{"id":1433,"date":"2023-06-05T14:51:41","date_gmt":"2023-06-05T14:51:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/rotation-of-axes\/"},"modified":"2023-06-05T14:51:41","modified_gmt":"2023-06-05T14:51:41","slug":"rotation-of-axes","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/rotation-of-axes\/","title":{"raw":"Rotation of Axes","rendered":"Rotation of Axes"},"content":{"raw":"\n<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Identify nondegenerate conic sections given their general form equations.<\/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>\nAs 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.\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183021\/CNX_Precalc_Figure_10_04_0012.jpg\" alt=\"\" width=\"975\" height=\"650\"> <b>Figure 1.<\/b> The nondegenerate conic sections[\/caption]\n\nEllipses, circles, hyperbolas, and parabolas are sometimes called the <strong>nondegenerate conic sections<\/strong>, in contrast to the <strong>degenerate conic sections<\/strong>, which are shown in Figure 2. 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.\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183025\/CNX_Precalc_Figure_10_04_002n2.jpg\" alt=\"\" width=\"975\" height=\"719\"> <b>Figure 2.<\/b> Degenerate conic sections[\/caption]\n<h2>Identifying Nondegenerate Conics in General Form<\/h2>\nIn 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.\n<div style=\"text-align: center;\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\nwhere [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.\n\nYou 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.\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 class=\"textbox\">\n<h3>A General Note: General Form of Conic Sections<\/h3>\nA <strong>nondegenerate conic section<\/strong> has the general form\n<p style=\"text-align: center;\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/p>\nwhere [latex]A,B[\/latex], and [latex]C[\/latex] are not all zero.\n\nThe table below 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.\n<table id=\"Table_10_04_02\" summary=\"..\">\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 class=\"textbox\">\n<h3>How To: Given the equation of a conic, identify the type of conic.<strong>\n<\/strong><\/h3>\n<ol>\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>\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 is 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 is a circle.<\/li>\n \t<li>If [latex]A[\/latex] and [latex]C[\/latex] are nonzero and have opposite signs, then the graph is a hyperbola.<\/li>\n \t<li>If either [latex]A[\/latex] or [latex]C[\/latex] is zero, then the graph is a parabola.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Identifying a Conic from Its General Form<\/h3>\nIdentify the graph of each of the following nondegenerate conic sections.\n<ol>\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[reveal-answer q=\"722128\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"722128\"]\n<ol>\n \t<li>Rewriting the general form, we have\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183027\/eq1_n2.jpg\" alt=\"\">\n[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.<\/li>\n \t<li>Rewriting the general form, we have\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183029\/eq2_n2.jpg\" alt=\"\">[latex]A=0[\/latex] and [latex]C=9[\/latex]. We can determine that the equation is a parabola, since [latex]A[\/latex] is zero.<\/li>\n \t<li>Rewriting the general form, we have\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183031\/eq3_n2.jpg\" alt=\"\">[latex]A=3[\/latex] and [latex]C=3[\/latex]. Because [latex]A=C[\/latex], the graph of this equation is a circle.<\/li>\n \t<li>Rewriting the general form, we have <img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183033\/eq42.jpg\" alt=\"\">[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.<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nIdentify the graph of each of the following nondegenerate conic sections.\n<ol>\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[reveal-answer q=\"185596\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"185596\"]\n<ol>\n \t<li>hyperbola<\/li>\n \t<li>ellipse<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<h2>Finding a New Representation of the Given Equation after Rotating through a Given Angle<\/h2>\nUntil 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]\\begin{align}\\left({x}^{\\prime },{y}^{\\prime }\\right)\\end{align}[\/latex] on the new plane defined by the new, rotated axes, called the <em>x'<\/em>-axis and <em>y'<\/em>-axis.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183035\/CNX_Precalc_Figure_10_04_0032.jpg\" alt=\"\" width=\"487\" height=\"441\"> <b>Figure 3.<\/b> The graph of the rotated ellipse [latex]{x}^{2}+{y}^{2}-xy - 15=0[\/latex][\/caption]We will find the relationships between [latex]x[\/latex] and [latex]y[\/latex] on the Cartesian plane with [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] on the new rotated plane.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183037\/CNX_Precalc_Figure_10_04_0042.jpg\" alt=\"\" width=\"487\" height=\"366\"> <b>Figure 4.<\/b> 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]\\theta [\/latex].[\/caption]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]\\begin{align}{i}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{j}^{\\prime }\\end{align}[\/latex]. The angle [latex]\\theta [\/latex] is known as the <strong>angle of rotation<\/strong>. We may write the new unit vectors in terms of the original ones.\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;{i}^{\\prime }=i\\cos \\theta +j\\sin \\theta \\\\ &amp;{j}^{\\prime }=-i\\sin \\theta +j\\cos \\theta \\end{align}[\/latex]<\/div>\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183039\/CNX_Precalc_Figure_10_04_0052.jpg\" alt=\"\" width=\"487\" height=\"364\"> <b>Figure 5.<\/b> Relationship between the old and new coordinate planes.[\/caption]\n\nConsider a vector<strong> [latex]u[\/latex] <\/strong>in the new coordinate plane. It may be represented in terms of its coordinate axes.\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime } \\\\ &amp;u={x}^{\\prime }\\left(i\\cos \\theta +j\\sin \\theta \\right)+{y}^{\\prime }\\left(-i\\sin \\theta +j\\cos \\theta \\right) &amp;&amp; \\text{Substitute}. \\\\ &amp;u=ix^{\\prime}\\cos \\theta +jx^{\\prime}\\sin \\theta -iy^{\\prime}\\sin \\theta +jy^{\\prime}\\cos \\theta &amp;&amp; \\text{Distribute}. \\\\ &amp;u=ix^{\\prime}\\cos \\theta -iy^{\\prime}\\sin \\theta +jx^{\\prime}\\sin \\theta +jy^{\\prime}\\cos \\theta &amp;&amp; \\text{Apply commutative property}. \\\\ &amp;u=\\left(x^{\\prime}\\cos \\theta -y^{\\prime}\\sin \\theta \\right)i+\\left(x^{\\prime}\\sin \\theta +y^{\\prime}\\cos \\theta \\right)j &amp;&amp; \\text{Factor by grouping}. \\end{align}[\/latex]<\/div>\nBecause [latex]\\begin{align}u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime }\\end{align}[\/latex], we have representations of [latex]x[\/latex] and [latex]y[\/latex] in terms of the new coordinate system.