{"id":1600,"date":"2025-07-25T04:02:34","date_gmt":"2025-07-25T04:02:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1600"},"modified":"2026-03-12T07:19:37","modified_gmt":"2026-03-12T07:19:37","slug":"rotation-of-axes-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rotation-of-axes-fresh-take\/","title":{"raw":"Rotation of Axes: Fresh Take","rendered":"Rotation of Axes: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Identify nondegenerate conic sections given their general form equations.<\/li>\r\n \t<li>Write equations of rotated conics in standard form.<\/li>\r\n \t<li>Identify conics without rotating axes.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Identifying Conic Sections from the General Equation<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"81\" data-end=\"143\">The <strong data-start=\"85\" data-end=\"119\">general second-degree equation<\/strong> for conic sections is [latex]Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0[\/latex].<\/p>\r\n<p data-start=\"192\" data-end=\"432\">Depending on the coefficients, this equation can represent a <strong data-start=\"253\" data-end=\"296\">circle, ellipse, parabola, or hyperbola<\/strong>. These are called <em data-start=\"315\" data-end=\"337\">nondegenerate conics<\/em>. Identifying which conic you have involves looking at the coefficients [latex]A,B,C[\/latex].<\/p>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: How to Identify the Conic<\/strong>\r\n<ol>\r\n \t<li data-start=\"484\" data-end=\"720\">\r\n<p data-start=\"487\" data-end=\"534\"><strong data-start=\"487\" data-end=\"532\">Check the Cross-Term ([latex]Bxy[\/latex])<\/strong><\/p>\r\n\r\n<ul data-start=\"538\" data-end=\"720\">\r\n \t<li data-start=\"538\" data-end=\"656\">\r\n<p data-start=\"540\" data-end=\"656\">If [latex]B\\ne 0[\/latex], the conic is rotated. Identification requires more advanced analysis (rotation of axes).<\/p>\r\n<\/li>\r\n \t<li data-start=\"660\" data-end=\"720\">\r\n<p data-start=\"662\" data-end=\"720\">If [latex]B=0[\/latex], proceed with simpler tests below.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"722\" data-end=\"862\">\r\n<p data-start=\"725\" data-end=\"737\"><strong data-start=\"725\" data-end=\"735\">Circle<\/strong><\/p>\r\n\r\n<ul data-start=\"741\" data-end=\"862\">\r\n \t<li data-start=\"741\" data-end=\"790\">\r\n<p data-start=\"743\" data-end=\"790\">[latex]A=C \\ne 0[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"794\" data-end=\"862\">\r\n<p data-start=\"796\" data-end=\"862\">Example: [latex]x^{2}+y^{2}-16=0[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"864\" data-end=\"1027\">\r\n<p data-start=\"867\" data-end=\"893\"><strong data-start=\"867\" data-end=\"891\">Ellipse (not circle)<\/strong><\/p>\r\n\r\n<ul data-start=\"897\" data-end=\"1027\">\r\n \t<li data-start=\"897\" data-end=\"959\">\r\n<p data-start=\"899\" data-end=\"959\">[latex]A \\ne C[\/latex], both positive.<\/p>\r\n<\/li>\r\n \t<li data-start=\"963\" data-end=\"1027\">\r\n<p data-start=\"965\" data-end=\"1027\">Example: [latex]\\dfrac{x^{2}}{9}+\\dfrac{y^{2}}{4}=1[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1029\" data-end=\"1167\">\r\n<p data-start=\"1032\" data-end=\"1046\"><strong data-start=\"1032\" data-end=\"1044\">Parabola<\/strong><\/p>\r\n\r\n<ul data-start=\"1050\" data-end=\"1167\">\r\n \t<li data-start=\"1050\" data-end=\"1126\">\r\n<p data-start=\"1052\" data-end=\"1126\">One squared term only (either [latex]A=0[\/latex] or [latex]C=0[\/latex]).<\/p>\r\n<\/li>\r\n \t<li data-start=\"1130\" data-end=\"1167\">\r\n<p data-start=\"1132\" data-end=\"1167\">Example: [latex]y^{2}=8x[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1169\" data-end=\"1321\">\r\n<p data-start=\"1172\" data-end=\"1187\"><strong data-start=\"1172\" data-end=\"1185\">Hyperbola<\/strong><\/p>\r\n\r\n<ul data-start=\"1573\" data-end=\"1647\">\r\n \t<li data-start=\"1191\" data-end=\"1253\">\r\n<p data-start=\"1193\" data-end=\"1253\">[latex]A[\/latex] and [latex]C[\/latex] have opposite signs.