=0,[\/latex] the conic section is a parabola<\/li>\n \t<li>[latex]&gt;0,[\/latex] the conic section is a hyperbola<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_10_04_05\" class=\"textbox examples\">\n<div id=\"fs-id2138546\">\n<div id=\"fs-id2138548\">\n<h3>Identifying the Conic without Rotating Axes<\/h3>\n<p id=\"fs-id2336006\">Identify the conic for each of the following without rotating axes.<\/p>\n\n<ol id=\"fs-id2336009\" type=\"a\">\n \t<li>[latex]5{x}^{2}+2\\sqrt{3}xy+2{y}^{2}-5=0[\/latex]<\/li>\n \t<li>[latex]5{x}^{2}+2\\sqrt{3}xy+12{y}^{2}-5=0[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id3068892\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id3068892\"]\n<ol id=\"fs-id3068892\" type=\"a\">\n \t<li>Let\u2019s begin by determining[latex]\\,A,B,[\/latex] and[latex]\\,C.[\/latex]\n<div id=\"fs-id1649564\" class=\"unnumbered aligncenter\">[latex]\\underset{A}{\\underbrace{5}}{x}^{2}+\\underset{B}{\\underbrace{2\\sqrt{3}}}xy+\\underset{C}{\\underbrace{2}}{y}^{2}-5=0[\/latex]<\/div>\n<p id=\"fs-id2462423\">Now, we find the discriminant.<\/p>\n\n<div id=\"fs-id2462426\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{B}^{2}-4AC={\\left(2\\sqrt{3}\\right)}^{2}-4\\left(5\\right)\\left(2\\right)\\hfill \\\\ \\text{ }=4\\left(3\\right)-40\\hfill \\\\ \\text{ }=12-40\\hfill \\\\ \\text{ }=-28&lt;0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1958098\">Therefore,[latex]\\,5{x}^{2}+2\\sqrt{3}xy+2{y}^{2}-5=0\\,[\/latex]represents an ellipse.<\/p>\n<\/li>\n \t<li>Again, let\u2019s begin by determining[latex]\\,A,B,[\/latex] and[latex]\\,C.[\/latex]\n<div id=\"fs-id2602274\" class=\"unnumbered aligncenter\">[latex]\\underset{A}{\\underbrace{5}}{x}^{2}+\\underset{B}{\\underbrace{2\\sqrt{3}}}xy+\\underset{C}{\\underbrace{12}}{y}^{2}-5=0[\/latex]<\/div>\n<p id=\"fs-id2748962\">Now, we find the discriminant.<\/p>\n\n<div id=\"fs-id2748965\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{B}^{2}-4AC={\\left(2\\sqrt{3}\\right)}^{2}-4\\left(5\\right)\\left(12\\right)\\hfill \\\\ \\text{ }=4\\left(3\\right)-240\\hfill \\\\ \\text{ }=12-240\\hfill \\\\ \\text{ }=-228&lt;0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2600622\">Therefore,[latex]\\,5{x}^{2}+2\\sqrt{3}xy+12{y}^{2}-5=0\\,[\/latex]represents an ellipse.[\/hidden-answer]<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2436719\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_04_03\">\n<div id=\"fs-id2436729\">\n<p id=\"fs-id2436730\">Identify the conic for each of the following without rotating axes.<\/p>\n\n<ol id=\"fs-id2436733\" type=\"a\">\n \t<li>[latex]{x}^{2}-9xy+3{y}^{2}-12=0[\/latex]<\/li>\n \t<li>[latex]10{x}^{2}-9xy+4{y}^{2}-4=0[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1753961\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1753961\"]\n<ol id=\"fs-id1753961\" type=\"a\">\n \t<li>hyperbola<\/li>\n \t<li>ellipse<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1951748\" class=\"precalculus media\">\n<p id=\"fs-id1951754\">Access this online resource for additional instruction and practice with conic sections and rotation of axes.<\/p>\n\n<ul id=\"fs-id1951759\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/introconic\">Introduction to Conic Sections<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1951769\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"fs-id1951776\" summary=\"..\">\n<tbody>\n<tr>\n<td>General Form equation of a conic section<\/td>\n<td>[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rotation of a conic section<\/td>\n<td>[latex]\\begin{array}{l}x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\hfill \\\\ y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta \\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Angle of rotation<\/td>\n<td>[latex]\\theta ,\\text{where }\\mathrm{cot}\\left(2\\theta \\right)=\\frac{A-C}{B}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id2430170\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id2430176\">\n \t<li>Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola.<\/li>\n \t<li>A nondegenerate conic section has the general form[latex]\\,A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0\\,[\/latex]where[latex]\\,A,B\\,[\/latex]and[latex]\\,C\\,[\/latex]are not all zero. The values of[latex]\\,A,B,[\/latex] and[latex]\\,C\\,[\/latex]determine the type of conic. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Example_10_04_01\">(Figure)<\/a>.<\/li>\n \t<li>Equations of conic sections with an[latex]\\,xy\\,[\/latex]term have been rotated about the origin. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Example_10_04_02\">(Figure)<\/a>.<\/li>\n \t<li>The general form can be transformed into an equation in the[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]coordinate system without the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]term. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Example_10_04_03\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Example_10_04_04\">(Figure)<\/a>.<\/li>\n \t<li>An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Example_10_04_05\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id2084607\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id2084610\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id2285327\">\n<div id=\"fs-id2285328\">\n<p id=\"fs-id2285329\">What effect does the[latex]\\,xy\\,[\/latex]term have on the graph of a conic section?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2285350\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2285350\"]\n<p id=\"fs-id2285350\">The[latex]\\,xy\\,[\/latex]term causes a rotation of the graph to occur.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2285371\">\n<div id=\"fs-id2285372\">\n<p id=\"fs-id2285373\">If the equation of a conic section is written in the form[latex]\\,A{x}^{2}+B{y}^{2}+Cx+Dy+E=0\\,[\/latex]and[latex]\\,AB=0,[\/latex] what can we conclude?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1400032\">\n<div id=\"fs-id1400033\">\n<p id=\"fs-id1400034\">If the equation of a conic section is written in the form[latex]\\,A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0,[\/latex]and[latex]\\,{B}^{2}-4AC&gt;0,[\/latex] what can we conclude?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1691756\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1691756\"]\n<p id=\"fs-id1691756\">The conic section is a hyperbola.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1691760\">\n<div id=\"fs-id1691761\">\n<p id=\"fs-id1691762\">Given the equation[latex]\\,a{x}^{2}+4x+3{y}^{2}-12=0,[\/latex] what can we conclude if[latex]\\,a&gt;0?[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1874649\">\n<div id=\"fs-id1874650\">\n<p id=\"fs-id1874651\">For the equation[latex]\\,A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0,[\/latex] the value of[latex]\\,\\theta \\,[\/latex]that satisfies[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)=\\frac{A-C}{B}\\,[\/latex]gives us what information?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2454774\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2454774\"]\n<p id=\"fs-id2454774\">It gives the angle of rotation of the axes in order to eliminate the[latex]\\,xy\\,[\/latex]term.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2100813\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id2100819\">For the following exercises, determine which conic section is represented based on the given equation.<\/p>\n\n<div id=\"fs-id2100822\">\n<div id=\"fs-id2100823\">\n<p id=\"fs-id2100824\">[latex]9{x}^{2}+4{y}^{2}+72x+36y-500=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2814657\">\n<div id=\"fs-id2814658\">\n<p id=\"fs-id2814659\">[latex]{x}^{2}-10x+4y-10=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1664771\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1664771\"]\n<p id=\"fs-id1664771\">[latex]AB=0,[\/latex] parabola<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1664792\">\n<div id=\"fs-id1664793\">\n<p id=\"fs-id1664794\">[latex]2{x}^{2}-2{y}^{2}+4x-6y-2=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2584248\">\n<div id=\"fs-id2584249\">\n<p id=\"fs-id2584250\">[latex]4{x}^{2}-{y}^{2}+8x-1=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2474747\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2474747\"]\n<p id=\"fs-id2474747\">[latex]AB=-4&lt;0,[\/latex] hyperbola<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2474775\">\n<div id=\"fs-id2474776\">\n<p id=\"fs-id2474777\">[latex]4{y}^{2}-5x+9y+1=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2927799\">\n<div id=\"fs-id2927800\">\n<p id=\"fs-id2927801\">[latex]2{x}^{2}+3{y}^{2}-8x-12y+2=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1277153\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1277153\"]\n<p id=\"fs-id1277153\">[latex]AB=6&gt;0,[\/latex] ellipse<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1277179\">\n<div id=\"fs-id1277180\">\n<p id=\"fs-id1277181\">[latex]4{x}^{2}+9xy+4{y}^{2}-36y-125=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1418850\">[latex]3{x}^{2}+6xy+3{y}^{2}-36y-125=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2433642\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2433642\"]\n<p id=\"fs-id2433642\">[latex]{B}^{2}-4AC=0,[\/latex] parabola<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2352771\">\n<div id=\"fs-id2352772\">\n<p id=\"fs-id2352774\">[latex]-3{x}^{2}+3\\sqrt{3}xy-4{y}^{2}+9=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2472798\">\n<div id=\"fs-id2472799\">\n<p id=\"fs-id2472800\">[latex]2{x}^{2}+4\\sqrt{3}xy+6{y}^{2}-6x-3=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2861569\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2861569\"]\n<p id=\"fs-id2861569\">[latex]{B}^{2}-4AC=0,[\/latex] parabola<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2155989\">\n<div id=\"fs-id2155990\">\n<p id=\"fs-id2155991\">[latex]-{x}^{2}+4\\sqrt{2}xy+2{y}^{2}-2y+1=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2161610\">\n<div id=\"fs-id2161611\">\n<p id=\"fs-id2161612\">[latex]8{x}^{2}+4\\sqrt{2}xy+4{y}^{2}-10x+1=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1940130\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1940130\"]\n<p id=\"fs-id1940130\">[latex]{B}^{2}-4AC=-96&lt;0,[\/latex] ellipse<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1940172\">For the following exercises, find a new representation of the given equation after rotating through the given angle.