\n<div style=\"text-align: center;\">[latex]\\begin{gathered}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\\\ \\text{and}\\\\ y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{gathered}[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Equations of Rotation<\/h3>\nIf 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]\\begin{align}\\left({x}^{\\prime },{y}^{\\prime }\\right)\\end{align}[\/latex]. We can use the following equations of rotation to define the relationship between [latex]\\begin{align}\\left(x,y\\right)\\end{align}[\/latex] and [latex]\\begin{align}\\left({x}^{\\prime },{y}^{\\prime }\\right):\\end{align}[\/latex]\n<p style=\"text-align: center;\">[latex]\\begin{gathered}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\\\ \\text{and}\\\\ y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{gathered}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the equation of a conic, find a new representation after rotating through an angle.<strong>\n<\/strong><\/h3>\n<ol id=\"fs-id1146233\">\n \t<li>Find [latex]x[\/latex] and [latex]y[\/latex] where [latex]\\begin{align}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\end{align}[\/latex] and [latex]\\begin{align}y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{align}[\/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]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] in standard form.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Finding a New Representation of an Equation after Rotating through a Given Angle<\/h3>\nFind 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^\\circ [\/latex].\n\n[reveal-answer q=\"303707\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"303707\"]\n\nFind [latex]x[\/latex] and [latex]y[\/latex], where [latex]\\begin{align}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\end{align}[\/latex] and [latex]\\begin{align} y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{align}[\/latex].\n\nBecause [latex]\\theta =45^\\circ [\/latex],\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;x={x}^{\\prime }\\cos \\left(45^\\circ \\right)-{y}^{\\prime }\\sin \\left(45^\\circ \\right) \\\\ &amp;x={x}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)-{y}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right) \\\\ &amp;x=\\frac{{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{2}} \\end{align}[\/latex]<\/p>\nand\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;y={x}^{\\prime }\\sin \\left(45^\\circ \\right)+{y}^{\\prime }\\cos \\left(45^\\circ \\right) \\\\ &amp;y={x}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)+{y}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right) \\\\ &amp;y=\\frac{{x}^{\\prime }+{y}^{\\prime }}{\\sqrt{2}} \\end{align}[\/latex]<\/p>\nSubstitute [latex]\\begin{align}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\end{align}[\/latex] and [latex]\\begin{align}y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{align}[\/latex] into [latex]2{x}^{2}-xy+2{y}^{2}-30=0[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{align} 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\\end{align}[\/latex]<\/p>\nSimplify.\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;2\\frac{\\left({x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }-{y}^{\\prime }\\right)}{2}-\\frac{\\left({x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }+{y}^{\\prime }\\right)}{2}+2\\frac{\\left({x}^{\\prime }+{y}^{\\prime }\\right)\\left({x}^{\\prime }+{y}^{\\prime }\\right)}{2}-30=0 &amp;&amp; \\text{FOIL method} \\\\ &amp;{x}^{\\prime }{}^{2}{-2{x}^{\\prime }y}^{\\prime }+{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}+{x}^{\\prime }{}^{2}+2{x}^{\\prime }{y}^{\\prime }+{y}^{\\prime }{}^{2}-30=0 &amp;&amp; \\text{Combine like terms}. \\\\ &amp;2{x}^{\\prime }{}^{2}+2{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}=30 &amp;&amp; \\text{Combine like terms}. \\\\ &amp;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) &amp;&amp; \\text{Multiply both sides by 2}. \\\\ &amp;4{x}^{\\prime }{}^{2}+4{y}^{\\prime }{}^{2}-\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)=60 &amp;&amp; \\text{Simplify}. \\\\ &amp;4{x}^{\\prime }{}^{2}+4{y}^{\\prime }{}^{2}-{x}^{\\prime }{}^{2}+{y}^{\\prime }{}^{2}=60 &amp;&amp; \\text{Distribute}. \\\\ &amp;\\frac{3{x}^{\\prime }{}^{2}}{60}+\\frac{5{y}^{\\prime }{}^{2}}{60}=\\frac{60}{60} &amp;&amp; \\text{Set equal to 1}. \\end{align}[\/latex]<\/p>\nWrite the equations with [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] in the standard form.\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{{{x}^{\\prime }}^{2}}{20}+\\frac{{{y}^{\\prime }}^{2}}{12}=1\\end{align}[\/latex]<\/p>\nThis equation is an ellipse. Figure 6&nbsp;shows the graph.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183042\/CNX_Precalc_Figure_10_04_0062.jpg\" alt=\"\" width=\"487\" height=\"441\"> <b>Figure 6<\/b>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<h2>Writing Equations of Rotated Conics in Standard Form<\/h2>\nNow 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\n<div style=\"text-align: center;\">[latex]\\cot \\left(2\\theta \\right)=\\frac{A-C}{B}[\/latex]<\/div>\nWe have learned already that any conic may be represented by the second degree equation\n<div style=\"text-align: center;\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\nwhere [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]\\cot \\left(2\\theta \\right)=\\frac{A-C}{B}[\/latex].\n<div>\n<ul>\n \t<li>If [latex]\\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^\\circ ,45^\\circ \\right)[\/latex].<\/li>\n \t<li>If [latex]\\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^\\circ ,90^\\circ \\right)[\/latex].<\/li>\n \t<li>If [latex]A=C[\/latex], then [latex]\\theta =45^\\circ [\/latex].<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an equation for a conic in the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] system, rewrite the equation without the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] term in terms of [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex], where the [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] axes are rotations of the standard axes by [latex]\\theta [\/latex] degrees.<\/h3>\n<ol>\n \t<li>Find [latex]\\cot \\left(2\\theta \\right)[\/latex].<\/li>\n \t<li>Find [latex]\\sin \\theta [\/latex] and [latex]\\cos \\theta [\/latex].