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1257\" data-end=\"1321\">\r\n<p data-start=\"1259\" data-end=\"1321\">Example: [latex]\\dfrac{x^{2}}{9}-\\dfrac{y^{2}}{4}=1[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">Identify the type of conic section represented by each equation.\r\na) [latex]4x^2 + 4y^2 - 16 = 0[\/latex]\r\nb) [latex]9x^2 + 16y^2 - 144 = 0[\/latex]\r\nc) [latex]y^2 - 8x = 0[\/latex]\r\nd) [latex]4x^2 - 9y^2 = 36[\/latex]\r\n[reveal-answer q=\"conic001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"conic001\"]\r\na) [latex]4x^2 + 4y^2 - 16 = 0[\/latex]\r\nIdentify the coefficients:\r\n[latex]A = 4, \\quad B = 0, \\quad C = 4[\/latex]\r\nSince [latex]A = C[\/latex] and [latex]B = 0[\/latex], this is a circle.b) [latex]9x^2 + 16y^2 - 144 = 0[\/latex]\r\nIdentify the coefficients:\r\n[latex]A = 9, \\quad B = 0, \\quad C = 16[\/latex]\r\nSince [latex]A \\neq C[\/latex], both are positive, and [latex]B = 0[\/latex], this is an ellipse.c) [latex]y^2 - 8x = 0[\/latex]\r\nIdentify the coefficients:\r\n[latex]A = 0, \\quad B = 0, \\quad C = 1[\/latex]\r\nOnly one squared term is present ([latex]C = 1[\/latex], [latex]A = 0[\/latex]), so this is a parabola.d) [latex]4x^2 - 9y^2 = 36[\/latex]\r\nIdentify the coefficients:\r\n[latex]A = 4, \\quad B = 0, \\quad C = -9[\/latex]\r\nSince [latex]A[\/latex] and [latex]C[\/latex] have opposite signs, this is a hyperbola.\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><\/div>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbhacebc-auD46ZWZxQo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/auD46ZWZxQo?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bbhacebc-auD46ZWZxQo\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661454&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bbhacebc-auD46ZWZxQo&vembed=0&video_id=auD46ZWZxQo&video_target=tpm-plugin-bbhacebc-auD46ZWZxQo'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Determining+What+Type+of+Conic+Section+from+General+Form_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermining What Type of Conic Section from General Form\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Equations of Rotated Conics in Standard Form<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"73\" data-end=\"416\">When the general conic equation has a cross-product term [latex]Bxy[\/latex], the conic is <strong data-start=\"163\" data-end=\"174\">rotated<\/strong> relative to the coordinate axes. To write its equation in standard form, we apply a <strong data-start=\"259\" data-end=\"279\">rotation of axes<\/strong>. This change of variables removes the [latex]xy[\/latex]-term so the conic can be recognized (circle, ellipse, parabola, or hyperbola).<\/p>\r\n<p data-start=\"418\" data-end=\"446\">The rotation formulas are:<\/p>\r\n\r\n<ul data-start=\"448\" data-end=\"549\">\r\n \t<li data-start=\"448\" data-end=\"498\">\r\n<p data-start=\"450\" data-end=\"498\">[latex]x = x'\\cos\\theta - y'\\sin\\theta[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"499\" data-end=\"549\">\r\n<p data-start=\"501\" data-end=\"549\">[latex]y = x'\\sin\\theta + y'\\cos\\theta[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<p data-start=\"551\" data-end=\"638\">The angle [latex]\\theta[\/latex] that eliminates the [latex]xy[\/latex]-term satisfies:<\/p>\r\n<p data-start=\"640\" data-end=\"688\">[latex]\\tan(2\\theta) = \\dfrac{B}{A-C}[\/latex].<\/p>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Writing Rotated Conics in Standard Form<\/strong>\r\n<ol>\r\n \t<li data-start=\"754\" data-end=\"953\">\r\n<p data-start=\"757\" data-end=\"786\"><strong data-start=\"757\" data-end=\"784\">Identify the Cross Term<\/strong><\/p>\r\n\r\n<ul data-start=\"790\" data-end=\"953\">\r\n \t<li data-start=\"790\" data-end=\"839\">\r\n<p data-start=\"792\" data-end=\"839\">If [latex]B=0[\/latex], no rotation is needed.