<\/p>\n\n<div id=\"fs-id1940176\">\n<div id=\"fs-id1940177\">\n<p id=\"fs-id1940178\">[latex]3{x}^{2}+xy+3{y}^{2}-5=0,\\theta =45\u00b0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2056389\">\n<div id=\"fs-id2056390\">\n<p id=\"fs-id2056391\">[latex]4{x}^{2}-xy+4{y}^{2}-2=0,\\theta =45\u00b0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1888348\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1888348\"]\n<p id=\"fs-id1888348\">[latex]7{{x}^{\\prime }}^{2}+9{{y}^{\\prime }}^{2}-4=0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1753645\">\n<div id=\"fs-id1753646\">\n<p id=\"fs-id1753647\">[latex]2{x}^{2}+8xy-1=0,\\theta =30\u00b0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1558064\">\n<div id=\"fs-id1558065\">\n<p id=\"fs-id1558066\">[latex]-2{x}^{2}+8xy+1=0,\\theta =45\u00b0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1799397\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1799397\"]\n<p id=\"fs-id1799397\">[latex]3{{x}^{\\prime }}^{2}+2{x}^{\\prime }{y}^{\\prime }-5{{y}^{\\prime }}^{2}+1=0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1587288\">\n<div id=\"fs-id1587289\">\n<p id=\"fs-id1587290\">[latex]4{x}^{2}+\\sqrt{2}xy+4{y}^{2}+y+2=0,\\theta =45\u00b0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2881099\">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-id1806812\">\n<div id=\"fs-id1806813\">\n<p id=\"fs-id1806814\">[latex]{x}^{2}+3\\sqrt{3}xy+4{y}^{2}+y-2=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2451648\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2451648\"]\n<p id=\"fs-id2451648\">[latex]\\theta ={60}^{\\circ },11{{x}^{\\prime }}^{2}-{{y}^{\\prime }}^{2}+\\sqrt{3}{x}^{\\prime }+{y}^{\\prime }-4=0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1663688\">\n<div id=\"fs-id1663689\">\n<p id=\"fs-id1663690\">[latex]4{x}^{2}+2\\sqrt{3}xy+6{y}^{2}+y-2=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2304092\">\n<div id=\"fs-id2304094\">\n<p id=\"fs-id2304095\">[latex]9{x}^{2}-3\\sqrt{3}xy+6{y}^{2}+4y-3=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1612806\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1612806\"]\n<p id=\"fs-id1612806\">[latex]\\theta ={150}^{\\circ },21{{x}^{\\prime }}^{2}+9{{y}^{\\prime }}^{2}+4{x}^{\\prime }-4\\sqrt{3}{y}^{\\prime }-6=0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2956037\">\n<div id=\"fs-id2956038\">\n<p id=\"fs-id2956040\">[latex]-3{x}^{2}-\\sqrt{3}xy-2{y}^{2}-x=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2691106\">\n<div id=\"fs-id2691107\">\n<p id=\"fs-id2691108\">[latex]16{x}^{2}+24xy+9{y}^{2}+6x-6y+2=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1154383\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1154383\"]\n<p id=\"fs-id1154383\">[latex]\\theta \\approx {36.9}^{\\circ },125{{x}^{\\prime }}^{2}+6{x}^{\\prime }-42{y}^{\\prime }+10=0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1263441\">\n<div id=\"fs-id1794643\">\n<p id=\"fs-id1794644\">[latex]{x}^{2}+4xy+4{y}^{2}+3x-2=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1794701\">\n<div id=\"fs-id1794702\">\n<p id=\"fs-id1794703\">[latex]{x}^{2}+4xy+{y}^{2}-2x+1=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1685135\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1685135\"]\n<p id=\"fs-id1685135\">[latex]\\theta ={45}^{\\circ },3{{x}^{\\prime }}^{2}-{{y}^{\\prime }}^{2}-\\sqrt{2}{x}^{\\prime }+\\sqrt{2}{y}^{\\prime }+1=0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1266703\">\n<div id=\"fs-id1266704\">\n<p id=\"fs-id1266705\">[latex]4{x}^{2}-2\\sqrt{3}xy+6{y}^{2}-1=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2821469\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id2821474\">For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation.<\/p>\n\n<div id=\"fs-id2821479\">\n<div id=\"fs-id2821480\">\n<p id=\"fs-id2821481\">[latex]y=-{x}^{2},\\theta =-{45}^{\\circ }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2164439\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2164439\"]\n<p id=\"fs-id2164439\">[latex]\\frac{\\sqrt{2}}{2}\\left({x}^{\\prime }+{y}^{\\prime }\\right)=\\frac{1}{2}{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}[\/latex]<\/p>\n<span id=\"fs-id2119635\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152820\/CNX_Precalc_Figure_10_04_201.jpg\" alt=\"\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2119646\">\n<div id=\"fs-id2787332\">\n<p id=\"fs-id2787334\">[latex]x={y}^{2},\\theta ={45}^{\\circ }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2787375\">\n<div id=\"fs-id2787376\">\n<p id=\"fs-id2787377\">[latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{1}=1,\\theta ={45}^{\\circ }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1264831\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1264831\"]\n<p id=\"fs-id1264831\">[latex]\\frac{{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}}{8}+\\frac{{\\left({x}^{\\prime }+{y}^{\\prime }\\right)}^{2}}{2}=1[\/latex]<\/p>\n<span id=\"fs-id1232420\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152830\/CNX_Precalc_Figure_10_04_203.jpg\" alt=\"\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1232431\">\n<div id=\"fs-id1232432\">\n<p id=\"fs-id1232433\">[latex]\\frac{{y}^{2}}{16}+\\frac{{x}^{2}}{9}=1,\\theta ={45}^{\\circ }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1403135\">\n<div id=\"fs-id1403136\">\n<p id=\"fs-id1403137\">[latex]{y}^{2}-{x}^{2}=1,\\theta ={45}^{\\circ }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2286260\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2286260\"]\n<p id=\"fs-id2286260\">[latex]\\frac{{\\left({x}^{\\prime }+{y}^{\\prime }\\right)}^{2}}{2}-\\frac{{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}}{2}=1[\/latex]<\/p>\n<span id=\"fs-id2245295\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152845\/CNX_Precalc_Figure_10_04_205.jpg\" alt=\"\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2245306\">\n<div id=\"fs-id2245307\">\n<p id=\"fs-id2245308\">[latex]y=\\frac{{x}^{2}}{2},\\theta ={30}^{\\circ }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2245359\">\n<div id=\"fs-id2245360\">\n<p id=\"fs-id2245361\">[latex]x={\\left(y-1\\right)}^{2},\\theta ={30}^{\\circ }[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2463278\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2463278\"]\n<p id=\"fs-id2463278\">[latex]\\frac{\\sqrt{3}}{2}{x}^{\\prime }-\\frac{1}{2}{y}^{\\prime }={\\left(\\frac{1}{2}{x}^{\\prime }+\\frac{\\sqrt{3}}{2}{y}^{\\prime }-1\\right)}^{2}[\/latex]<\/p>\n<span id=\"fs-id2322951\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152856\/CNX_Precalc_Figure_10_04_207.jpg\" alt=\"\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2322962\">\n<div id=\"fs-id2322963\">\n<p id=\"fs-id2322964\">[latex]\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{4}=1,\\theta ={30}^{\\circ }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2199206\">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-id1999288\">\n<div id=\"fs-id1999289\">\n<p id=\"fs-id1999290\">[latex]xy=9[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"1999308\"]Show Solution[\/reveal-answer][hidden-answer a=\"1999308\"]<span id=\"fs-id1999314\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152901\/CNX_Precalc_Figure_10_04_209.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1999324\">\n<div id=\"fs-id1999325\">\n<p id=\"fs-id1999326\">[latex]{x}^{2}+10xy+{y}^{2}-6=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1673790\">\n<div id=\"fs-id1673791\">\n<p id=\"fs-id1673792\">[latex]{x}^{2}-10xy+{y}^{2}-24=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"2061208\"]Show Solution[\/reveal-answer][hidden-answer a=\"2061208\"]<span id=\"fs-id2061214\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152903\/CNX_Precalc_Figure_10_04_211.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id2061224\">\n<div id=\"fs-id2061225\">\n<p id=\"fs-id2061226\">[latex]4{x}^{2}-3\\sqrt{3}xy+{y}^{2}-22=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2757196\">\n<div id=\"fs-id2757197\">\n<p id=\"fs-id2757198\">[latex]6{x}^{2}+2\\sqrt{3}xy+4{y}^{2}-21=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"2757257\"]Show Solution[\/reveal-answer][hidden-answer a=\"2757257\"]<span id=\"fs-id2757262\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152910\/CNX_Precalc_Figure_10_04_213.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1617309\">\n<div id=\"fs-id1617310\">\n<p id=\"fs-id1617311\">[latex]11{x}^{2}+10\\sqrt{3}xy+{y}^{2}-64=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1617367\">\n<div id=\"fs-id1617368\">\n<p id=\"fs-id1617370\">[latex]21{x}^{2}+2\\sqrt{3}xy+19{y}^{2}-18=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"2020873\"]Show Solution[\/reveal-answer][hidden-answer a=\"2020873\"]<span id=\"fs-id2020878\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152918\/CNX_Precalc_Figure_10_04_215.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id2020889\">\n<div id=\"fs-id2020890\">\n<p id=\"fs-id2020891\">[latex]16{x}^{2}+24xy+9{y}^{2}-130x+90y=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2289143\">\n<div id=\"fs-id2289144\">\n<p id=\"fs-id2289146\">[latex]16{x}^{2}+24xy+9{y}^{2}-60x+80y=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"2106826\"]Show Solution[\/reveal-answer][hidden-answer a=\"2106826\"]<span id=\"fs-id2106831\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152927\/CNX_Precalc_Figure_10_04_217.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id2106842\">\n<div id=\"fs-id2106843\">\n<p id=\"fs-id2106844\">[latex]13{x}^{2}-6\\sqrt{3}xy+7{y}^{2}-16=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1492652\">\n<div id=\"fs-id1492653\">\n<p id=\"fs-id1492654\">[latex]4{x}^{2}-4xy+{y}^{2}-8\\sqrt{5}x-16\\sqrt{5}y=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"2430876\"]Show Solution[\/reveal-answer][hidden-answer a=\"2430876\"]<span id=\"fs-id2430882\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152935\/CNX_Precalc_Figure_10_04_219.jpg\" alt=\"\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<p id=\"fs-id2430892\">For the following exercises, determine the angle of rotation in order to eliminate the[latex]\\,xy\\,[\/latex]term. Then graph the new set of axes.<\/p>\n\n<div id=\"fs-id2430912\">\n<div id=\"fs-id2430913\">\n<p id=\"fs-id2430914\">[latex]6{x}^{2}-5\\sqrt{3}xy+{y}^{2}+10x-12y=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2261896\">\n<div id=\"fs-id2261897\">\n<p id=\"fs-id2261898\">[latex]6{x}^{2}-5xy+6{y}^{2}+20x-y=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1722813\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1722813\"]\n<p id=\"fs-id1722813\">[latex]\\theta ={45}^{\\circ }[\/latex]<\/p>\n<span id=\"fs-id1722842\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152947\/CNX_Precalc_Figure_10_04_221.jpg\" alt=\"\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1722853\">\n<div id=\"fs-id1722854\">\n<p id=\"fs-id1722855\">[latex]6{x}^{2}-8\\sqrt{3}xy+14{y}^{2}+10x-3y=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2613786\">\n<div id=\"fs-id2613787\">\n<p id=\"fs-id2613788\">[latex]4{x}^{2}+6\\sqrt{3}xy+10{y}^{2}+20x-40y=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2320678\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2320678\"]\n<p id=\"fs-id2320678\">[latex]\\theta ={60}^{\\circ }[\/latex]<\/p>\n<span id=\"fs-id2320706\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152953\/CNX_Precalc_Figure_10_04_223.jpg\" alt=\"\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2320717\">\n<div id=\"fs-id2320718\">\n<p id=\"fs-id1244573\">[latex]8{x}^{2}+3xy+4{y}^{2}+2x-4=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1244632\">\n<div id=\"fs-id1244633\">\n<p id=\"fs-id1244634\">[latex]16{x}^{2}+24xy+9{y}^{2}+20x-44y=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1651852\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1651852\"]\n<p id=\"fs-id1651852\">[latex]\\theta \\approx {36.9}^{\\circ }[\/latex]<\/p>\n<span id=\"fs-id1651881\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152955\/CNX_Precalc_Figure_10_04_225.jpg\" alt=\"\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1651892\">For the following exercises, determine the value of[latex]\\,k\\,[\/latex]based on the given equation.