<\/li>\n \t<li>Substitute [latex]\\sin \\theta [\/latex] and [latex]\\cos \\theta [\/latex] into [latex]\\begin{align}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\end{align}[\/latex] and [latex]\\begin{align} y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{align}[\/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]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] in the standard form with respect to the rotated axes.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: 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>\nRewrite the equation [latex]8{x}^{2}-12xy+17{y}^{2}=20[\/latex] in the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] system without an [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] term.\n\n[reveal-answer q=\"952851\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"952851\"]\n\nFirst, we find [latex]\\cot \\left(2\\theta \\right)[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{align}8{x}^{2}-12xy+17{y}^{2}=20\\Rightarrow A=8,B=-12\\text{ and }C=17\\end{align}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;\\cot \\left(2\\theta \\right)=\\frac{A-C}{B}=\\frac{8 - 17}{-12} \\\\ &amp;\\cot \\left(2\\theta \\right)=\\frac{-9}{-12}=\\frac{3}{4} \\end{align}[\/latex]<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183044\/CNX_Precalc_Figure_10_04_0072.jpg\" alt=\"\" width=\"487\" height=\"328\"> <b>Figure 7<\/b>[\/caption]\n\n<div style=\"text-align: center;\"><\/div>\n<p style=\"text-align: center;\">[latex]\\cot \\left(2\\theta \\right)=\\frac{3}{4}=\\frac{\\text{adjacent}}{\\text{opposite}}[\/latex]<\/p>\nSo the hypotenuse is\n<p style=\"text-align: center;\">[latex]\\begin{gathered} {3}^{2}+{4}^{2}={h}^{2}\\\\ 9+16={h}^{2}\\\\ 25={h}^{2}\\\\ h=5\\end{gathered}[\/latex]<\/p>\nNext, we find [latex]\\sin \\text{ }\\theta [\/latex] and [latex]\\cos \\text{ }\\theta [\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{align} \\sin \\theta &amp;=\\sqrt{\\frac{1-\\cos \\left(2\\theta \\right)}{2}} \\\\ &amp;=\\sqrt{\\frac{1-\\frac{3}{5}}{2}} \\\\ &amp;=\\sqrt{\\frac{\\frac{5}{5}-\\frac{3}{5}}{2}} \\\\ &amp;=\\sqrt{\\frac{5 - 3}{5}\\cdot \\frac{1}{2}} \\\\ &amp;=\\sqrt{\\frac{2}{10}} \\\\ &amp;=\\sqrt{\\frac{1}{5}} \\\\ &amp;=\\frac{1}{\\sqrt{5}} \\\\[2mm] \\cos \\theta &amp;=\\sqrt{\\frac{1+\\cos \\left(2\\theta \\right)}{2}} \\\\ &amp;=\\sqrt{\\frac{1+\\frac{3}{5}}{2}} \\\\ &amp;=\\sqrt{\\frac{\\frac{5}{5}+\\frac{3}{5}}{2}} \\\\ &amp;=\\sqrt{\\frac{5+3}{5}\\cdot \\frac{1}{2}} \\\\ &amp;=\\sqrt{\\frac{8}{10}} \\\\ &amp;=\\sqrt{\\frac{4}{5}} \\\\ &amp;=\\frac{2}{\\sqrt{5}} \\end{align}[\/latex]<\/p>\nSubstitute the values of [latex]\\sin \\text{ }\\theta [\/latex] and [latex]\\cos \\text{ }\\theta [\/latex] into [latex]\\begin{align}x={x}^{\\prime }\\cos \\text{ }\\theta -{y}^{\\prime }\\sin \\text{ }\\theta \\end{align}[\/latex] and [latex]\\begin{align}y={x}^{\\prime }\\sin \\text{ }\\theta +{y}^{\\prime }\\cos \\text{ }\\theta \\end{align}[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\\\ &amp;x={x}^{\\prime }\\left(\\frac{2}{\\sqrt{5}}\\right)-{y}^{\\prime }\\left(\\frac{1}{\\sqrt{5}}\\right) \\\\ &amp;x=\\frac{2{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{5}} \\end{align}[\/latex]<\/p>\nand\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\\\ &amp;y={x}^{\\prime }\\left(\\frac{1}{\\sqrt{5}}\\right)+{y}^{\\prime }\\left(\\frac{2}{\\sqrt{5}}\\right) \\\\ &amp;y=\\frac{{x}^{\\prime }+2{y}^{\\prime }}{\\sqrt{5}} \\end{align}[\/latex]<\/p>\nSubstitute the expressions for [latex]x[\/latex] and [latex]y[\/latex] into in the given equation, and then simplify.\n<p style=\"text-align: center;\">[latex]\\begin{gathered}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 \\\\ 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 \\\\ 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 \\\\ 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 \\\\ 25{x}^{\\prime }{}^{2}+100{y}^{\\prime }{}^{2}=100 \\\\ \\frac{25}{100}{x}^{\\prime }{}^{2}+\\frac{100}{100}{y}^{\\prime }{}^{2}=\\frac{100}{100} \\end{gathered}[\/latex]<\/p>\nWrite the equations with [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] in the standard form with respect to the new coordinate system.\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{{{x}^{\\prime }}^{2}}{4}+\\frac{{{y}^{\\prime }}^{2}}{1}=1\\end{align}[\/latex]<\/p>\nFigure 8 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\/3675\/2018\/09\/27183046\/CNX_Precalc_Figure_10_04_0082.jpg\" alt=\"\" width=\"487\" height=\"217\"> <b>Figure 8<\/b>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nRewrite the [latex]13{x}^{2}-6\\sqrt{3}xy+7{y}^{2}=16[\/latex] in the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] system without the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] term.\n\n[reveal-answer q=\"452974\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"452974\"]\n\n[latex]\\begin{align}\\frac{{{x}^{\\prime }}^{2}}{4}+\\frac{{{y}^{\\prime }}^{2}}{1}=1\\end{align}[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 4: Graphing an Equation That Has No <em>x\u2032y\u2032<\/em> Terms<\/h3>\nGraph the following equation relative to the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] system:\n<p style=\"text-align: center;\">[latex]{x}^{2}+12xy - 4{y}^{2}=30[\/latex]<\/p>\n[reveal-answer q=\"16063\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"16063\"]\n\nFirst, we find [latex]\\cot \\left(2\\theta \\right)[\/latex].\n<p style=\"text-align: center;\">[latex]{x}^{2}+12xy - 4{y}^{2}=20\\Rightarrow A=1,B=12,\\text{ and }C=-4[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;\\cot \\left(2\\theta \\right)=\\frac{A-C}{B} \\\\ &amp;\\cot \\left(2\\theta \\right)=\\frac{1-\\left(-4\\right)}{12} \\\\ &amp;\\cot \\left(2\\theta \\right)=\\frac{5}{12} \\end{align}[\/latex]<\/p>\nBecause [latex]\\cot \\left(2\\theta \\right)=\\frac{5}{12}[\/latex], we can draw a reference triangle as in Figure 9.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183048\/CNX_Precalc_Figure_10_04_0092.jpg\" alt=\"\" width=\"487\" height=\"591\"> <b>Figure 9<\/b>[\/caption]\n\n<div style=\"text-align: center;\"><\/div>\n<p style=\"text-align: center;\">[latex]\\cot \\left(2\\theta \\right)=\\frac{5}{12}=\\frac{\\text{adjacent}}{\\text{opposite}}[\/latex]<\/p>\nThus, the hypotenuse is\n<p style=\"text-align: center;\">[latex]\\begin{gathered} {5}^{2}+{12}^{2}={h}^{2}\\\\ 25+144={h}^{2}\\\\ 169={h}^{2}\\\\ h=13\\end{gathered}[\/latex]<\/p>\nNext, we find [latex]\\sin \\theta [\/latex] and [latex]\\cos \\theta [\/latex]. We will use half-angle identities.