<\/p>\r\n<\/li>\r\n \t<li data-start=\"843\" data-end=\"953\">\r\n<p data-start=\"845\" data-end=\"953\">If [latex]B\\ne 0[\/latex], compute [latex]\\theta[\/latex] using [latex]\\tan(2\\theta)=\\dfrac{B}{A-C}[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"955\" data-end=\"1130\">\r\n<p data-start=\"958\" data-end=\"987\"><strong data-start=\"958\" data-end=\"985\">Apply Rotation Formulas<\/strong><\/p>\r\n\r\n<ul data-start=\"991\" data-end=\"1130\">\r\n \t<li data-start=\"991\" data-end=\"1060\">\r\n<p data-start=\"993\" data-end=\"1060\">Substitute [latex]x, y[\/latex] in terms of [latex]x', y'[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1064\" data-end=\"1130\">\r\n<p data-start=\"1066\" data-end=\"1130\">Expand and simplify to eliminate the [latex]x'y'[\/latex]-term.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1132\" data-end=\"1475\">\r\n<p data-start=\"1135\" data-end=\"1168\"><strong data-start=\"1135\" data-end=\"1166\">Recognize the Standard Form<\/strong><\/p>\r\n\r\n<ul data-start=\"1172\" data-end=\"1475\">\r\n \t<li data-start=\"1172\" data-end=\"1475\">\r\n<p data-start=\"1174\" data-end=\"1239\">Once simplified, the equation will match a conic standard form:<\/p>\r\n\r\n<ul data-start=\"1245\" data-end=\"1475\">\r\n \t<li data-start=\"1245\" data-end=\"1318\">\r\n<p data-start=\"1247\" data-end=\"1318\">Ellipse: [latex]\\dfrac{x'^{2}}{a^{2}}+\\dfrac{y'^{2}}{b^{2}}=1[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1324\" data-end=\"1399\">\r\n<p data-start=\"1326\" data-end=\"1399\">Hyperbola: [latex]\\dfrac{x'^{2}}{a^{2}}-\\dfrac{y'^{2}}{b^{2}}=1[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"1405\" data-end=\"1475\">\r\n<p data-start=\"1407\" data-end=\"1475\">Parabola: [latex]y'^{2}=4px'[\/latex] or [latex]x'^{2}=4py'[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div><section class=\"textbox example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Rewrite [latex]xy = 4[\/latex] in standard form by rotating the axes.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[reveal-answer q=\"rotate002\"]Show Solution[\/reveal-answer] [hidden-answer a=\"rotate002\"]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Step 1: Identify coefficients<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Rewrite as [latex]0x^2 + xy + 0y^2 - 4 = 0[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[latex]A = 0, \\quad B = 1, \\quad C = 0[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Step 2: Find the rotation angle<\/p>\r\n[latex] \\begin{aligned} \\tan(2\\theta) &amp;= \\frac{B}{A - C} = \\frac{1}{0 - 0} \\end{aligned} [\/latex]\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">This is undefined, so [latex]2\\theta = 90\u00b0[\/latex], giving [latex]\\theta = 45\u00b0[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Step 3: Apply rotation formulas<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">For [latex]\\theta = 45\u00b0[\/latex]: [latex] \\cos 45\u00b0 = \\sin 45\u00b0 = \\frac{\\sqrt{2}}{2} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">The rotation formulas are:\r\n[latex] \\begin{aligned} x &amp;= x'\\cos 45\u00b0 - y'\\sin 45\u00b0 = \\frac{\\sqrt{2}}{2}(x' - y') \\\\ y &amp;= x'\\sin 45\u00b0 + y'\\cos 45\u00b0 = \\frac{\\sqrt{2}}{2}(x' + y') \\end{aligned} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Step 4: Substitute into the original equation<\/p>\r\n[latex] \\begin{aligned} xy &amp;= 4 \\\\ \\left[\\frac{\\sqrt{2}}{2}(x' - y')\\right] \\left[\\frac{\\sqrt{2}}{2}(x' + y')\\right] &amp;= 4 \\\\ \\frac{1}{2}(x' - y')(x' + y') &amp;= 4 \\\\ \\frac{1}{2}(x'^2 - y'^2) &amp;= 4 \\\\ x'^2 - y'^2 &amp;= 8 \\end{aligned} [\/latex]\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Step 5: Write in standard form [latex] \\frac{x'^2}{8} - \\frac{y'^2}{8} = 1 [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">This is a <strong>hyperbola<\/strong> in the rotated coordinate system, with [latex]a^2 = b^2 = 8[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><\/p>\r\n\r\n<\/section><\/div>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-egbhfbda-L3phw2wFWS4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/L3phw2wFWS4?