<\/p>\n\n<div id=\"fs-id1550256\">\n<div id=\"fs-id1550257\">\n<p id=\"fs-id1550258\">Given[latex]\\,4{x}^{2}+kxy+16{y}^{2}+8x+24y-48=0,[\/latex] find[latex]\\,k\\,[\/latex]for the graph to be a parabola.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2601769\">\n<div id=\"fs-id2601770\">\n<p id=\"fs-id2601771\">Given[latex]\\,2{x}^{2}+kxy+12{y}^{2}+10x-16y+28=0,[\/latex] find[latex]\\,k\\,[\/latex]for the graph to be an ellipse.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1233480\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1233480\"]\n<p id=\"fs-id1233480\">[latex]-4\\sqrt{6}&lt;k&lt;4\\sqrt{6}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1233515\">\n<div id=\"fs-id1233516\">\n<p id=\"fs-id1233517\">Given[latex]\\,3{x}^{2}+kxy+4{y}^{2}-6x+20y+128=0,[\/latex] find[latex]\\,k\\,[\/latex]for the graph to be a hyperbola.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1088231\">\n<div id=\"fs-id1088232\">\n<p id=\"fs-id1088234\">Given[latex]\\,k{x}^{2}+8xy+8{y}^{2}-12x+16y+18=0,[\/latex] find[latex]\\,k\\,[\/latex]for the graph to be a parabola.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1924178\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1924178\"]\n<p id=\"fs-id1924178\">[latex]k=2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1924196\">\n<div id=\"fs-id1924197\">\n<p id=\"fs-id1924198\">Given[latex]\\,6{x}^{2}+12xy+k{y}^{2}+16x+10y+4=0,[\/latex] find[latex]\\,k\\,[\/latex]for the graph to be an ellipse.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id2107200\">\n \t<dt>angle of rotation<\/dt>\n \t<dd id=\"fs-id2118805\">an acute angle formed by a set of axes rotated from the Cartesian plane where, if[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)&gt;0,[\/latex]then[latex]\\,\\theta \\,[\/latex]is between[latex]\\,\\left(0\u00b0,45\u00b0\\right);[\/latex]if[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)&lt;0,[\/latex]then[latex]\\,\\theta \\,[\/latex]is between[latex]\\,\\left(45\u00b0,90\u00b0\\right);\\,[\/latex]and if[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)=0,[\/latex]then[latex]\\,\\theta =45\u00b0[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id2165541\">\n \t<dt>degenerate conic sections<\/dt>\n \t<dd id=\"fs-id2165546\">any of the possible shapes formed when a plane intersects a double cone through the apex. Types of degenerate conic sections include a point, a line, and intersecting lines.<\/dd>\n<\/dl>\n<dl id=\"fs-id1840460\">\n \t<dt>nondegenerate conic section<\/dt>\n \t<dd id=\"fs-id1840465\">a shape formed by the intersection of a plane with a double right cone such that the plane does not pass through the apex; nondegenerate conics include circles, ellipses, hyperbolas, and parabolas<\/dd>\n<\/dl>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>arning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Identify nondegenerate conic sections given their general form equations.<\/li>\n<li>Use rotation of axes formulas.<\/li>\n<li>Write equations of rotated conics in standard form.<\/li>\n<li>Identify conics without rotating axes.<\/li>\n<\/ul>\n<\/div>\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\/19152642\/CNX_Precalc_Figure_10_04_001.jpg\" alt=\"\" width=\"975\" height=\"650\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1. <\/strong>The nondegenerate conic sections<\/figcaption><\/figure>\n<p id=\"fs-id1141495\">As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a <em>cone<\/em>. The way in which we slice the cone will determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane perpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not perpendicular to the axis of symmetry. A parabola is formed by slicing the plane through the top or bottom of the double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone. See <a class=\"autogenerated-content\" href=\"#Figure_10_04_001\">(Figure)<\/a>.<span id=\"fs-id1471958\"><\/span><\/p>\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\/19152645\/CNX_Precalc_Figure_10_04_002n.jpg\" alt=\"\" width=\"975\" height=\"719\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2. <\/strong>Degenerate conic sections<\/figcaption><\/figure>\n<p id=\"fs-id2467642\">Ellipses, circles, hyperbolas, and parabolas are sometimes called the nondegenerate conic sections, in contrast to the degenerate conic sections, which are shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_002\">(Figure)<\/a>. A degenerate conic results when a plane intersects the double cone and passes through the apex. Depending on the angle of the plane, three types of degenerate conic sections are possible: a point, a line, or two intersecting lines.<span id=\"fs-id2295508\"><\/span><\/p>\n<div id=\"fs-id1741499\" class=\"bc-section section\">\n<h3>Identifying Nondegenerate Conics in General Form<\/h3>\n<p id=\"fs-id2726428\">In previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections. In this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below.<\/p>\n<div id=\"fs-id1264739\" class=\"unnumbered aligncenter\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\n<p id=\"fs-id1306881\">where[latex]\\,A,B,[\/latex] and [latex]\\,C\\,[\/latex]are not all zero. We can use the values of the coefficients to identify which type conic is represented by a given equation.<\/p>\n<p id=\"fs-id1675947\">You may notice that the general form equation has an[latex]\\,xy\\,[\/latex]term that we have not seen in any of the standard form equations. As we will discuss later, the[latex]\\,xy\\,[\/latex]term rotates the conic whenever[latex]\\text{ }B\\text{ }[\/latex]is not equal to zero.<\/p>\n<table id=\"Table_10_04_01\" summary=\"..\">\n<thead>\n<tr>\n<th><strong>Conic Sections<\/strong><\/th>\n<th><strong>Example<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>ellipse<\/td>\n<td>[latex]4{x}^{2}+9{y}^{2}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>circle<\/td>\n<td>[latex]4{x}^{2}+4{y}^{2}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>hyperbola<\/td>\n<td>[latex]4{x}^{2}-9{y}^{2}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>parabola<\/td>\n<td>[latex]4{x}^{2}=9y\\text{ or }4{y}^{2}=9x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>one line<\/td>\n<td>[latex]4x+9y=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>intersecting lines<\/td>\n<td>[latex]\\left(x-4\\right)\\left(y+4\\right)=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>parallel lines<\/td>\n<td>[latex]\\left(x-4\\right)\\left(x-9\\right)=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>a point<\/td>\n<td>[latex]4{x}^{2}+4{y}^{2}=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>no graph<\/td>\n<td>[latex]4{x}^{2}+4{y}^{2}=\\,-\\,1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1532180\" class=\"textbox key-takeaways\">\n<h3>General Form of Conic Sections<\/h3>\n<p id=\"fs-id1784761\">A conic section has the general form<\/p>\n<div>[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\n<p id=\"fs-id1264927\">where[latex]\\,A,B,[\/latex] and[latex]\\,C\\,[\/latex]are not all zero.<\/p>\n<p id=\"fs-id1400290\"><a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Table_10_04_02\">(Figure)<\/a> summarizes the different conic sections where[latex]\\,B=0,[\/latex] and[latex]\\,A\\,[\/latex]and[latex]\\,C\\,[\/latex]are nonzero real numbers. This indicates that the conic has not been rotated.<\/p>\n<table id=\"Table_10_04_02\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>ellipse<\/strong><\/td>\n<td>[latex]A{x}^{2}+C{y}^{2}+Dx+Ey+F=0,\\text{ }A\\ne C\\text{ and }AC>0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>circle<\/strong><\/td>\n<td>[latex]A{x}^{2}+C{y}^{2}+Dx+Ey+F=0,\\text{ }A=C[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>hyperbola<\/strong><\/td>\n<td>[latex]A{x}^{2}-C{y}^{2}+Dx+Ey+F=0\\text{ or }-A{x}^{2}+C{y}^{2}+Dx+Ey+F=0,[\/latex]where[latex]\\,A\\,[\/latex]and[latex]\\,C\\,[\/latex]are positive<\/td>\n<\/tr>\n<tr>\n<td><strong>parabola<\/strong><\/td>\n<td>[latex]A{x}^{2}+Dx+Ey+F=0\\text{ or }C{y}^{2}+Dx+Ey+F=0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1200060\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1128527\"><strong>Given the equation of a conic, identify the type of conic.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1121724\" type=\"1\">\n<li>Rewrite the equation in the general form, [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0.[\/latex]<\/li>\n<li>Identify the values of[latex]\\,A\\,[\/latex]and[latex]\\,C\\,[\/latex]from the general form.\n<ol id=\"fs-id2773969\" type=\"a\">\n<li>If[latex]\\,A\\,[\/latex]and[latex]\\,C\\,[\/latex] are nonzero, have the same sign, and are not equal to each other, then the graph may be an ellipse.<\/li>\n<li>If[latex]\\,A\\,[\/latex]and[latex]\\,C\\,[\/latex]are equal and nonzero and have the same sign, then the graph may be a circle.<\/li>\n<li>If[latex]\\,A\\,[\/latex]and[latex]\\,C\\,[\/latex]are nonzero and have opposite signs, then the graph may be a hyperbola.<\/li>\n<li>If either[latex]\\,A\\,[\/latex]or[latex]\\,C\\,[\/latex] is zero, then the graph may be a parabola.<\/li>\n<\/ol>\n<p>If <em>B<\/em> = 0, the conic section will have a vertical and\/or horizontal axes. If <em>B<\/em> does not equal 0, as shown below, the conic section is rotated.&nbsp; Notice the phrase \u201cmay be\u201d in the definitions. That is because the equation may not represent a conic section at all, depending on the values of <em>A<\/em>, <em>B<\/em>, <em>C<\/em>, <em>D<\/em>, <em>E<\/em>, and <em>F<\/em>. For example, the degenerate case of a circle or an ellipse is a point:<\/p>\n<p>[latex]A{x}^{2}+B{y}^{2}=0\\text{,}[\/latex]when <em>A<\/em> and <em>B<\/em> have the same sign.<\/p>\n<p>The degenerate case of a hyperbola is two intersecting straight lines:[latex]A{x}^{2}+B{y}^{2}=0\\text{,}[\/latex]when A and B have opposite signs.<\/p>\n<p>On the other hand, the equation, [latex]A{x}^{2}+B{y}^{2}+1=0\\text{,}[\/latex] when A and B are positive does not represent a graph at all, since there are no real ordered pairs which satisfy it.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_10_04_01\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1926798\">\n<h3>Identifying a Conic from Its General Form<\/h3>\n<p id=\"fs-id1963747\">Identify the graph of each of the following nondegenerate conic sections.