\n<p style=\"text-align: center;\">[latex]\\begin{align} \\sin \\theta &amp;=\\sqrt{\\frac{1-\\cos \\left(2\\theta \\right)}{2}} \\\\ &amp;=\\sqrt{\\frac{1-\\frac{5}{13}}{2}} \\\\ &amp;=\\sqrt{\\frac{\\frac{13}{13}-\\frac{5}{13}}{2}} \\\\&amp;=\\sqrt{\\frac{8}{13}\\cdot \\frac{1}{2}} \\\\ &amp;=\\frac{2}{\\sqrt{13}} \\\\ \\cos \\theta &amp;=\\sqrt{\\frac{1+\\cos \\left(2\\theta \\right)}{2}} \\\\ &amp;=\\sqrt{\\frac{1+\\frac{5}{13}}{2}} \\\\ &amp;=\\sqrt{\\frac{\\frac{13}{13}+\\frac{5}{13}}{2}} \\\\ &amp;=\\sqrt{\\frac{18}{13}\\cdot \\frac{1}{2}} \\\\ &amp;=\\frac{3}{\\sqrt{13}} \\end{align}[\/latex]<\/p>\nNow we find [latex]x[\/latex] and [latex]y[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\\\ &amp;x={x}^{\\prime }\\left(\\frac{3}{\\sqrt{13}}\\right)-{y}^{\\prime }\\left(\\frac{2}{\\sqrt{13}}\\right) \\\\ &amp;x=\\frac{3{x}^{\\prime }-2{y}^{\\prime }}{\\sqrt{13}} \\end{align}[\/latex]<\/p>\nand\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\\\ &amp;y={x}^{\\prime }\\left(\\frac{2}{\\sqrt{13}}\\right)+{y}^{\\prime }\\left(\\frac{3}{\\sqrt{13}}\\right) \\\\ &amp;y=\\frac{2{x}^{\\prime }+3{y}^{\\prime }}{\\sqrt{13}} \\end{align}[\/latex]<\/p>\nNow we substitute [latex]\\begin{align}x=\\frac{3{x}^{\\prime }-2{y}^{\\prime }}{\\sqrt{13}}\\end{align}[\/latex] and [latex]\\begin{align}y=\\frac{2{x}^{\\prime }+3{y}^{\\prime }}{\\sqrt{13}}\\end{align}[\/latex] into [latex]\\begin{align}{x}^{2}+12xy - 4{y}^{2}=30\\end{align}[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;{\\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 \\\\ &amp;\\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 &amp;&amp; \\text{Factor}. \\\\ &amp;\\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 &amp;&amp; \\text{Multiply}. \\\\ &amp;\\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 &amp;&amp; \\text{Distribute}. \\\\ &amp;\\left(\\frac{1}{13}\\right)\\left[65{x}^{\\prime }{}^{2}-104{y}^{\\prime }{}^{2}\\right]=30 &amp;&amp; \\text{Combine like terms}. \\\\ &amp;65{x}^{\\prime }{}^{2}-104{y}^{\\prime }{}^{2}=390 &amp;&amp; \\text{Multiply}. \\\\ &amp;\\frac{{x}^{\\prime }{}^{2}}{6}-\\frac{4{y}^{\\prime }{}^{2}}{15}=1 &amp;&amp; \\text{Divide by 390}. \\end{align}[\/latex]<\/p>\nFigure 10 shows the graph of the hyperbola [latex]\\begin{align}\\frac{{{x}^{\\prime }}^{2}}{6}-\\frac{4{{y}^{\\prime }}^{2}}{15}=1\\end{align}[\/latex].\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183050\/CNX_Precalc_Figure_10_04_0102.jpg\" alt=\"\" width=\"487\" height=\"441\"> <b>Figure 10<\/b>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<h2>Identifying Conics without Rotating Axes<\/h2>\nNow 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\n<div style=\"text-align: center;\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\nIf we apply the rotation formulas to this equation we get the form\n<div style=\"text-align: center;\">[latex]\\begin{align}{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\\end{align}[\/latex]<\/div>\nIt may be shown that [latex]\\begin{align}{B}^{2}-4AC={{B}^{\\prime }}^{2}-4{A}^{\\prime }{C}^{\\prime }\\end{align}[\/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.\n<div class=\"textbox\">\n<h3>A General Note: Using the Discriminant to Identify a Conic<\/h3>\nIf the equation [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex] is transformed by rotating axes into the equation [latex]\\begin{align}{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\\end{align}[\/latex], then [latex]\\begin{align}{B}^{2}-4AC={{B}^{\\prime }}^{2}-4{A}^{\\prime }{C}^{\\prime }\\end{align}[\/latex].\n\nThe 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.\n\nIf the discriminant, [latex]{B}^{2}-4AC[\/latex], is\n<ul>\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 class=\"textbox shaded\">\n<h3>Example 5: Identifying the Conic without Rotating Axes<\/h3>\nIdentify the conic for each of the following without rotating axes.\n<ol>\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[reveal-answer q=\"255593\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"255593\"]\n<ol>\n \t<li>Let\u2019s begin by determining [latex]A,B[\/latex], and [latex]C[\/latex].\n<div style=\"text-align: center;\">[latex]\\underset{A}{\\underbrace{5}}{x}^{2}+\\underset{B}{\\underbrace{2\\sqrt{3}}}xy+\\underset{C}{\\underbrace{2}}{y}^{2}-5=0[\/latex]<\/div>\nNow, we find the discriminant.\n<div style=\"text-align: center;\">[latex]\\begin{align}{B}^{2}-4AC&amp;={\\left(2\\sqrt{3}\\right)}^{2}-4\\left(5\\right)\\left(2\\right) \\\\ &amp;=4\\left(3\\right)-40 \\\\ &amp;=12 - 40 \\\\ &amp;=-28&lt;0 \\end{align}[\/latex]<\/div>\nTherefore, [latex]5{x}^{2}+2\\sqrt{3}xy+2{y}^{2}-5=0[\/latex] represents an ellipse.<\/li>\n \t<li>Again, let\u2019s begin by determining [latex]A,B[\/latex], and [latex]C[\/latex].\n<div style=\"text-align: center;\">[latex]\\underset{A}{\\underbrace{5}}{x}^{2}+\\underset{B}{\\underbrace{2\\sqrt{3}}}xy+\\underset{C}{\\underbrace{12}}{y}^{2}-5=0[\/latex]<\/div>\nNow, we find the discriminant.\n<div style=\"text-align: center;\">[latex]\\begin{align}{B}^{2}-4AC&amp;={\\left(2\\sqrt{3}\\right)}^{2}-4\\left(5\\right)\\left(12\\right) \\\\ &amp;=4\\left(3\\right)-240 \\\\ &amp;=12 - 240 \\\\ &amp;=-228&lt;0 \\end{align}[\/latex]<\/div>\nTherefore, [latex]5{x}^{2}+2\\sqrt{3}xy+12{y}^{2}-5=0[\/latex] represents an ellipse.<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nIdentify the conic for each of the following without rotating axes.\n<ol>\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[reveal-answer q=\"67134\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"67134\"]\n<ol>\n \t<li>hyperbola<\/li>\n \t<li>ellipse<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<h2>Key Equations<\/h2>\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{align}&amp;x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\\\ &amp;y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{align}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Angle of rotation<\/td>\n<td>[latex]\\theta ,\\text{ where }\\cot \\left(2\\theta \\right)=\\frac{A-C}{B}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\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.<\/li>\n \t<li>Equations of conic sections with an [latex]xy[\/latex] term have been rotated about the origin.<\/li>\n \t<li>The general form can be transformed into an equation in the [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] coordinate system without the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] term.<\/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.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id2107200\" class=\"definition\">\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]\\cot \\left(2\\theta \\right)&gt;0[\/latex], then [latex]\\theta [\/latex] is between [latex]\\left(0^\\circ ,45^\\circ \\right)[\/latex]; if [latex]\\cot \\left(2\\theta \\right)&lt;0[\/latex], then [latex]\\theta [\/latex] is between [latex]\\left(45^\\circ ,90^\\circ \\right)[\/latex]; and if [latex]\\cot \\left(2\\theta \\right)=0[\/latex], then [latex]\\theta =45^\\circ [\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id2165541\" class=\"definition\">\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\" class=\"definition\">\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","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify nondegenerate conic sections given their general form equations.