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-egbhfbda-L3phw2wFWS4\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661455&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-egbhfbda-L3phw2wFWS4&vembed=0&video_id=L3phw2wFWS4&video_target=tpm-plugin-egbhfbda-L3phw2wFWS4'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Rotation+of+Conics_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cRotation of Conics\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Identifying Conics without Rotating Axes<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"69\" data-end=\"271\">Even if a conic equation includes a cross-product term [latex]Bxy[\/latex], it is possible to identify the type of conic <strong data-start=\"189\" data-end=\"227\">without actually rotating the axes<\/strong>. This is done using the <strong data-start=\"252\" data-end=\"268\">discriminant<\/strong>: <span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u0394<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">B<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mord mathnormal\">C <\/span><\/span><\/span><\/span><\/span>where the general conic equation is [latex]Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0[\/latex].<\/p>\r\n<p data-start=\"398\" data-end=\"443\">The discriminant reveals the type of conic:<\/p>\r\n\r\n<ul data-start=\"445\" data-end=\"632\">\r\n \t<li data-start=\"445\" data-end=\"547\">\r\n<p data-start=\"447\" data-end=\"547\">[latex]\\Delta &lt; 0[\/latex] \u2192 ellipse (if [latex]A=C[\/latex] and [latex]B=0[\/latex], it\u2019s a circle).<\/p>\r\n<\/li>\r\n \t<li data-start=\"548\" data-end=\"589\">\r\n<p data-start=\"550\" data-end=\"589\">[latex]\\Delta = 0[\/latex] \u2192 parabola.<\/p>\r\n<\/li>\r\n \t<li data-start=\"590\" data-end=\"632\">\r\n<p data-start=\"592\" data-end=\"632\">[latex]\\Delta &gt; 0[\/latex] \u2192 hyperbola.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Identifying Conics<\/strong>\r\n<ol>\r\n \t<li data-start=\"677\" data-end=\"784\">\r\n<p data-start=\"680\" data-end=\"707\"><strong data-start=\"680\" data-end=\"705\">Write in General Form<\/strong><\/p>\r\n\r\n<ul data-start=\"711\" data-end=\"784\">\r\n \t<li data-start=\"711\" data-end=\"784\">\r\n<p data-start=\"713\" data-end=\"784\">Ensure the equation is in [latex]Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"786\" data-end=\"861\">\r\n<p data-start=\"789\" data-end=\"819\"><strong data-start=\"789\" data-end=\"817\">Compute the Discriminant<\/strong><\/p>\r\n\r\n<ul data-start=\"823\" data-end=\"861\">\r\n \t<li data-start=\"823\" data-end=\"861\">\r\n<p data-start=\"825\" data-end=\"861\">[latex]\\Delta = B^{2}-4AC[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"863\" data-end=\"1036\">\r\n<p data-start=\"866\" data-end=\"892\"><strong data-start=\"866\" data-end=\"890\">Interpret the Result<\/strong><\/p>\r\n\r\n<ul data-start=\"896\" data-end=\"1036\">\r\n \t<li data-start=\"896\" data-end=\"939\">\r\n<p data-start=\"898\" data-end=\"939\">If [latex]\\Delta &lt; 0[\/latex] \u2192 ellipse.<\/p>\r\n<\/li>\r\n \t<li data-start=\"943\" data-end=\"987\">\r\n<p data-start=\"945\" data-end=\"987\">If [latex]\\Delta = 0[\/latex] \u2192 parabola.<\/p>\r\n<\/li>\r\n \t<li data-start=\"991\" data-end=\"1036\">\r\n<p data-start=\"993\" data-end=\"1036\">If [latex]\\Delta &gt; 0[\/latex] \u2192 hyperbola.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1038\" data-end=\"1175\">\r\n<p data-start=\"1041\" data-end=\"1067\"><strong data-start=\"1041\" data-end=\"1065\">Special Case: Circle<\/strong><\/p>\r\n\r\n<ul data-start=\"1682\" data-end=\"1806\">\r\n \t<li data-start=\"1071\" data-end=\"1175\">\r\n<p data-start=\"1073\" data-end=\"1175\">If [latex]\\Delta &lt; 0[\/latex] and [latex]A=C[\/latex], with [latex]B=0[\/latex], the conic is a circle.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div><section class=\"textbox example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Use the discriminant to identify each conic section.