<\/p>\n<ol id=\"fs-id1113018\" type=\"a\">\n<li>[latex]4{x}^{2}-9{y}^{2}+36x+36y-125=0[\/latex]<\/li>\n<li>[latex]9{y}^{2}+16x+36y-10=0[\/latex]<\/li>\n<li>[latex]3{x}^{2}+3{y}^{2}-2x-6y-4=0[\/latex]<\/li>\n<li>[latex]-25{x}^{2}-4{y}^{2}+100x+16y+20=0[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1582186\" type=\"a\">\n<li>Rewriting the general form, we have<span id=\"eq1\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152651\/eq1_n.jpg\" alt=\"\" \/><\/span>\n<p id=\"fs-id2570988\">[latex]A=4\\,[\/latex]and[latex]\\,C=-9,[\/latex] so we observe that[latex]\\,A\\,[\/latex]and[latex]\\,C\\,[\/latex]have opposite signs. The graph of this equation is a hyperbola.<\/p>\n<\/li>\n<li>Rewriting the general form, we have<br \/>\n<span id=\"eq2\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152657\/eq2_n.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1459350\">[latex]A=0\\,[\/latex]and[latex]\\,C=9.\\,[\/latex]We can determine that the equation is a parabola, since[latex]\\,A\\,[\/latex]is zero.<\/p>\n<\/li>\n<li>Rewriting the general form, we have <span id=\"eq3\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152715\/eq3_n.jpg\" alt=\"\" \/><\/span>\n<p id=\"fs-id1154673\">[latex]A=3\\,[\/latex]and[latex]\\,C=3.\\,[\/latex]Because[latex]\\,A=C,[\/latex] the graph of this equation is a circle.<\/p>\n<\/li>\n<li>Rewriting the general form, we have<br \/>\n<span id=\"eq4\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152734\/eq4.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id2587849\">[latex]A=-25\\,[\/latex]and[latex]\\,C=-4.\\,[\/latex]Because[latex]\\,AC>0\\,[\/latex]and[latex]\\,A\\ne C,[\/latex] the graph of this equation is an ellipse.<\/details>\n<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2814436\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1295024\">\n<p id=\"fs-id1342273\">Identify the graph of each of the following nondegenerate conic sections.<\/p>\n<ol id=\"fs-id1081247\" type=\"a\">\n<li>[latex]16{y}^{2}-{x}^{2}+x-4y-9=0[\/latex]<\/li>\n<li>[latex]16{x}^{2}+4{y}^{2}+16x+49y-81=0[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id2335243\" type=\"a\">\n<li>hyperbola<\/li>\n<li>ellipse<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1319873\" class=\"bc-section section\">\n<h4>Finding a New Representation of the Given Equation after Rotating through a Given Angle<\/h4>\n<p id=\"fs-id1378740\">Until now, we have looked at equations of conic sections without an[latex]\\,xy\\,[\/latex]term, which aligns the graphs with the <em>x<\/em>&#8211; and <em>y<\/em>-axes. When we add an[latex]\\,xy\\,[\/latex]term, we are rotating the conic about the origin. If the <em>x<\/em>&#8211; and <em>y<\/em>-axes are rotated through an angle, say[latex]\\,\\theta ,[\/latex]then every point on the plane may be thought of as having two representations:[latex]\\,\\left(x,y\\right)\\,[\/latex]on the Cartesian plane with the original <em>x<\/em>-axis and <em>y<\/em>-axis, and[latex]\\,\\left({x}^{\\prime },{y}^{\\prime }\\right)\\,[\/latex]on the new plane defined by the new, rotated axes, called the <em>x&#8217;<\/em>-axis and <em>y&#8217;<\/em>-axis. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_003\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_10_04_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\/19152737\/CNX_Precalc_Figure_10_04_003.jpg\" alt=\"\" width=\"487\" height=\"441\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3. <\/strong>The graph of the rotated ellipse[latex]\\,{x}^{2}+{y}^{2}\u2013xy\u201315=0[\/latex]<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1352584\">We will find the relationships between[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]on the Cartesian plane with[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]on the new rotated plane. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_004\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_10_04_004\" 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\/19152741\/CNX_Precalc_Figure_10_04_004.jpg\" alt=\"\" width=\"487\" height=\"366\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4. <\/strong>The Cartesian plane with x- and y-axes and the resulting x\u2032\u2212 and y\u2032\u2212axes formed by a rotation by an angle[latex]\\text{ }\\theta .[\/latex]<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1333360\">The original coordinate <em>x<\/em>&#8211; and <em>y<\/em>-axes have unit vectors[latex]\\,i\\,[\/latex]and[latex]\\,j\\,.[\/latex]The rotated coordinate axes have unit vectors[latex]\\,{i}^{\\prime }\\,[\/latex]and[latex]\\,{j}^{\\prime }.[\/latex]The angle[latex]\\,\\theta \\,[\/latex]is known as the angle of rotation. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_005\">(Figure)<\/a>. We may write the new unit vectors in terms of the original ones.<\/p>\n<div id=\"fs-id2753493\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{i}^{\\prime }=\\mathrm{cos}\\text{ }\\theta i+\\mathrm{sin}\\text{ }\\theta j\\hfill \\\\ {j}^{\\prime }=-\\mathrm{sin}\\text{ }\\theta i+\\mathrm{cos}\\text{ }\\theta j\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"Figure_10_04_005\" 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\/19152747\/CNX_Precalc_Figure_10_04_005.jpg\" alt=\"\" width=\"487\" height=\"364\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5. <\/strong>Relationship between the old and new coordinate planes.<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id2129430\">Consider a vector<strong>[latex]\\,u\\,[\/latex]<\/strong>in the new coordinate plane. It may be represented in terms of its coordinate axes.<\/p>\n<div id=\"fs-id1260569\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime }\\hfill & \\hfill \\\\ u={x}^{\\prime }\\left(i\\text{ }\\mathrm{cos}\\text{ }\\theta +j\\text{ }\\mathrm{sin}\\text{ }\\theta \\right)+{y}^{\\prime }\\left(-i\\text{ }\\mathrm{sin}\\text{ }\\theta +j\\text{ }\\mathrm{cos}\\text{ }\\theta \\right)\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Substitute}.\\hfill \\\\ u=ix\\text{'}\\text{ }\\mathrm{cos}\\text{ }\\theta +jx\\text{'}\\text{ }\\mathrm{sin}\\text{ }\\theta -iy\\text{'}\\text{ }\\mathrm{sin}\\text{ }\\theta +jy\\text{'}\\text{ }\\mathrm{cos}\\text{ }\\theta \\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Distribute}.\\hfill \\\\ u=ix\\text{'}\\text{ }\\mathrm{cos}\\text{ }\\theta -iy\\text{'}\\text{ }\\mathrm{sin}\\text{ }\\theta +jx\\text{'}\\text{ }\\mathrm{sin}\\text{ }\\theta +jy\\text{'}\\text{ }\\mathrm{cos}\\text{ }\\theta \\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Apply commutative property}.\\hfill \\\\ u=\\left(x\\text{'}\\text{ }\\mathrm{cos}\\text{ }\\theta -y\\text{'}\\text{ }\\mathrm{sin}\\text{ }\\theta \\right)i+\\left(x\\text{'}\\text{ }\\mathrm{sin}\\text{ }\\theta +y\\text{'}\\text{ }\\mathrm{cos}\\text{ }\\theta \\right)j\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Factor by grouping}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1319681\">Because[latex]\\,u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime },[\/latex] we have representations of[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]in terms of the new coordinate system.<\/p>\n<div id=\"fs-id1339818\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{c}x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\\\ \\text{and}\\\\ y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1741266\" class=\"textbox key-takeaways\">\n<h3>Equations of Rotation<\/h3>\n<p id=\"fs-id1385513\">If a point[latex]\\,\\left(x,y\\right)\\,[\/latex]on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle[latex]\\,\\theta \\,[\/latex]from the positive <em>x<\/em>-axis, then the coordinates of the point with respect to the new axes are[latex]\\,\\left({x}^{\\prime },{y}^{\\prime }\\right).\\,[\/latex]We can use the following equations of rotation to define the relationship between[latex]\\,\\left(x,y\\right)\\,[\/latex]and[latex]\\,\\left({x}^{\\prime },{y}^{\\prime }\\right):[\/latex]<\/p>\n<div id=\"Equation_10_04_02\">[latex]x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta[\/latex]<\/div>\n<p id=\"fs-id2281028\">and<\/p>\n<div id=\"Equation_10_04_03\">[latex]y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1355297\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1560707\"><strong>Given the equation of a conic, find a new representation after rotating through an angle.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1146233\" type=\"1\">\n<li>Find[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]where[latex]\\,x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta .[\/latex]<\/li>\n<li>Substitute the expression for[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]into in the given equation, then simplify.<\/li>\n<li>Write the equations with[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]in standard form.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_10_04_02\" class=\"textbox examples\">\n<div id=\"fs-id1357778\">\n<div id=\"fs-id2354314\">\n<h3>Finding a New Representation of an Equation after Rotating through a Given Angle<\/h3>\n<p id=\"fs-id1541593\">Find a new representation of the equation[latex]\\,2{x}^{2}-xy+2{y}^{2}-30=0\\,[\/latex]after rotating through an angle of[latex]\\,\\theta =45\u00b0.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1503906\">Find[latex]\\,x\\,[\/latex]and[latex]\\,y,[\/latex]where[latex]\\,x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta .[\/latex]<\/p>\n<p id=\"fs-id1290570\">Because[latex]\\,\\theta =45\u00b0,[\/latex]<\/p>\n<div id=\"fs-id1989238\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ x={x}^{\\prime }\\mathrm{cos}\\left(45\u00b0\\right)-{y}^{\\prime }\\mathrm{sin}\\left(45\u00b0\\right)\\hfill \\\\ x={x}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)-{y}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)\\hfill \\\\ x=\\frac{{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{2}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2113047\">and<\/p>\n<div id=\"fs-id2091380\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\\\ \\begin{array}{l}y={x}^{\\prime }\\mathrm{sin}\\left(45\u00b0\\right)+{y}^{\\prime }\\mathrm{cos}\\left(45\u00b0\\right)\\hfill \\\\ y={x}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)+{y}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)\\hfill \\\\ y=\\frac{{x}^{\\prime }+{y}^{\\prime }}{\\sqrt{2}}\\hfill \\end{array}\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1674091\">Substitute[latex]\\,x={x}^{\\prime }\\mathrm{cos}\\theta -{y}^{\\prime }\\mathrm{sin}\\theta \\,[\/latex]and[latex]\\,y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta \\,[\/latex]into[latex]\\,2{x}^{2}-xy+2{y}^{2}-30=0.[\/latex]<\/p>\n<div id=\"fs-id1375279\" class=\"unnumbered aligncenter\">[latex]2{\\left(\\frac{{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{2}}\\right)}^{2}-\\left(\\frac{{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{2}}\\right)\\left(\\frac{{x}^{\\prime }+{y}^{\\prime }}{\\sqrt{2}}\\right)+2{\\left(\\frac{{x}^{\\prime }+{y}^{\\prime }}{\\sqrt{2}}\\right)}^{2}-30=0[\/latex]<\/div>\n<p id=\"fs-id1901790\">Simplify.