<\/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<p>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.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183021\/CNX_Precalc_Figure_10_04_0012.jpg\" alt=\"\" width=\"975\" height=\"650\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1.<\/b> The nondegenerate conic sections<\/p>\n<\/div>\n<p>Ellipses, circles, hyperbolas, and parabolas are sometimes called the <strong>nondegenerate conic sections<\/strong>, in contrast to the <strong>degenerate conic sections<\/strong>, which are shown in Figure 2. 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.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183025\/CNX_Precalc_Figure_10_04_002n2.jpg\" alt=\"\" width=\"975\" height=\"719\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2.<\/b> Degenerate conic sections<\/p>\n<\/div>\n<h2>Identifying Nondegenerate Conics in General Form<\/h2>\n<p>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 style=\"text-align: center;\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\n<p>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>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 class=\"textbox\">\n<h3>A General Note: General Form of Conic Sections<\/h3>\n<p>A <strong>nondegenerate conic section<\/strong> has the general form<\/p>\n<p style=\"text-align: center;\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/p>\n<p>where [latex]A,B[\/latex], and [latex]C[\/latex] are not all zero.<\/p>\n<p>The table below 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<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 class=\"textbox\">\n<h3>How To: Given the equation of a conic, identify the type of conic.<strong><br \/>\n<\/strong><\/h3>\n<ol>\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>\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 is 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 is a circle.<\/li>\n<li>If [latex]A[\/latex] and [latex]C[\/latex] are nonzero and have opposite signs, then the graph is a hyperbola.<\/li>\n<li>If either [latex]A[\/latex] or [latex]C[\/latex] is zero, then the graph is a parabola.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Identifying a Conic from Its General Form<\/h3>\n<p>Identify the graph of each of the following nondegenerate conic sections.<\/p>\n<ol>\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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q722128\">Show Solution<\/span><\/p>\n<div id=\"q722128\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Rewriting the general form, we have<br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183027\/eq1_n2.jpg\" alt=\"\" \/><br \/>\n[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.<\/li>\n<li>Rewriting the general form, we have<br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183029\/eq2_n2.jpg\" alt=\"\" \/>[latex]A=0[\/latex] and [latex]C=9[\/latex]. We can determine that the equation is a parabola, since [latex]A[\/latex] is zero.<\/li>\n<li>Rewriting the general form, we have<br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183031\/eq3_n2.jpg\" alt=\"\" \/>[latex]A=3[\/latex] and [latex]C=3[\/latex]. Because [latex]A=C[\/latex], the graph of this equation is a circle.<\/li>\n<li>Rewriting the general form, we have <img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183033\/eq42.jpg\" alt=\"\" \/>[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.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Identify the graph of each of the following nondegenerate conic sections.<\/p>\n<ol>\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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q185596\">Show Solution<\/span><\/p>\n<div id=\"q185596\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>hyperbola<\/li>\n<li>ellipse<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Finding a New Representation of the Given Equation after Rotating through a Given Angle<\/h2>\n<p>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]\\begin{align}\\left({x}^{\\prime },{y}^{\\prime }\\right)\\end{align}[\/latex] on the new plane defined by the new, rotated axes, called the <em>x&#8217;<\/em>-axis and <em>y&#8217;<\/em>-axis.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183035\/CNX_Precalc_Figure_10_04_0032.jpg\" alt=\"\" width=\"487\" height=\"441\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3.<\/b> The graph of the rotated ellipse [latex]{x}^{2}+{y}^{2}-xy - 15=0[\/latex]<\/p>\n<\/div>\n<p>We will find the relationships between [latex]x[\/latex] and [latex]y[\/latex] on the Cartesian plane with [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] on the new rotated plane.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183037\/CNX_Precalc_Figure_10_04_0042.jpg\" alt=\"\" width=\"487\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4.<\/b> 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]\\theta [\/latex].<\/p>\n<\/div>\n<p>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]\\begin{align}{i}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{j}^{\\prime }\\end{align}[\/latex]. The angle [latex]\\theta[\/latex] is known as the <strong>angle of rotation<\/strong>. We may write the new unit vectors in terms of the original ones.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}&{i}^{\\prime }=i\\cos \\theta +j\\sin \\theta \\\\ &{j}^{\\prime }=-i\\sin \\theta +j\\cos \\theta \\end{align}[\/latex]<\/div>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183039\/CNX_Precalc_Figure_10_04_0052.jpg\" alt=\"\" width=\"487\" height=\"364\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5.<\/b> Relationship between the old and new coordinate planes.<\/p>\n<\/div>\n<p>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 style=\"text-align: center;\">[latex]\\begin{align}&u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime } \\\\ &u={x}^{\\prime }\\left(i\\cos \\theta +j\\sin \\theta \\right)+{y}^{\\prime }\\left(-i\\sin \\theta +j\\cos \\theta \\right) && \\text{Substitute}. \\\\ &u=ix^{\\prime}\\cos \\theta +jx^{\\prime}\\sin \\theta -iy^{\\prime}\\sin \\theta +jy^{\\prime}\\cos \\theta && \\text{Distribute}. \\\\ &u=ix^{\\prime}\\cos \\theta -iy^{\\prime}\\sin \\theta +jx^{\\prime}\\sin \\theta +jy^{\\prime}\\cos \\theta && \\text{Apply commutative property}. \\\\ &u=\\left(x^{\\prime}\\cos \\theta -y^{\\prime}\\sin \\theta \\right)i+\\left(x^{\\prime}\\sin \\theta +y^{\\prime}\\cos \\theta \\right)j && \\text{Factor by grouping}. \\end{align}[\/latex]<\/div>\n<p>Because [latex]\\begin{align}u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime }\\end{align}[\/latex], we have representations of [latex]x[\/latex] and [latex]y[\/latex] in terms of the new coordinate system.