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]9x^2 + 16y^2 + 24x - 32y - 36 = 0[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]x^2 - 6x + 4y + 9 = 0[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) [latex]4x^2 - 9y^2 + 8xy = 12[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[reveal-answer q=\"disc001\"]Show Solution[\/reveal-answer] [hidden-answer a=\"disc001\"]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The discriminant is [latex]\\Delta = B^2 - 4AC[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]9x^2 + 16y^2 + 24x - 32y - 36 = 0[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">Identify coefficients: [latex]A = 9, \\quad B = 0, \\quad C = 16[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">Calculate the discriminant: [latex] \\begin{aligned} \\Delta &amp;= B^2 - 4AC \\ &amp;= 0^2 - 4(9)(16) \\ &amp;= 0 - 576 \\ &amp;= -576 \\end{aligned} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since [latex]\\Delta &lt; 0[\/latex], this is an ellipse.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]x^2 - 6x + 4y + 9 = 0[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Rewrite: [latex]x^2 + 0xy + 0y^2 - 6x + 4y + 9 = 0[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[latex]A = 1, \\quad B = 0, \\quad C = 0[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">Calculate the discriminant: [latex] \\begin{aligned} \\Delta &amp;= 0^2 - 4(1)(0) \\ &amp;= 0 \\end{aligned} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since [latex]\\Delta = 0[\/latex], this is a parabola.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) [latex]4x^2 - 9y^2 + 8xy = 12[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Rewrite: [latex]4x^2 + 8xy - 9y^2 - 12 = 0[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[latex]A = 4, \\quad B = 8, \\quad C = -9[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">Calculate the discriminant: [latex] \\begin{aligned} \\Delta &amp;= B^2 - 4AC \\ &amp;= 8^2 - 4(4)(-9) \\ &amp;= 64 - (-144) \\ &amp;= 64 + 144 \\ &amp;= 208 \\end{aligned} [\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since [latex]\\Delta &gt; 0[\/latex], this is a hyperbola.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><\/div>\r\n<div><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-haahfhfa-sVWuWSPRHX8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/sVWuWSPRHX8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-haahfhfa-sVWuWSPRHX8\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661456&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-haahfhfa-sVWuWSPRHX8&vembed=0&video_id=sVWuWSPRHX8&video_target=tpm-plugin-haahfhfa-sVWuWSPRHX8'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Using+the+Discriminant+and+Coefficients+to+Identify+a+Conic_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cUsing the Discriminant and Coefficients to Identify a Conic\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\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<\/section>\n<h2>Identifying Conic Sections from the General Equation<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"81\" data-end=\"143\">The <strong data-start=\"85\" data-end=\"119\">general second-degree equation<\/strong> for conic sections is [latex]Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0[\/latex].<\/p>\n<p data-start=\"192\" data-end=\"432\">Depending on the coefficients, this equation can represent a <strong data-start=\"253\" data-end=\"296\">circle, ellipse, parabola, or hyperbola<\/strong>. These are called <em data-start=\"315\" data-end=\"337\">nondegenerate conics<\/em>. Identifying which conic you have involves looking at the coefficients [latex]A,B,C[\/latex].<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: How to Identify the Conic<\/strong><\/p>\n<ol>\n<li data-start=\"484\" data-end=\"720\">\n<p data-start=\"487\" data-end=\"534\"><strong data-start=\"487\" data-end=\"532\">Check the Cross-Term ([latex]Bxy[\/latex])<\/strong><\/p>\n<ul data-start=\"538\" data-end=\"720\">\n<li data-start=\"538\" data-end=\"656\">\n<p data-start=\"540\" data-end=\"656\">If [latex]B\\ne 0[\/latex], the conic is rotated. Identification requires more advanced analysis (rotation of axes).<\/p>\n<\/li>\n<li data-start=\"660\" data-end=\"720\">\n<p data-start=\"662\" data-end=\"720\">If [latex]B=0[\/latex], proceed with simpler tests below.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"722\" data-end=\"862\">\n<p data-start=\"725\" data-end=\"737\"><strong data-start=\"725\" data-end=\"735\">Circle<\/strong><\/p>\n<ul data-start=\"741\" data-end=\"862\">\n<li data-start=\"741\" data-end=\"790\">\n<p data-start=\"743\" data-end=\"790\">[latex]A=C \\ne 0[\/latex].