<\/p>\n<div id=\"fs-id1901793\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\overline{)2}\\frac{\\left({x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }-{y}^{\\prime }\\right)}{\\overline{)2}}-\\frac{\\left({x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }+{y}^{\\prime }\\right)}{2}+\\overline{)2}\\frac{\\left({x}^{\\prime }+{y}^{\\prime }\\right)\\left({x}^{\\prime }+{y}^{\\prime }\\right)}{\\overline{)2}}-30=0\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{FOIL method}\\hfill \\\\ \\text{ }{x}^{\\prime }{}^{2}{\\overline{)-2{x}^{\\prime }y}}^{\\prime }+{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}+{x}^{\\prime }{}^{2}\\overline{)+2{x}^{\\prime }{y}^{\\prime }}+{y}^{\\prime }{}^{2}-30=0\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Combine like terms}.\\hfill \\\\ \\text{ }2{x}^{\\prime }{}^{2}+2{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}=30\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Combine like terms}.\\hfill \\\\ \\text{ }2\\left(2{x}^{\\prime }{}^{2}+2{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}\\right)=2\\left(30\\right)\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Multiply both sides by 2}.\\hfill \\\\ \\text{ }4{x}^{\\prime }{}^{2}+4{y}^{\\prime }{}^{2}-\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)=60\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Simplify}.\\hfill \\\\ \\text{ }4{x}^{\\prime }{}^{2}+4{y}^{\\prime }{}^{2}-{x}^{\\prime }{}^{2}+{y}^{\\prime }{}^{2}=60\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Distribute}.\\hfill \\\\ \\text{ }\\frac{3{x}^{\\prime }{}^{2}}{60}+\\frac{5{y}^{\\prime }{}^{2}}{60}=\\frac{60}{60}\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Set equal to 1}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2230049\">Write the equations with[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]in the standard form.<\/p>\n<div id=\"fs-id2465469\" class=\"unnumbered aligncenter\">[latex]\\frac{{{x}^{\\prime }}^{2}}{20}+\\frac{{{y}^{\\prime }}^{2}}{12}=1[\/latex]<\/div>\n<p>This equation is an ellipse. <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_006\">(Figure)<\/a> shows the graph.<\/p>\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\/19152753\/CNX_Precalc_Figure_10_04_006.jpg\" alt=\"\" width=\"487\" height=\"441\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1923420\"><\/details>\n<p><span id=\"fs-id1194819\"><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1194834\" class=\"bc-section section\">\n<h3>Writing Equations of Rotated Conics in Standard Form<\/h3>\n<p id=\"fs-id1194839\">Now that we can find the standard form of a conic when we are given an angle of rotation, we will learn how to transform the equation of a conic given in the form[latex]\\,A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0\\,[\/latex]into standard form by rotating the axes. To do so, we will rewrite the general form as an equation in the[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]coordinate system without the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]term, by rotating the axes by a measure of[latex]\\,\\theta \\,[\/latex]that satisfies<\/p>\n<div id=\"Equation_10_04_04\">[latex]\\mathrm{cot}\\left(2\\theta \\right)=\\frac{A-C}{B}[\/latex]<\/div>\n<p id=\"fs-id1186178\">We have learned already that any conic may be represented by the second degree equation<\/p>\n<div id=\"fs-id1186181\" class=\"unnumbered aligncenter\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\n<p id=\"fs-id2702021\">where[latex]\\,A,B,[\/latex]and[latex]\\,C\\,[\/latex]are not all zero. However, if[latex]\\,B\\ne 0,[\/latex] then we have an[latex]\\,xy\\,[\/latex]term that prevents us from rewriting the equation in standard form. To eliminate it, we can rotate the axes by an acute angle[latex]\\,\\theta \\,[\/latex]where[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)=\\frac{A-C}{B}.[\/latex]<\/p>\n<ul id=\"fs-id2132367\">\n<li>If[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)>0,[\/latex] then[latex]\\,2\\theta \\,[\/latex] is in the first quadrant, and[latex]\\,\\theta \\,[\/latex] is between[latex]\\,\\left(0\u00b0,45\u00b0\\right).[\/latex]<\/li>\n<li>If[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)<0,[\/latex] then[latex]\\,2\\theta \\,[\/latex] is in the second quadrant, and[latex]\\,\\theta \\,[\/latex] is between[latex]\\,\\left(45\u00b0,90\u00b0\\right).[\/latex]<\/li>\n<li>If[latex]\\,A=C,[\/latex] then[latex]\\,\\theta =45\u00b0.[\/latex]<\/li>\n<\/ul>\n<div id=\"fs-id1503870\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1503876\"><strong>Given an equation for a conic in the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]system, rewrite the equation without the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]term in terms of[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime },[\/latex]where the[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]axes are rotations of the standard axes by[latex]\\,\\theta \\,[\/latex]degrees.<\/strong><\/p>\n<ol id=\"fs-id2274393\" type=\"1\">\n<li>Find[latex]\\,\\mathrm{cot}\\left(2\\theta \\right).[\/latex]<\/li>\n<li>Find[latex]\\,\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,\\mathrm{cos}\\text{ }\\theta .[\/latex]<\/li>\n<li>Substitute[latex]\\,\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,\\mathrm{cos}\\text{ }\\theta \\,[\/latex]into[latex]\\,x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta .[\/latex]<\/li>\n<li>Substitute the expression for[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]into in the given equation, and then simplify.<\/li>\n<li>Write the equations with[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]in the standard form with respect to the rotated axes.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_10_04_03\" class=\"textbox examples\">\n<div id=\"fs-id1414093\">\n<div id=\"fs-id1414095\">\n<h3>Rewriting an Equation with respect to the <em>x\u2032<\/em> and <em>y\u2032<\/em> axes without the <em>x\u2032y\u2032<\/em> Term<\/h3>\n<p id=\"fs-id1259372\">Rewrite the equation[latex]\\,8{x}^{2}-12xy+17{y}^{2}=20\\,[\/latex]in the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]system without an[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]term.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1741772\">First, we find[latex]\\,\\mathrm{cot}\\left(2\\theta \\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_007\">(Figure)<\/a>.<\/p>\n<div id=\"fs-id1403863\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}8{x}^{2}-12xy+17{y}^{2}=20\u21d2A=8,\\,B=-12\\,\\text{and}\\,C=17\\hfill \\\\ \\text{ }\\,\\,\\,\\mathrm{cot}\\left(2\\theta \\right)=\\frac{A-C}{B}=\\frac{8-17}{-12}\\hfill \\\\ \\text{ }\\,\\,\\,\\mathrm{cot}\\left(2\\theta \\right)=\\frac{-9}{-12}=\\frac{3}{4}\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"Figure_10_04_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\/19152755\/CNX_Precalc_Figure_10_04_007.jpg\" alt=\"\" width=\"487\" height=\"328\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 7.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1901664\" class=\"unnumbered aligncenter\">[latex]\\mathrm{cot}\\left(2\\theta \\right)=\\frac{3}{4}=\\frac{\\text{adjacent}}{\\text{opposite}}[\/latex]<\/div>\n<p id=\"fs-id2206549\">So the hypotenuse is<\/p>\n<div id=\"fs-id2206552\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{r}\\hfill {3}^{2}+{4}^{2}={h}^{2}\\\\ \\hfill 9+16={h}^{2}\\\\ \\hfill 25={h}^{2}\\\\ \\hfill h=5\\,\\,\\,\\end{array}[\/latex]<\/div>\n<p id=\"fs-id2175665\">Next, we find[latex]\\,\\mathrm{sin}\\text{ }\\theta[\/latex] and [latex]\\mathrm{cos}\\text{ }\\theta .[\/latex]<\/p>\n<div id=\"fs-id1316548\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ \\mathrm{sin}\\text{ }\\theta =\\sqrt{\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}}=\\sqrt{\\frac{1-\\frac{3}{5}}{2}}=\\sqrt{\\frac{\\frac{5}{5}-\\frac{3}{5}}{2}}=\\sqrt{\\frac{5-3}{5}\\cdot \\frac{1}{2}}=\\sqrt{\\frac{2}{10}}=\\sqrt{\\frac{1}{5}}\\hfill \\end{array}\\hfill \\\\ \\mathrm{sin}\\text{ }\\theta =\\frac{1}{\\sqrt{5}}\\hfill \\\\ \\mathrm{cos}\\text{ }\\theta =\\sqrt{\\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}}=\\sqrt{\\frac{1+\\frac{3}{5}}{2}}=\\sqrt{\\frac{\\frac{5}{5}+\\frac{3}{5}}{2}}=\\sqrt{\\frac{5+3}{5}\\cdot \\frac{1}{2}}=\\sqrt{\\frac{8}{10}}=\\sqrt{\\frac{4}{5}}\\hfill \\\\ \\mathrm{cos}\\text{ }\\theta =\\frac{2}{\\sqrt{5}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1888188\">Substitute the values of[latex]\\,\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,\\mathrm{cos}\\text{ }\\theta \\,[\/latex]into[latex]\\,x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta .[\/latex]<\/p>\n<div id=\"fs-id1587444\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\begin{array}{l}x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\hfill \\\\ x={x}^{\\prime }\\left(\\frac{2}{\\sqrt{5}}\\right)-{y}^{\\prime }\\left(\\frac{1}{\\sqrt{5}}\\right)\\hfill \\\\ x=\\frac{2{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{5}}\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2691278\">and<\/p>\n<div id=\"fs-id2691281\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta \\hfill \\end{array}\\hfill \\\\ y={x}^{\\prime }\\left(\\frac{1}{\\sqrt{5}}\\right)+{y}^{\\prime }\\left(\\frac{2}{\\sqrt{5}}\\right)\\hfill \\\\ y=\\frac{{x}^{\\prime }+2{y}^{\\prime }}{\\sqrt{5}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2290144\">Substitute the expressions for[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]into in the given equation, and then simplify.<\/p>\n<div id=\"fs-id2030209\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }8{\\left(\\frac{2{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{5}}\\right)}^{2}-12\\left(\\frac{2{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{5}}\\right)\\left(\\frac{{x}^{\\prime }+2{y}^{\\prime }}{\\sqrt{5}}\\right)+17{\\left(\\frac{{x}^{\\prime }+2{y}^{\\prime }}{\\sqrt{5}}\\right)}^{2}=20\\text{ }\\hfill \\\\ \\text{ }8\\left(\\frac{\\left(2{x}^{\\prime }-{y}^{\\prime }\\right)\\left(2{x}^{\\prime }-{y}^{\\prime }\\right)}{5}\\right)-12\\left(\\frac{\\left(2{x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }+2{y}^{\\prime }\\right)}{5}\\right)+17\\left(\\frac{\\left({x}^{\\prime }+2{y}^{\\prime }\\right)\\left({x}^{\\prime }+2{y}^{\\prime }\\right)}{5}\\right)=20\\text{ }\\hfill \\\\ \\text{ }8\\left(4{x}^{\\prime }{}^{2}-4{x}^{\\prime }{y}^{\\prime }+{y}^{\\prime }{}^{2}\\right)-12\\left(2{x}^{\\prime }{}^{2}+3{x}^{\\prime }{y}^{\\prime }-2{y}^{\\prime }{}^{2}\\right)+17\\left({x}^{\\prime }{}^{2}+4{x}^{\\prime }{y}^{\\prime }+4{y}^{\\prime }{}^{2}\\right)=100\\hfill \\\\ 32{x}^{\\prime }{}^{2}-32{x}^{\\prime }{y}^{\\prime }+8{y}^{\\prime }{}^{2}-24{x}^{\\prime }{}^{2}-36{x}^{\\prime }{y}^{\\prime }+24{y}^{\\prime }{}^{2}+17{x}^{\\prime }{}^{2}+68{x}^{\\prime }{y}^{\\prime }+68{y}^{\\prime }{}^{2}=100\\hfill \\\\ \\text{ }25{x}^{\\prime }{}^{2}+100{y}^{\\prime }{}^{2}=100\\text{ }\\hfill \\\\ \\text{ }\\frac{25}{100}{x}^{\\prime }{}^{2}+\\frac{100}{100}{y}^{\\prime }{}^{2}=\\frac{100}{100} \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1805836\">Write the equations with[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]in the standard form with respect to the new coordinate system.