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\\\ \\text{and}\\\\ y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{gathered}[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Equations of Rotation<\/h3>\n<p>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]\\begin{align}\\left({x}^{\\prime },{y}^{\\prime }\\right)\\end{align}[\/latex]. We can use the following equations of rotation to define the relationship between [latex]\\begin{align}\\left(x,y\\right)\\end{align}[\/latex] and [latex]\\begin{align}\\left({x}^{\\prime },{y}^{\\prime }\\right):\\end{align}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\\\ \\text{and}\\\\ y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{gathered}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the equation of a conic, find a new representation after rotating through an angle.<strong><br \/>\n<\/strong><\/h3>\n<ol id=\"fs-id1146233\">\n<li>Find [latex]x[\/latex] and [latex]y[\/latex] where [latex]\\begin{align}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\end{align}[\/latex] and [latex]\\begin{align}y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{align}[\/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]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] in standard form.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Finding a New Representation of an Equation after Rotating through a Given Angle<\/h3>\n<p>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^\\circ[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q303707\">Show Solution<\/span><\/p>\n<div id=\"q303707\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find [latex]x[\/latex] and [latex]y[\/latex], where [latex]\\begin{align}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\end{align}[\/latex] and [latex]\\begin{align} y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{align}[\/latex].<\/p>\n<p>Because [latex]\\theta =45^\\circ[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &x={x}^{\\prime }\\cos \\left(45^\\circ \\right)-{y}^{\\prime }\\sin \\left(45^\\circ \\right) \\\\ &x={x}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)-{y}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right) \\\\ &x=\\frac{{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{2}} \\end{align}[\/latex]<\/p>\n<p>and<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&y={x}^{\\prime }\\sin \\left(45^\\circ \\right)+{y}^{\\prime }\\cos \\left(45^\\circ \\right) \\\\ &y={x}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)+{y}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right) \\\\ &y=\\frac{{x}^{\\prime }+{y}^{\\prime }}{\\sqrt{2}} \\end{align}[\/latex]<\/p>\n<p>Substitute [latex]\\begin{align}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\end{align}[\/latex] and [latex]\\begin{align}y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{align}[\/latex] into [latex]2{x}^{2}-xy+2{y}^{2}-30=0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} 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\\end{align}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&2\\frac{\\left({x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }-{y}^{\\prime }\\right)}{2}-\\frac{\\left({x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }+{y}^{\\prime }\\right)}{2}+2\\frac{\\left({x}^{\\prime }+{y}^{\\prime }\\right)\\left({x}^{\\prime }+{y}^{\\prime }\\right)}{2}-30=0 && \\text{FOIL method} \\\\ &{x}^{\\prime }{}^{2}{-2{x}^{\\prime }y}^{\\prime }+{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}+{x}^{\\prime }{}^{2}+2{x}^{\\prime }{y}^{\\prime }+{y}^{\\prime }{}^{2}-30=0 && \\text{Combine like terms}. \\\\ &2{x}^{\\prime }{}^{2}+2{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}=30 && \\text{Combine like terms}. \\\\ &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) && \\text{Multiply both sides by 2}. \\\\ &4{x}^{\\prime }{}^{2}+4{y}^{\\prime }{}^{2}-\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)=60 && \\text{Simplify}. \\\\ &4{x}^{\\prime }{}^{2}+4{y}^{\\prime }{}^{2}-{x}^{\\prime }{}^{2}+{y}^{\\prime }{}^{2}=60 && \\text{Distribute}. \\\\ &\\frac{3{x}^{\\prime }{}^{2}}{60}+\\frac{5{y}^{\\prime }{}^{2}}{60}=\\frac{60}{60} && \\text{Set equal to 1}. \\end{align}[\/latex]<\/p>\n<p>Write the equations with [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] in the standard form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{{{x}^{\\prime }}^{2}}{20}+\\frac{{{y}^{\\prime }}^{2}}{12}=1\\end{align}[\/latex]<\/p>\n<p>This equation is an ellipse. Figure 6&nbsp;shows the graph.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183042\/CNX_Precalc_Figure_10_04_0062.jpg\" alt=\"\" width=\"487\" height=\"441\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Writing Equations of Rotated Conics in Standard Form<\/h2>\n<p>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 style=\"text-align: center;\">[latex]\\cot \\left(2\\theta \\right)=\\frac{A-C}{B}[\/latex]<\/div>\n<p>We have learned already that any conic may be represented by the second degree equation<\/p>\n<div style=\"text-align: center;\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\n<p>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]\\cot \\left(2\\theta \\right)=\\frac{A-C}{B}[\/latex].<\/p>\n<div>\n<ul>\n<li>If [latex]\\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^\\circ ,45^\\circ \\right)[\/latex].<\/li>\n<li>If [latex]\\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^\\circ ,90^\\circ \\right)[\/latex].<\/li>\n<li>If [latex]A=C[\/latex], then [latex]\\theta =45^\\circ[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an equation for a conic in the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] system, rewrite the equation without the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] term in terms of [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex], where the [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] axes are rotations of the standard axes by [latex]\\theta[\/latex] degrees.<\/h3>\n<ol>\n<li>Find [latex]\\cot \\left(2\\theta \\right)[\/latex].<\/li>\n<li>Find [latex]\\sin \\theta[\/latex] and [latex]\\cos \\theta[\/latex].<\/li>\n<li>Substitute [latex]\\sin \\theta[\/latex] and [latex]\\cos \\theta[\/latex] into [latex]\\begin{align}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\end{align}[\/latex] and [latex]\\begin{align} y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{align}[\/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]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] in the standard form with respect to the rotated axes.