<\/p>\n<\/li>\n<li data-start=\"794\" data-end=\"862\">\n<p data-start=\"796\" data-end=\"862\">Example: [latex]x^{2}+y^{2}-16=0[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"864\" data-end=\"1027\">\n<p data-start=\"867\" data-end=\"893\"><strong data-start=\"867\" data-end=\"891\">Ellipse (not circle)<\/strong><\/p>\n<ul data-start=\"897\" data-end=\"1027\">\n<li data-start=\"897\" data-end=\"959\">\n<p data-start=\"899\" data-end=\"959\">[latex]A \\ne C[\/latex], both positive.<\/p>\n<\/li>\n<li data-start=\"963\" data-end=\"1027\">\n<p data-start=\"965\" data-end=\"1027\">Example: [latex]\\dfrac{x^{2}}{9}+\\dfrac{y^{2}}{4}=1[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1029\" data-end=\"1167\">\n<p data-start=\"1032\" data-end=\"1046\"><strong data-start=\"1032\" data-end=\"1044\">Parabola<\/strong><\/p>\n<ul data-start=\"1050\" data-end=\"1167\">\n<li data-start=\"1050\" data-end=\"1126\">\n<p data-start=\"1052\" data-end=\"1126\">One squared term only (either [latex]A=0[\/latex] or [latex]C=0[\/latex]).<\/p>\n<\/li>\n<li data-start=\"1130\" data-end=\"1167\">\n<p data-start=\"1132\" data-end=\"1167\">Example: [latex]y^{2}=8x[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1169\" data-end=\"1321\">\n<p data-start=\"1172\" data-end=\"1187\"><strong data-start=\"1172\" data-end=\"1185\">Hyperbola<\/strong><\/p>\n<ul data-start=\"1573\" data-end=\"1647\">\n<li data-start=\"1191\" data-end=\"1253\">\n<p data-start=\"1193\" data-end=\"1253\">[latex]A[\/latex] and [latex]C[\/latex] have opposite signs.<\/p>\n<\/li>\n<li data-start=\"1257\" data-end=\"1321\">\n<p data-start=\"1259\" data-end=\"1321\">Example: [latex]\\dfrac{x^{2}}{9}-\\dfrac{y^{2}}{4}=1[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">Identify the type of conic section represented by each equation.<br \/>\na) [latex]4x^2 + 4y^2 - 16 = 0[\/latex]<br \/>\nb) [latex]9x^2 + 16y^2 - 144 = 0[\/latex]<br \/>\nc) [latex]y^2 - 8x = 0[\/latex]<br \/>\nd) [latex]4x^2 - 9y^2 = 36[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qconic001\">Show Solution<\/button><\/p>\n<div id=\"qconic001\" class=\"hidden-answer\" style=\"display: none\">\na) [latex]4x^2 + 4y^2 - 16 = 0[\/latex]<br \/>\nIdentify the coefficients:<br \/>\n[latex]A = 4, \\quad B = 0, \\quad C = 4[\/latex]<br \/>\nSince [latex]A = C[\/latex] and [latex]B = 0[\/latex], this is a circle.b) [latex]9x^2 + 16y^2 - 144 = 0[\/latex]<br \/>\nIdentify the coefficients:<br \/>\n[latex]A = 9, \\quad B = 0, \\quad C = 16[\/latex]<br \/>\nSince [latex]A \\neq C[\/latex], both are positive, and [latex]B = 0[\/latex], this is an ellipse.c) [latex]y^2 - 8x = 0[\/latex]<br \/>\nIdentify the coefficients:<br \/>\n[latex]A = 0, \\quad B = 0, \\quad C = 1[\/latex]<br \/>\nOnly one squared term is present ([latex]C = 1[\/latex], [latex]A = 0[\/latex]), so this is a parabola.d) [latex]4x^2 - 9y^2 = 36[\/latex]<br \/>\nIdentify the coefficients:<br \/>\n[latex]A = 4, \\quad B = 0, \\quad C = -9[\/latex]<br \/>\nSince [latex]A[\/latex] and [latex]C[\/latex] have opposite signs, this is a hyperbola.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbhacebc-auD46ZWZxQo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/auD46ZWZxQo?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bbhacebc-auD46ZWZxQo\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661454&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bbhacebc-auD46ZWZxQo&#38;vembed=0&#38;video_id=auD46ZWZxQo&#38;video_target=tpm-plugin-bbhacebc-auD46ZWZxQo\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Determining+What+Type+of+Conic+Section+from+General+Form_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermining What Type of Conic Section from General Form\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Equations of Rotated Conics in Standard Form<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"73\" data-end=\"416\">When the general conic equation has a cross-product term [latex]Bxy[\/latex], the conic is <strong data-start=\"163\" data-end=\"174\">rotated<\/strong> relative to the coordinate axes. To write its equation in standard form, we apply a <strong data-start=\"259\" data-end=\"279\">rotation of axes<\/strong>. This change of variables removes the [latex]xy[\/latex]-term so the conic can be recognized (circle, ellipse, parabola, or hyperbola).