<\/p>\n<div id=\"fs-id2044859\" class=\"unnumbered aligncenter\">[latex]\\frac{{{x}^{\\prime }}^{2}}{4}+\\frac{{{y}^{\\prime }}^{2}}{1}=1[\/latex]<\/div>\n<p><a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_008\">(Figure)<\/a> shows the graph of the ellipse.<\/p>\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\/19152809\/CNX_Precalc_Figure_10_04_008.jpg\" alt=\"\" width=\"487\" height=\"217\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 8.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1419500\"><span id=\"fs-id1419508\"><\/span><\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2008343\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id2008352\">\n<p id=\"fs-id2008353\">Rewrite the[latex]\\,13{x}^{2}-6\\sqrt{3}xy+7{y}^{2}=16\\,[\/latex]in the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]system without the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]term.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2579717\">[latex]\\frac{{{x}^{\\prime }}^{2}}{4}+\\frac{{{y}^{\\prime }}^{2}}{1}=1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_10_04_04\" class=\"textbox examples\">\n<div id=\"fs-id2753537\">\n<div id=\"fs-id2050743\">\n<h3>Graphing an Equation That Has No <em>x\u2032y\u2032<\/em> Terms<\/h3>\n<p id=\"fs-id2050753\">Graph the following equation relative to the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]system:<\/p>\n<div id=\"fs-id2067868\" class=\"unnumbered aligncenter\">[latex]{x}^{2}+12xy-4{y}^{2}=30[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1770490\">First, we find[latex]\\,\\mathrm{cot}\\left(2\\theta \\right).[\/latex]<\/p>\n<div id=\"fs-id2102392\" class=\"unnumbered aligncenter\">[latex]{x}^{2}+12xy-4{y}^{2}=20\u21d2A=1,\\text{ }B=12,\\text{and }C=-4[\/latex]<\/div>\n<div id=\"fs-id1649246\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{cot}\\left(2\\theta \\right)=\\frac{A-C}{B}\\hfill \\\\ \\mathrm{cot}\\left(2\\theta \\right)=\\frac{1-\\left(-4\\right)}{12}\\hfill \\\\ \\mathrm{cot}\\left(2\\theta \\right)=\\frac{5}{12}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2349896\">Because[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)=\\frac{5}{12},[\/latex] we can draw a reference triangle as in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_009\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_10_04_009\" 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\/19152811\/CNX_Precalc_Figure_10_04_009.jpg\" alt=\"\" width=\"487\" height=\"591\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 9.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id2637723\" class=\"unnumbered aligncenter\">[latex]\\mathrm{cot}\\left(2\\theta \\right)=\\frac{5}{12}=\\frac{\\text{adjacent}}{\\text{opposite}}[\/latex]<\/div>\n<p id=\"fs-id2309119\">Thus, the hypotenuse is<\/p>\n<div id=\"fs-id2309122\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{r}\\hfill {5}^{2}+{12}^{2}={h}^{2}\\\\ \\hfill 25+144={h}^{2}\\\\ \\hfill 169={h}^{2}\\\\ \\hfill h=13\\end{array}[\/latex]<\/div>\n<p id=\"fs-id2292950\">Next, we find[latex]\\,\\mathrm{sin}\\text{ }\\theta \\,[\/latex]and[latex]\\,\\mathrm{cos}\\text{ }\\theta .\\,[\/latex]We will use half-angle identities.<\/p>\n<div id=\"fs-id2594568\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ \\mathrm{sin}\\text{ }\\theta =\\sqrt{\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}}=\\sqrt{\\frac{1-\\frac{5}{13}}{2}}=\\sqrt{\\frac{\\frac{13}{13}-\\frac{5}{13}}{2}}=\\sqrt{\\frac{8}{13}\\cdot \\frac{1}{2}}=\\frac{2}{\\sqrt{13}}\\hfill \\end{array}\\hfill \\\\ \\mathrm{cos}\\text{ }\\theta =\\sqrt{\\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}}=\\sqrt{\\frac{1+\\frac{5}{13}}{2}}=\\sqrt{\\frac{\\frac{13}{13}+\\frac{5}{13}}{2}}=\\sqrt{\\frac{18}{13}\\cdot \\frac{1}{2}}=\\frac{3}{\\sqrt{13}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1405820\">Now we find[latex]\\,x\\,[\/latex]and[latex]\\,y\\text{.\\hspace{0.17em}}[\/latex]<\/p>\n<div id=\"fs-id1936549\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\hfill \\\\ x={x}^{\\prime }\\left(\\frac{3}{\\sqrt{13}}\\right)-{y}^{\\prime }\\left(\\frac{2}{\\sqrt{13}}\\right)\\hfill \\\\ x=\\frac{3{x}^{\\prime }-2{y}^{\\prime }}{\\sqrt{13}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1576589\">and<\/p>\n<div id=\"fs-id1576592\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta \\hfill \\\\ y={x}^{\\prime }\\left(\\frac{2}{\\sqrt{13}}\\right)+{y}^{\\prime }\\left(\\frac{3}{\\sqrt{13}}\\right)\\hfill \\\\ y=\\frac{2{x}^{\\prime }+3{y}^{\\prime }}{\\sqrt{13}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2290414\">Now we substitute[latex]\\,x=\\frac{3{x}^{\\prime }-2{y}^{\\prime }}{\\sqrt{13}}\\,[\/latex]and[latex]\\,y=\\frac{2{x}^{\\prime }+3{y}^{\\prime }}{\\sqrt{13}}\\,[\/latex]into[latex]\\,{x}^{2}+12xy-4{y}^{2}=30.[\/latex]<\/p>\n<div id=\"fs-id2067955\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{llll}\\text{ }{\\left(\\frac{3{x}^{\\prime }-2{y}^{\\prime }}{\\sqrt{13}}\\right)}^{2}+12\\left(\\frac{3{x}^{\\prime }-2{y}^{\\prime }}{\\sqrt{13}}\\right)\\left(\\frac{2{x}^{\\prime }+3{y}^{\\prime }}{\\sqrt{13}}\\right)-4{\\left(\\frac{2{x}^{\\prime }+3{y}^{\\prime }}{\\sqrt{13}}\\right)}^{2}=30\\hfill & \\hfill & \\hfill & \\hfill \\\\ \\text{ }\\,\\,\\left(\\frac{1}{13}\\right)\\left[{\\left(3{x}^{\\prime }-2{y}^{\\prime }\\right)}^{2}+12\\left(3{x}^{\\prime }-2{y}^{\\prime }\\right)\\left(2{x}^{\\prime }+3{y}^{\\prime }\\right)-4{\\left(2{x}^{\\prime }+3{y}^{\\prime }\\right)}^{2}\\right]=30 \\hfill & \\hfill & \\hfill & \\text{Factor}.\\hfill \\\\ \\left(\\frac{1}{13}\\right)\\left[9{x}^{\\prime }{}^{2}-12{x}^{\\prime }{y}^{\\prime }+4{y}^{\\prime }{}^{2}+12\\left(6{x}^{\\prime }{}^{2}+5{x}^{\\prime }{y}^{\\prime }-6{y}^{\\prime }{}^{2}\\right)-4\\left(4{x}^{\\prime }{}^{2}+12{x}^{\\prime }{y}^{\\prime }+9{y}^{\\prime }{}^{2}\\right)\\right]=30\\hfill & \\hfill & \\hfill & \\text{Multiply}.\\hfill \\\\ \\text{ }\\left(\\frac{1}{13}\\right)\\left[9{x}^{\\prime }{}^{2}-12{x}^{\\prime }{y}^{\\prime }+4{y}^{\\prime }{}^{2}+72{x}^{\\prime }{}^{2}+60{x}^{\\prime }{y}^{\\prime }-72{y}^{\\prime }{}^{2}-16{x}^{\\prime }{}^{2}-48{x}^{\\prime }{y}^{\\prime }-36{y}^{\\prime }{}^{2}\\right]=30\\hfill & \\hfill & \\hfill & \\text{Distribute}.\\hfill \\\\ \\text{ }\\,\\text{ }\\left(\\frac{1}{13}\\right)\\left[65{x}^{\\prime }{}^{2}-104{y}^{\\prime }{}^{2}\\right]=30\\hfill & \\hfill & \\hfill & \\text{Combine like terms}.\\hfill \\\\ \\text{ }65{x}^{\\prime }{}^{2}-104{y}^{\\prime }{}^{2}=390\\hfill & \\hfill & \\hfill & \\text{Multiply}.\\text{ }\\hfill \\\\ \\text{ }\\frac{{x}^{\\prime }{}^{2}}{6}-\\frac{4{y}^{\\prime }{}^{2}}{15}=1 \\hfill & \\hfill & \\hfill & \\text{Divide by 390}.\\hfill \\end{array}[\/latex]<\/div>\n<p><a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Figure_10_04_010\">(Figure)<\/a> shows the graph of the hyperbola[latex]\\,\\frac{{{x}^{\\prime }}^{2}}{6}-\\frac{4{{y}^{\\prime }}^{2}}{15}=1.\\text{ }[\/latex]<\/p>\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\/19152818\/CNX_Precalc_Figure_10_04_010.jpg\" alt=\"\" width=\"487\" height=\"441\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 10.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1500365\"><\/details>\n<p><span id=\"fs-id2011757\"><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2011771\" class=\"bc-section section\">\n<h3>Identifying Conics without Rotating Axes<\/h3>\n<p id=\"fs-id2822590\">Now we have come full circle. How do we identify the type of conic described by an equation? What happens when the axes are rotated? Recall, the general form of a conic is<\/p>\n<div id=\"fs-id2822595\" class=\"unnumbered aligncenter\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\n<p id=\"fs-id1966856\">If we apply the rotation formulas to this equation we get the form<\/p>\n<div id=\"fs-id1966860\" class=\"unnumbered aligncenter\">[latex]{A}^{\\prime }{{x}^{\\prime }}^{2}+{B}^{\\prime }{x}^{\\prime }{y}^{\\prime }+{C}^{\\prime }{{y}^{\\prime }}^{2}+{D}^{\\prime }{x}^{\\prime }+{E}^{\\prime }{y}^{\\prime }+{F}^{\\prime }=0[\/latex]<\/div>\n<p id=\"fs-id1587522\">It may be shown that[latex]\\,{B}^{2}-4AC={{B}^{\\prime }}^{2}-4{A}^{\\prime }{C}^{\\prime }.\\,[\/latex]The expression does not vary after rotation, so we call the expression invariant<strong>.<\/strong> The discriminant,[latex]\\,{B}^{2}-4AC,[\/latex] is invariant and remains unchanged after rotation. Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section.<\/p>\n<div id=\"fs-id1971164\" class=\"textbox key-takeaways\">\n<h3>Using the Discriminant to Identify a Conic<\/h3>\n<p id=\"fs-id1971173\">If the equation[latex]\\,A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0\\,[\/latex]is transformed by rotating axes into the equation[latex]\\,{A}^{\\prime }{{x}^{\\prime }}^{2}+{B}^{\\prime }{x}^{\\prime }{y}^{\\prime }+{C}^{\\prime }{{y}^{\\prime }}^{2}+{D}^{\\prime }{x}^{\\prime }+{E}^{\\prime }{y}^{\\prime }+{F}^{\\prime }=0,[\/latex] then[latex]\\,{B}^{2}-4AC={{B}^{\\prime }}^{2}-4{A}^{\\prime }{C}^{\\prime }.[\/latex]<\/p>\n<p id=\"fs-id1527850\">The equation[latex]\\,A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0\\,[\/latex]is an ellipse, a parabola, or a hyperbola, or a degenerate case of one of these.<\/p>\n<p id=\"fs-id1246926\">If the discriminant,[latex]\\,{B}^{2}-4AC,[\/latex]is<\/p>\n<ul id=\"fs-id1246959\">\n<li>[latex]<0,[\/latex] the conic section is an ellipse<\/li>\n<li>[latex]=0,[\/latex] the conic section is a parabola<\/li>\n<li>[latex]>0,[\/latex] the conic section is a hyperbola<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_10_04_05\" class=\"textbox examples\">\n<div id=\"fs-id2138546\">\n<div id=\"fs-id2138548\">\n<h3>Identifying the Conic without Rotating Axes<\/h3>\n<p id=\"fs-id2336006\">Identify the conic for each of the following without rotating axes.<\/p>\n<ol id=\"fs-id2336009\" type=\"a\">\n<li>[latex]5{x}^{2}+2\\sqrt{3}xy+2{y}^{2}-5=0[\/latex]<\/li>\n<li>[latex]5{x}^{2}+2\\sqrt{3}xy+12{y}^{2}-5=0[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id3068892\" type=\"a\">\n<li>Let\u2019s begin by determining[latex]\\,A,B,[\/latex] and[latex]\\,C.[\/latex]\n<div id=\"fs-id1649564\" class=\"unnumbered aligncenter\">[latex]\\underset{A}{\\underbrace{5}}{x}^{2}+\\underset{B}{\\underbrace{2\\sqrt{3}}}xy+\\underset{C}{\\underbrace{2}}{y}^{2}-5=0[\/latex]<\/div>\n<p id=\"fs-id2462423\">Now, we find the discriminant.<\/p>\n<div id=\"fs-id2462426\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{B}^{2}-4AC={\\left(2\\sqrt{3}\\right)}^{2}-4\\left(5\\right)\\left(2\\right)\\hfill \\\\ \\text{ }=4\\left(3\\right)-40\\hfill \\\\ \\text{ }=12-40\\hfill \\\\ \\text{ }=-28<0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1958098\">Therefore,[latex]\\,5{x}^{2}+2\\sqrt{3}xy+2{y}^{2}-5=0\\,[\/latex]represents an ellipse.