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: 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>Rewrite the equation [latex]8{x}^{2}-12xy+17{y}^{2}=20[\/latex] in the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] system without an [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] term.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q952851\">Show Solution<\/span><\/p>\n<div id=\"q952851\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we find [latex]\\cot \\left(2\\theta \\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}8{x}^{2}-12xy+17{y}^{2}=20\\Rightarrow A=8,B=-12\\text{ and }C=17\\end{align}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &\\cot \\left(2\\theta \\right)=\\frac{A-C}{B}=\\frac{8 - 17}{-12} \\\\ &\\cot \\left(2\\theta \\right)=\\frac{-9}{-12}=\\frac{3}{4} \\end{align}[\/latex]<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183044\/CNX_Precalc_Figure_10_04_0072.jpg\" alt=\"\" width=\"487\" height=\"328\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7<\/b><\/p>\n<\/div>\n<div style=\"text-align: center;\"><\/div>\n<p style=\"text-align: center;\">[latex]\\cot \\left(2\\theta \\right)=\\frac{3}{4}=\\frac{\\text{adjacent}}{\\text{opposite}}[\/latex]<\/p>\n<p>So the hypotenuse is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered} {3}^{2}+{4}^{2}={h}^{2}\\\\ 9+16={h}^{2}\\\\ 25={h}^{2}\\\\ h=5\\end{gathered}[\/latex]<\/p>\n<p>Next, we find [latex]\\sin \\text{ }\\theta[\/latex] and [latex]\\cos \\text{ }\\theta[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} \\sin \\theta &=\\sqrt{\\frac{1-\\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}} \\\\ &=\\frac{1}{\\sqrt{5}} \\\\[2mm] \\cos \\theta &=\\sqrt{\\frac{1+\\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}} \\\\ &=\\frac{2}{\\sqrt{5}} \\end{align}[\/latex]<\/p>\n<p>Substitute the values of [latex]\\sin \\text{ }\\theta[\/latex] and [latex]\\cos \\text{ }\\theta[\/latex] into [latex]\\begin{align}x={x}^{\\prime }\\cos \\text{ }\\theta -{y}^{\\prime }\\sin \\text{ }\\theta \\end{align}[\/latex] and [latex]\\begin{align}y={x}^{\\prime }\\sin \\text{ }\\theta +{y}^{\\prime }\\cos \\text{ }\\theta \\end{align}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\\\ &x={x}^{\\prime }\\left(\\frac{2}{\\sqrt{5}}\\right)-{y}^{\\prime }\\left(\\frac{1}{\\sqrt{5}}\\right) \\\\ &x=\\frac{2{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{5}} \\end{align}[\/latex]<\/p>\n<p>and<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\\\ &y={x}^{\\prime }\\left(\\frac{1}{\\sqrt{5}}\\right)+{y}^{\\prime }\\left(\\frac{2}{\\sqrt{5}}\\right) \\\\ &y=\\frac{{x}^{\\prime }+2{y}^{\\prime }}{\\sqrt{5}} \\end{align}[\/latex]<\/p>\n<p>Substitute the expressions for [latex]x[\/latex] and [latex]y[\/latex] into in the given equation, and then simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}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 \\\\ 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 \\\\ 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 \\\\ 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 \\\\ 25{x}^{\\prime }{}^{2}+100{y}^{\\prime }{}^{2}=100 \\\\ \\frac{25}{100}{x}^{\\prime }{}^{2}+\\frac{100}{100}{y}^{\\prime }{}^{2}=\\frac{100}{100} \\end{gathered}[\/latex]<\/p>\n<p>Write the equations with [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] in the standard form with respect to the new coordinate system.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{{{x}^{\\prime }}^{2}}{4}+\\frac{{{y}^{\\prime }}^{2}}{1}=1\\end{align}[\/latex]<\/p>\n<p>Figure 8 shows the graph of the ellipse.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183046\/CNX_Precalc_Figure_10_04_0082.jpg\" alt=\"\" width=\"487\" height=\"217\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Rewrite the [latex]13{x}^{2}-6\\sqrt{3}xy+7{y}^{2}=16[\/latex] in the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] system without the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] term.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q452974\">Show Solution<\/span><\/p>\n<div id=\"q452974\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{align}\\frac{{{x}^{\\prime }}^{2}}{4}+\\frac{{{y}^{\\prime }}^{2}}{1}=1\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 4: Graphing an Equation That Has No <em>x\u2032y\u2032<\/em> Terms<\/h3>\n<p>Graph the following equation relative to the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] system:<\/p>\n<p style=\"text-align: center;\">[latex]{x}^{2}+12xy - 4{y}^{2}=30[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q16063\">Show Solution<\/span><\/p>\n<div id=\"q16063\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we find [latex]\\cot \\left(2\\theta \\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]{x}^{2}+12xy - 4{y}^{2}=20\\Rightarrow A=1,B=12,\\text{ and }C=-4[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&\\cot \\left(2\\theta \\right)=\\frac{A-C}{B} \\\\ &\\cot \\left(2\\theta \\right)=\\frac{1-\\left(-4\\right)}{12} \\\\ &\\cot \\left(2\\theta \\right)=\\frac{5}{12} \\end{align}[\/latex]<\/p>\n<p>Because [latex]\\cot \\left(2\\theta \\right)=\\frac{5}{12}[\/latex], we can draw a reference triangle as in Figure 9.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183048\/CNX_Precalc_Figure_10_04_0092.jpg\" alt=\"\" width=\"487\" height=\"591\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 9<\/b><\/p>\n<\/div>\n<div style=\"text-align: center;\"><\/div>\n<p style=\"text-align: center;\">[latex]\\cot \\left(2\\theta \\right)=\\frac{5}{12}=\\frac{\\text{adjacent}}{\\text{opposite}}[\/latex]<\/p>\n<p>Thus, the hypotenuse is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered} {5}^{2}+{12}^{2}={h}^{2}\\\\ 25+144={h}^{2}\\\\ 169={h}^{2}\\\\ h=13\\end{gathered}[\/latex]<\/p>\n<p>Next, we find [latex]\\sin \\theta[\/latex] and [latex]\\cos \\theta[\/latex]. We will use half-angle identities.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} \\sin \\theta &=\\sqrt{\\frac{1-\\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}} \\\\ \\cos \\theta &=\\sqrt{\\frac{1+\\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}} \\end{align}[\/latex]<\/p>\n<p>Now we find [latex]x[\/latex] and [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\\\ &x={x}^{\\prime }\\left(\\frac{3}{\\sqrt{13}}\\right)-{y}^{\\prime }\\left(\\frac{2}{\\sqrt{13}}\\right) \\\\ &x=\\frac{3{x}^{\\prime }-2{y}^{\\prime }}{\\sqrt{13}} \\end{align}[\/latex]<\/p>\n<p>and<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\\\ &y={x}^{\\prime }\\left(\\frac{2}{\\sqrt{13}}\\right)+{y}^{\\prime }\\left(\\frac{3}{\\sqrt{13}}\\right) \\\\ &y=\\frac{2{x}^{\\prime }+3{y}^{\\prime }}{\\sqrt{13}} \\end{align}[\/latex]<\/p>\n<p>Now we substitute [latex]\\begin{align}x=\\frac{3{x}^{\\prime }-2{y}^{\\prime }}{\\sqrt{13}}\\end{align}[\/latex] and [latex]\\begin{align}y=\\frac{2{x}^{\\prime }+3{y}^{\\prime }}{\\sqrt{13}}\\end{align}[\/latex] into [latex]\\begin{align}{x}^{2}+12xy - 4{y}^{2}=30\\end{align}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &{\\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 \\\\ &\\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 && \\text{Factor}. \\\\ &\\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 && \\text{Multiply}. \\\\ &\\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 && \\text{Distribute}. \\\\ &\\left(\\frac{1}{13}\\right)\\left[65{x}^{\\prime }{}^{2}-104{y}^{\\prime }{}^{2}\\right]=30 && \\text{Combine like terms}. \\\\ &65{x}^{\\prime }{}^{2}-104{y}^{\\prime }{}^{2}=390 && \\text{Multiply}. \\\\ &\\frac{{x}^{\\prime }{}^{2}}{6}-\\frac{4{y}^{\\prime }{}^{2}}{15}=1 && \\text{Divide by 390}. \\end{align}[\/latex]<\/p>\n<p>Figure 10 shows the graph of the hyperbola [latex]\\begin{align}\\frac{{{x}^{\\prime }}^{2}}{6}-\\frac{4{{y}^{\\prime }}^{2}}{15}=1\\end{align}[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183050\/CNX_Precalc_Figure_10_04_0102.jpg\" alt=\"\" width=\"487\" height=\"441\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Identifying Conics without Rotating Axes<\/h2>\n<p>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 style=\"text-align: center;\">[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\n<p>If we apply the rotation formulas to this equation we get the form<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}{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\\end{align}[\/latex]<\/div>\n<p>It may be shown that [latex]\\begin{align}{B}^{2}-4AC={{B}^{\\prime }}^{2}-4{A}^{\\prime }{C}^{\\prime }\\end{align}[\/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 class=\"textbox\">\n<h3>A General Note: Using the Discriminant to Identify a Conic<\/h3>\n<p>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]\\begin{align}{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\\end{align}[\/latex], then [latex]\\begin{align}{B}^{2}-4AC={{B}^{\\prime }}^{2}-4{A}^{\\prime }{C}^{\\prime }\\end{align}[\/latex].<\/p>\n<p>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>If the discriminant, [latex]{B}^{2}-4AC[\/latex], is<\/p>\n<ul>\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 class=\"textbox shaded\">\n<h3>Example 5: Identifying the Conic without Rotating Axes<\/h3>\n<p>Identify the conic for each of the following without rotating axes.<\/p>\n<ol>\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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q255593\">Show Solution<\/span><\/p>\n<div id=\"q255593\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Let\u2019s begin by determining [latex]A,B[\/latex], and [latex]C[\/latex].\n<div style=\"text-align: center;\">[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>Now, we find the discriminant.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}{B}^{2}-4AC&={\\left(2\\sqrt{3}\\right)}^{2}-4\\left(5\\right)\\left(2\\right) \\\\ &=4\\left(3\\right)-40 \\\\ &=12 - 40 \\\\ &=-28<0 \\end{align}[\/latex]<\/div>\n<p>Therefore, [latex]5{x}^{2}+2\\sqrt{3}xy+2{y}^{2}-5=0[\/latex] represents an ellipse.<\/li>\n<li>Again, let\u2019s begin by determining [latex]A,B[\/latex], and [latex]C[\/latex].\n<div style=\"text-align: center;\">[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>Now, we find the discriminant.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}{B}^{2}-4AC&={\\left(2\\sqrt{3}\\right)}^{2}-4\\left(5\\right)\\left(12\\right) \\\\ &=4\\left(3\\right)-240 \\\\ &=12 - 240 \\\\ &=-228<0 \\end{align}[\/latex]<\/div>\n<p>Therefore, [latex]5{x}^{2}+2\\sqrt{3}xy+12{y}^{2}-5=0[\/latex] represents an ellipse.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Identify the conic for each of the following without rotating axes.<\/p>\n<ol>\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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q67134\">Show Solution<\/span><\/p>\n<div id=\"q67134\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>hyperbola<\/li>\n<li>ellipse<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Key Equations<\/h2>\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{align}&x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\\\ &y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{align}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Angle of rotation<\/td>\n<td>[latex]\\theta ,\\text{ where }\\cot \\left(2\\theta \\right)=\\frac{A-C}{B}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\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.<\/li>\n<li>Equations of conic sections with an [latex]xy[\/latex] term have been rotated about the origin.<\/li>\n<li>The general form can be transformed into an equation in the [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] coordinate system without the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] term.<\/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.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id2107200\" class=\"definition\">\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]\\cot \\left(2\\theta \\right)>0[\/latex], then [latex]\\theta[\/latex] is between [latex]\\left(0^\\circ ,45^\\circ \\right)[\/latex]; if [latex]\\cot \\left(2\\theta \\right)<0[\/latex], then [latex]\\theta[\/latex] is between [latex]\\left(45^\\circ ,90^\\circ \\right)[\/latex]; and if [latex]\\cot \\left(2\\theta \\right)=0[\/latex], then [latex]\\theta =45^\\circ[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id2165541\" class=\"definition\">\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\" class=\"definition\">\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\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1433\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":503070,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1433","chapter","type-chapter","status-publish","hentry"],"part":1429,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1433","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/users\/503070"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1433\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/parts\/1429"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1433\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/media?parent=1433"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapter-type?post=1433"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/contributor?post=1433"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/license?post=1433"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}