<\/p>\n<\/li>\n<li>Again, let\u2019s begin by determining[latex]\\,A,B,[\/latex] and[latex]\\,C.[\/latex]\n<div id=\"fs-id2602274\" class=\"unnumbered aligncenter\">[latex]\\underset{A}{\\underbrace{5}}{x}^{2}+\\underset{B}{\\underbrace{2\\sqrt{3}}}xy+\\underset{C}{\\underbrace{12}}{y}^{2}-5=0[\/latex]<\/div>\n<p id=\"fs-id2748962\">Now, we find the discriminant.<\/p>\n<div id=\"fs-id2748965\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{B}^{2}-4AC={\\left(2\\sqrt{3}\\right)}^{2}-4\\left(5\\right)\\left(12\\right)\\hfill \\\\ \\text{ }=4\\left(3\\right)-240\\hfill \\\\ \\text{ }=12-240\\hfill \\\\ \\text{ }=-228<0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2600622\">Therefore,[latex]\\,5{x}^{2}+2\\sqrt{3}xy+12{y}^{2}-5=0\\,[\/latex]represents an ellipse.<\/details>\n<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2436719\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_10_04_03\">\n<div id=\"fs-id2436729\">\n<p id=\"fs-id2436730\">Identify the conic for each of the following without rotating axes.<\/p>\n<ol id=\"fs-id2436733\" type=\"a\">\n<li>[latex]{x}^{2}-9xy+3{y}^{2}-12=0[\/latex]<\/li>\n<li>[latex]10{x}^{2}-9xy+4{y}^{2}-4=0[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1753961\" type=\"a\">\n<li>hyperbola<\/li>\n<li>ellipse<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1951748\" class=\"precalculus media\">\n<p id=\"fs-id1951754\">Access this online resource for additional instruction and practice with conic sections and rotation of axes.<\/p>\n<ul id=\"fs-id1951759\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/introconic\">Introduction to Conic Sections<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1951769\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"fs-id1951776\" summary=\"..\">\n<tbody>\n<tr>\n<td>General Form equation of a conic section<\/td>\n<td>[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rotation of a conic section<\/td>\n<td>[latex]\\begin{array}{l}x={x}^{\\prime }\\mathrm{cos}\\text{ }\\theta -{y}^{\\prime }\\mathrm{sin}\\text{ }\\theta \\hfill \\\\ y={x}^{\\prime }\\mathrm{sin}\\text{ }\\theta +{y}^{\\prime }\\mathrm{cos}\\text{ }\\theta \\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Angle of rotation<\/td>\n<td>[latex]\\theta ,\\text{where }\\mathrm{cot}\\left(2\\theta \\right)=\\frac{A-C}{B}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id2430170\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id2430176\">\n<li>Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola.<\/li>\n<li>A nondegenerate conic section has the general form[latex]\\,A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0\\,[\/latex]where[latex]\\,A,B\\,[\/latex]and[latex]\\,C\\,[\/latex]are not all zero. The values of[latex]\\,A,B,[\/latex] and[latex]\\,C\\,[\/latex]determine the type of conic. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Example_10_04_01\">(Figure)<\/a>.<\/li>\n<li>Equations of conic sections with an[latex]\\,xy\\,[\/latex]term have been rotated about the origin. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Example_10_04_02\">(Figure)<\/a>.<\/li>\n<li>The general form can be transformed into an equation in the[latex]\\,{x}^{\\prime }\\,[\/latex]and[latex]\\,{y}^{\\prime }\\,[\/latex]coordinate system without the[latex]\\,{x}^{\\prime }{y}^{\\prime }\\,[\/latex]term. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Example_10_04_03\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Example_10_04_04\">(Figure)<\/a>.<\/li>\n<li>An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=3352&amp;action=edit#Example_10_04_05\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id2084607\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id2084610\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id2285327\">\n<div id=\"fs-id2285328\">\n<p id=\"fs-id2285329\">What effect does the[latex]\\,xy\\,[\/latex]term have on the graph of a conic section?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2285350\">The[latex]\\,xy\\,[\/latex]term causes a rotation of the graph to occur.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2285371\">\n<div id=\"fs-id2285372\">\n<p id=\"fs-id2285373\">If the equation of a conic section is written in the form[latex]\\,A{x}^{2}+B{y}^{2}+Cx+Dy+E=0\\,[\/latex]and[latex]\\,AB=0,[\/latex] what can we conclude?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1400032\">\n<div id=\"fs-id1400033\">\n<p id=\"fs-id1400034\">If the equation of a conic section is written in the form[latex]\\,A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0,[\/latex]and[latex]\\,{B}^{2}-4AC>0,[\/latex] what can we conclude?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1691756\">The conic section is a hyperbola.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1691760\">\n<div id=\"fs-id1691761\">\n<p id=\"fs-id1691762\">Given the equation[latex]\\,a{x}^{2}+4x+3{y}^{2}-12=0,[\/latex] what can we conclude if[latex]\\,a>0?[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1874649\">\n<div id=\"fs-id1874650\">\n<p id=\"fs-id1874651\">For the equation[latex]\\,A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0,[\/latex] the value of[latex]\\,\\theta \\,[\/latex]that satisfies[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)=\\frac{A-C}{B}\\,[\/latex]gives us what information?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2454774\">It gives the angle of rotation of the axes in order to eliminate the[latex]\\,xy\\,[\/latex]term.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2100813\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id2100819\">For the following exercises, determine which conic section is represented based on the given equation.<\/p>\n<div id=\"fs-id2100822\">\n<div id=\"fs-id2100823\">\n<p id=\"fs-id2100824\">[latex]9{x}^{2}+4{y}^{2}+72x+36y-500=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2814657\">\n<div id=\"fs-id2814658\">\n<p id=\"fs-id2814659\">[latex]{x}^{2}-10x+4y-10=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1664771\">[latex]AB=0,[\/latex] parabola<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1664792\">\n<div id=\"fs-id1664793\">\n<p id=\"fs-id1664794\">[latex]2{x}^{2}-2{y}^{2}+4x-6y-2=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2584248\">\n<div id=\"fs-id2584249\">\n<p id=\"fs-id2584250\">[latex]4{x}^{2}-{y}^{2}+8x-1=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2474747\">[latex]AB=-4<0,[\/latex] hyperbola<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2474775\">\n<div id=\"fs-id2474776\">\n<p id=\"fs-id2474777\">[latex]4{y}^{2}-5x+9y+1=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2927799\">\n<div id=\"fs-id2927800\">\n<p id=\"fs-id2927801\">[latex]2{x}^{2}+3{y}^{2}-8x-12y+2=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1277153\">[latex]AB=6>0,[\/latex] ellipse<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1277179\">\n<div id=\"fs-id1277180\">\n<p id=\"fs-id1277181\">[latex]4{x}^{2}+9xy+4{y}^{2}-36y-125=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1418850\">[latex]3{x}^{2}+6xy+3{y}^{2}-36y-125=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2433642\">[latex]{B}^{2}-4AC=0,[\/latex] parabola<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2352771\">\n<div id=\"fs-id2352772\">\n<p id=\"fs-id2352774\">[latex]-3{x}^{2}+3\\sqrt{3}xy-4{y}^{2}+9=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2472798\">\n<div id=\"fs-id2472799\">\n<p id=\"fs-id2472800\">[latex]2{x}^{2}+4\\sqrt{3}xy+6{y}^{2}-6x-3=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2861569\">[latex]{B}^{2}-4AC=0,[\/latex] parabola<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2155989\">\n<div id=\"fs-id2155990\">\n<p id=\"fs-id2155991\">[latex]-{x}^{2}+4\\sqrt{2}xy+2{y}^{2}-2y+1=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2161610\">\n<div id=\"fs-id2161611\">\n<p id=\"fs-id2161612\">[latex]8{x}^{2}+4\\sqrt{2}xy+4{y}^{2}-10x+1=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1940130\">[latex]{B}^{2}-4AC=-96<0,[\/latex] ellipse<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1940172\">For the following exercises, find a new representation of the given equation after rotating through the given angle.<\/p>\n<div id=\"fs-id1940176\">\n<div id=\"fs-id1940177\">\n<p id=\"fs-id1940178\">[latex]3{x}^{2}+xy+3{y}^{2}-5=0,\\theta =45\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2056389\">\n<div id=\"fs-id2056390\">\n<p id=\"fs-id2056391\">[latex]4{x}^{2}-xy+4{y}^{2}-2=0,\\theta =45\u00b0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1888348\">[latex]7{{x}^{\\prime }}^{2}+9{{y}^{\\prime }}^{2}-4=0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1753645\">\n<div id=\"fs-id1753646\">\n<p id=\"fs-id1753647\">[latex]2{x}^{2}+8xy-1=0,\\theta =30\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1558064\">\n<div id=\"fs-id1558065\">\n<p id=\"fs-id1558066\">[latex]-2{x}^{2}+8xy+1=0,\\theta =45\u00b0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1799397\">[latex]3{{x}^{\\prime }}^{2}+2{x}^{\\prime }{y}^{\\prime }-5{{y}^{\\prime }}^{2}+1=0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1587288\">\n<div id=\"fs-id1587289\">\n<p id=\"fs-id1587290\">[latex]4{x}^{2}+\\sqrt{2}xy+4{y}^{2}+y+2=0,\\theta =45\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2881099\">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-id1806812\">\n<div id=\"fs-id1806813\">\n<p id=\"fs-id1806814\">[latex]{x}^{2}+3\\sqrt{3}xy+4{y}^{2}+y-2=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2451648\">[latex]\\theta ={60}^{\\circ },11{{x}^{\\prime }}^{2}-{{y}^{\\prime }}^{2}+\\sqrt{3}{x}^{\\prime }+{y}^{\\prime }-4=0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1663688\">\n<div id=\"fs-id1663689\">\n<p id=\"fs-id1663690\">[latex]4{x}^{2}+2\\sqrt{3}xy+6{y}^{2}+y-2=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2304092\">\n<div id=\"fs-id2304094\">\n<p id=\"fs-id2304095\">[latex]9{x}^{2}-3\\sqrt{3}xy+6{y}^{2}+4y-3=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1612806\">[latex]\\theta ={150}^{\\circ },21{{x}^{\\prime }}^{2}+9{{y}^{\\prime }}^{2}+4{x}^{\\prime }-4\\sqrt{3}{y}^{\\prime }-6=0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2956037\">\n<div id=\"fs-id2956038\">\n<p id=\"fs-id2956040\">[latex]-3{x}^{2}-\\sqrt{3}xy-2{y}^{2}-x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2691106\">\n<div id=\"fs-id2691107\">\n<p id=\"fs-id2691108\">[latex]16{x}^{2}+24xy+9{y}^{2}+6x-6y+2=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1154383\">[latex]\\theta \\approx {36.9}^{\\circ },125{{x}^{\\prime }}^{2}+6{x}^{\\prime }-42{y}^{\\prime }+10=0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1263441\">\n<div id=\"fs-id1794643\">\n<p id=\"fs-id1794644\">[latex]{x}^{2}+4xy+4{y}^{2}+3x-2=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1794701\">\n<div id=\"fs-id1794702\">\n<p id=\"fs-id1794703\">[latex]{x}^{2}+4xy+{y}^{2}-2x+1=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1685135\">[latex]\\theta ={45}^{\\circ },3{{x}^{\\prime }}^{2}-{{y}^{\\prime }}^{2}-\\sqrt{2}{x}^{\\prime }+\\sqrt{2}{y}^{\\prime }+1=0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1266703\">\n<div id=\"fs-id1266704\">\n<p id=\"fs-id1266705\">[latex]4{x}^{2}-2\\sqrt{3}xy+6{y}^{2}-1=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2821469\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id2821474\">For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation.<\/p>\n<div id=\"fs-id2821479\">\n<div id=\"fs-id2821480\">\n<p id=\"fs-id2821481\">[latex]y=-{x}^{2},\\theta =-{45}^{\\circ }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2164439\">[latex]\\frac{\\sqrt{2}}{2}\\left({x}^{\\prime }+{y}^{\\prime }\\right)=\\frac{1}{2}{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}[\/latex]<\/p>\n<p><span id=\"fs-id2119635\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152820\/CNX_Precalc_Figure_10_04_201.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2119646\">\n<div id=\"fs-id2787332\">\n<p id=\"fs-id2787334\">[latex]x={y}^{2},\\theta ={45}^{\\circ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2787375\">\n<div id=\"fs-id2787376\">\n<p id=\"fs-id2787377\">[latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{1}=1,\\theta ={45}^{\\circ }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1264831\">[latex]\\frac{{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}}{8}+\\frac{{\\left({x}^{\\prime }+{y}^{\\prime }\\right)}^{2}}{2}=1[\/latex]<\/p>\n<p><span id=\"fs-id1232420\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152830\/CNX_Precalc_Figure_10_04_203.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1232431\">\n<div id=\"fs-id1232432\">\n<p id=\"fs-id1232433\">[latex]\\frac{{y}^{2}}{16}+\\frac{{x}^{2}}{9}=1,\\theta ={45}^{\\circ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1403135\">\n<div id=\"fs-id1403136\">\n<p id=\"fs-id1403137\">[latex]{y}^{2}-{x}^{2}=1,\\theta ={45}^{\\circ }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2286260\">[latex]\\frac{{\\left({x}^{\\prime }+{y}^{\\prime }\\right)}^{2}}{2}-\\frac{{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}}{2}=1[\/latex]<\/p>\n<p><span id=\"fs-id2245295\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152845\/CNX_Precalc_Figure_10_04_205.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2245306\">\n<div id=\"fs-id2245307\">\n<p id=\"fs-id2245308\">[latex]y=\\frac{{x}^{2}}{2},\\theta ={30}^{\\circ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2245359\">\n<div id=\"fs-id2245360\">\n<p id=\"fs-id2245361\">[latex]x={\\left(y-1\\right)}^{2},\\theta ={30}^{\\circ }[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2463278\">[latex]\\frac{\\sqrt{3}}{2}{x}^{\\prime }-\\frac{1}{2}{y}^{\\prime }={\\left(\\frac{1}{2}{x}^{\\prime }+\\frac{\\sqrt{3}}{2}{y}^{\\prime }-1\\right)}^{2}[\/latex]<\/p>\n<p><span id=\"fs-id2322951\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152856\/CNX_Precalc_Figure_10_04_207.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2322962\">\n<div id=\"fs-id2322963\">\n<p id=\"fs-id2322964\">[latex]\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{4}=1,\\theta ={30}^{\\circ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2199206\">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-id1999288\">\n<div id=\"fs-id1999289\">\n<p id=\"fs-id1999290\">[latex]xy=9[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1999314\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152901\/CNX_Precalc_Figure_10_04_209.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1999324\">\n<div id=\"fs-id1999325\">\n<p id=\"fs-id1999326\">[latex]{x}^{2}+10xy+{y}^{2}-6=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1673790\">\n<div id=\"fs-id1673791\">\n<p id=\"fs-id1673792\">[latex]{x}^{2}-10xy+{y}^{2}-24=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id2061214\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152903\/CNX_Precalc_Figure_10_04_211.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2061224\">\n<div id=\"fs-id2061225\">\n<p id=\"fs-id2061226\">[latex]4{x}^{2}-3\\sqrt{3}xy+{y}^{2}-22=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2757196\">\n<div id=\"fs-id2757197\">\n<p id=\"fs-id2757198\">[latex]6{x}^{2}+2\\sqrt{3}xy+4{y}^{2}-21=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id2757262\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152910\/CNX_Precalc_Figure_10_04_213.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1617309\">\n<div id=\"fs-id1617310\">\n<p id=\"fs-id1617311\">[latex]11{x}^{2}+10\\sqrt{3}xy+{y}^{2}-64=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1617367\">\n<div id=\"fs-id1617368\">\n<p id=\"fs-id1617370\">[latex]21{x}^{2}+2\\sqrt{3}xy+19{y}^{2}-18=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id2020878\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152918\/CNX_Precalc_Figure_10_04_215.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2020889\">\n<div id=\"fs-id2020890\">\n<p id=\"fs-id2020891\">[latex]16{x}^{2}+24xy+9{y}^{2}-130x+90y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2289143\">\n<div id=\"fs-id2289144\">\n<p id=\"fs-id2289146\">[latex]16{x}^{2}+24xy+9{y}^{2}-60x+80y=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id2106831\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152927\/CNX_Precalc_Figure_10_04_217.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2106842\">\n<div id=\"fs-id2106843\">\n<p id=\"fs-id2106844\">[latex]13{x}^{2}-6\\sqrt{3}xy+7{y}^{2}-16=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1492652\">\n<div id=\"fs-id1492653\">\n<p id=\"fs-id1492654\">[latex]4{x}^{2}-4xy+{y}^{2}-8\\sqrt{5}x-16\\sqrt{5}y=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id2430882\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152935\/CNX_Precalc_Figure_10_04_219.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2430892\">For the following exercises, determine the angle of rotation in order to eliminate the[latex]\\,xy\\,[\/latex]term. Then graph the new set of axes.<\/p>\n<div id=\"fs-id2430912\">\n<div id=\"fs-id2430913\">\n<p id=\"fs-id2430914\">[latex]6{x}^{2}-5\\sqrt{3}xy+{y}^{2}+10x-12y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2261896\">\n<div id=\"fs-id2261897\">\n<p id=\"fs-id2261898\">[latex]6{x}^{2}-5xy+6{y}^{2}+20x-y=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1722813\">[latex]\\theta ={45}^{\\circ }[\/latex]<\/p>\n<p><span id=\"fs-id1722842\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152947\/CNX_Precalc_Figure_10_04_221.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1722853\">\n<div id=\"fs-id1722854\">\n<p id=\"fs-id1722855\">[latex]6{x}^{2}-8\\sqrt{3}xy+14{y}^{2}+10x-3y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2613786\">\n<div id=\"fs-id2613787\">\n<p id=\"fs-id2613788\">[latex]4{x}^{2}+6\\sqrt{3}xy+10{y}^{2}+20x-40y=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2320678\">[latex]\\theta ={60}^{\\circ }[\/latex]<\/p>\n<p><span id=\"fs-id2320706\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152953\/CNX_Precalc_Figure_10_04_223.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2320717\">\n<div id=\"fs-id2320718\">\n<p id=\"fs-id1244573\">[latex]8{x}^{2}+3xy+4{y}^{2}+2x-4=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1244632\">\n<div id=\"fs-id1244633\">\n<p id=\"fs-id1244634\">[latex]16{x}^{2}+24xy+9{y}^{2}+20x-44y=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1651852\">[latex]\\theta \\approx {36.9}^{\\circ }[\/latex]<\/p>\n<p><span id=\"fs-id1651881\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152955\/CNX_Precalc_Figure_10_04_225.jpg\" alt=\"\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1651892\">For the following exercises, determine the value of[latex]\\,k\\,[\/latex]based on the given equation.<\/p>\n<div id=\"fs-id1550256\">\n<div id=\"fs-id1550257\">\n<p id=\"fs-id1550258\">Given[latex]\\,4{x}^{2}+kxy+16{y}^{2}+8x+24y-48=0,[\/latex] find[latex]\\,k\\,[\/latex]for the graph to be a parabola.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2601769\">\n<div id=\"fs-id2601770\">\n<p id=\"fs-id2601771\">Given[latex]\\,2{x}^{2}+kxy+12{y}^{2}+10x-16y+28=0,[\/latex] find[latex]\\,k\\,[\/latex]for the graph to be an ellipse.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1233480\">[latex]-4\\sqrt{6}<k<4\\sqrt{6}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1233515\">\n<div id=\"fs-id1233516\">\n<p id=\"fs-id1233517\">Given[latex]\\,3{x}^{2}+kxy+4{y}^{2}-6x+20y+128=0,[\/latex] find[latex]\\,k\\,[\/latex]for the graph to be a hyperbola.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1088231\">\n<div id=\"fs-id1088232\">\n<p id=\"fs-id1088234\">Given[latex]\\,k{x}^{2}+8xy+8{y}^{2}-12x+16y+18=0,[\/latex] find[latex]\\,k\\,[\/latex]for the graph to be a parabola.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1924178\">[latex]k=2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1924196\">\n<div id=\"fs-id1924197\">\n<p id=\"fs-id1924198\">Given[latex]\\,6{x}^{2}+12xy+k{y}^{2}+16x+10y+4=0,[\/latex] find[latex]\\,k\\,[\/latex]for the graph to be an ellipse.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id2107200\">\n<dt>angle of rotation<\/dt>\n<dd id=\"fs-id2118805\">an acute angle formed by a set of axes rotated from the Cartesian plane where, if[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)>0,[\/latex]then[latex]\\,\\theta \\,[\/latex]is between[latex]\\,\\left(0\u00b0,45\u00b0\\right);[\/latex]if[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)<0,[\/latex]then[latex]\\,\\theta \\,[\/latex]is between[latex]\\,\\left(45\u00b0,90\u00b0\\right);\\,[\/latex]and if[latex]\\,\\mathrm{cot}\\left(2\\theta \\right)=0,[\/latex]then[latex]\\,\\theta =45\u00b0[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id2165541\">\n<dt>degenerate conic sections<\/dt>\n<dd id=\"fs-id2165546\">any of the possible shapes formed when a plane intersects a double cone through the apex. Types of degenerate conic sections include a point, a line, and intersecting lines.<\/dd>\n<\/dl>\n<dl id=\"fs-id1840460\">\n<dt>nondegenerate conic section<\/dt>\n<dd id=\"fs-id1840465\">a shape formed by the intersection of a plane with a double right cone such that the plane does not pass through the apex; nondegenerate conics include circles, ellipses, hyperbolas, and parabolas<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":291,"menu_order":5,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-194","chapter","type-chapter","status-publish","hentry"],"part":185,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/194","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\/194\/revisions"}],"predecessor-version":[{"id":195,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/194\/revisions\/195"}],"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\/194\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=194"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=194"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=194"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}