<\/p>\n<p data-start=\"418\" data-end=\"446\">The rotation formulas are:<\/p>\n<ul data-start=\"448\" data-end=\"549\">\n<li data-start=\"448\" data-end=\"498\">\n<p data-start=\"450\" data-end=\"498\">[latex]x = x'\\cos\\theta - y'\\sin\\theta[\/latex]<\/p>\n<\/li>\n<li data-start=\"499\" data-end=\"549\">\n<p data-start=\"501\" data-end=\"549\">[latex]y = x'\\sin\\theta + y'\\cos\\theta[\/latex]<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"551\" data-end=\"638\">The angle [latex]\\theta[\/latex] that eliminates the [latex]xy[\/latex]-term satisfies:<\/p>\n<p data-start=\"640\" data-end=\"688\">[latex]\\tan(2\\theta) = \\dfrac{B}{A-C}[\/latex].<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Writing Rotated Conics in Standard Form<\/strong><\/p>\n<ol>\n<li data-start=\"754\" data-end=\"953\">\n<p data-start=\"757\" data-end=\"786\"><strong data-start=\"757\" data-end=\"784\">Identify the Cross Term<\/strong><\/p>\n<ul data-start=\"790\" data-end=\"953\">\n<li data-start=\"790\" data-end=\"839\">\n<p data-start=\"792\" data-end=\"839\">If [latex]B=0[\/latex], no rotation is needed.<\/p>\n<\/li>\n<li data-start=\"843\" data-end=\"953\">\n<p data-start=\"845\" data-end=\"953\">If [latex]B\\ne 0[\/latex], compute [latex]\\theta[\/latex] using [latex]\\tan(2\\theta)=\\dfrac{B}{A-C}[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"955\" data-end=\"1130\">\n<p data-start=\"958\" data-end=\"987\"><strong data-start=\"958\" data-end=\"985\">Apply Rotation Formulas<\/strong><\/p>\n<ul data-start=\"991\" data-end=\"1130\">\n<li data-start=\"991\" data-end=\"1060\">\n<p data-start=\"993\" data-end=\"1060\">Substitute [latex]x, y[\/latex] in terms of [latex]x', y'[\/latex].<\/p>\n<\/li>\n<li data-start=\"1064\" data-end=\"1130\">\n<p data-start=\"1066\" data-end=\"1130\">Expand and simplify to eliminate the [latex]x'y'[\/latex]-term.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1132\" data-end=\"1475\">\n<p data-start=\"1135\" data-end=\"1168\"><strong data-start=\"1135\" data-end=\"1166\">Recognize the Standard Form<\/strong><\/p>\n<ul data-start=\"1172\" data-end=\"1475\">\n<li data-start=\"1172\" data-end=\"1475\">\n<p data-start=\"1174\" data-end=\"1239\">Once simplified, the equation will match a conic standard form:<\/p>\n<ul data-start=\"1245\" data-end=\"1475\">\n<li data-start=\"1245\" data-end=\"1318\">\n<p data-start=\"1247\" data-end=\"1318\">Ellipse: [latex]\\dfrac{x'^{2}}{a^{2}}+\\dfrac{y'^{2}}{b^{2}}=1[\/latex]<\/p>\n<\/li>\n<li data-start=\"1324\" data-end=\"1399\">\n<p data-start=\"1326\" data-end=\"1399\">Hyperbola: [latex]\\dfrac{x'^{2}}{a^{2}}-\\dfrac{y'^{2}}{b^{2}}=1[\/latex]<\/p>\n<\/li>\n<li data-start=\"1405\" data-end=\"1475\">\n<p data-start=\"1407\" data-end=\"1475\">Parabola: [latex]y'^{2}=4px'[\/latex] or [latex]x'^{2}=4py'[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<div>\n<section class=\"textbox example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Rewrite [latex]xy = 4[\/latex] in standard form by rotating the axes.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qrotate002\">Show Solution<\/button> <\/p>\n<div id=\"qrotate002\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Step 1: Identify coefficients<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Rewrite as [latex]0x^2 + xy + 0y^2 - 4 = 0[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">[latex]A = 0, \\quad B = 1, \\quad C = 0[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Step 2: Find the rotation angle<\/p>\n<p>[latex]\\begin{aligned} \\tan(2\\theta) &= \\frac{B}{A - C} = \\frac{1}{0 - 0} \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">This is undefined, so [latex]2\\theta = 90\u00b0[\/latex], giving [latex]\\theta = 45\u00b0[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Step 3: Apply rotation formulas<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">For [latex]\\theta = 45\u00b0[\/latex]: [latex]\\cos 45\u00b0 = \\sin 45\u00b0 = \\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">The rotation formulas are:<br \/>\n[latex]\\begin{aligned} x &= x'\\cos 45\u00b0 - y'\\sin 45\u00b0 = \\frac{\\sqrt{2}}{2}(x' - y') \\\\ y &= x'\\sin 45\u00b0 + y'\\cos 45\u00b0 = \\frac{\\sqrt{2}}{2}(x' + y') \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Step 4: Substitute into the original equation<\/p>\n<p>[latex]\\begin{aligned} xy &= 4 \\\\ \\left[\\frac{\\sqrt{2}}{2}(x' - y')\\right] \\left[\\frac{\\sqrt{2}}{2}(x' + y')\\right] &= 4 \\\\ \\frac{1}{2}(x' - y')(x' + y') &= 4 \\\\ \\frac{1}{2}(x'^2 - y'^2) &= 4 \\\\ x'^2 - y'^2 &= 8 \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Step 5: Write in standard form [latex]\\frac{x'^2}{8} - \\frac{y'^2}{8} = 1[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">This is a <strong>hyperbola<\/strong> in the rotated coordinate system, with [latex]a^2 = b^2 = 8[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-egbhfbda-L3phw2wFWS4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/L3phw2wFWS4?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-egbhfbda-L3phw2wFWS4\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661455&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-egbhfbda-L3phw2wFWS4&#38;vembed=0&#38;video_id=L3phw2wFWS4&#38;video_target=tpm-plugin-egbhfbda-L3phw2wFWS4\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Rotation+of+Conics_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cRotation of Conics\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Identifying Conics without Rotating Axes<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"69\" data-end=\"271\">Even if a conic equation includes a cross-product term [latex]Bxy[\/latex], it is possible to identify the type of conic <strong data-start=\"189\" data-end=\"227\">without actually rotating the axes<\/strong>. This is done using the <strong data-start=\"252\" data-end=\"268\">discriminant<\/strong>: <span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u0394<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">B<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mord mathnormal\">C <\/span><\/span><\/span><\/span><\/span>where the general conic equation is [latex]Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0[\/latex].<\/p>\n<p data-start=\"398\" data-end=\"443\">The discriminant reveals the type of conic:<\/p>\n<ul data-start=\"445\" data-end=\"632\">\n<li data-start=\"445\" data-end=\"547\">\n<p data-start=\"447\" data-end=\"547\">[latex]\\Delta < 0[\/latex] \u2192 ellipse (if [latex]A=C[\/latex] and [latex]B=0[\/latex], it\u2019s a circle).<\/p>\n<\/li>\n<li data-start=\"548\" data-end=\"589\">\n<p data-start=\"550\" data-end=\"589\">[latex]\\Delta = 0[\/latex] \u2192 parabola.<\/p>\n<\/li>\n<li data-start=\"590\" data-end=\"632\">\n<p data-start=\"592\" data-end=\"632\">[latex]\\Delta > 0[\/latex] \u2192 hyperbola.<\/p>\n<\/li>\n<\/ul>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Identifying Conics<\/strong><\/p>\n<ol>\n<li data-start=\"677\" data-end=\"784\">\n<p data-start=\"680\" data-end=\"707\"><strong data-start=\"680\" data-end=\"705\">Write in General Form<\/strong><\/p>\n<ul data-start=\"711\" data-end=\"784\">\n<li data-start=\"711\" data-end=\"784\">\n<p data-start=\"713\" data-end=\"784\">Ensure the equation is in [latex]Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"786\" data-end=\"861\">\n<p data-start=\"789\" data-end=\"819\"><strong data-start=\"789\" data-end=\"817\">Compute the Discriminant<\/strong><\/p>\n<ul data-start=\"823\" data-end=\"861\">\n<li data-start=\"823\" data-end=\"861\">\n<p data-start=\"825\" data-end=\"861\">[latex]\\Delta = B^{2}-4AC[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"863\" data-end=\"1036\">\n<p data-start=\"866\" data-end=\"892\"><strong data-start=\"866\" data-end=\"890\">Interpret the Result<\/strong><\/p>\n<ul data-start=\"896\" data-end=\"1036\">\n<li data-start=\"896\" data-end=\"939\">\n<p data-start=\"898\" data-end=\"939\">If [latex]\\Delta < 0[\/latex] \u2192 ellipse.<\/p>\n<\/li>\n<li data-start=\"943\" data-end=\"987\">\n<p data-start=\"945\" data-end=\"987\">If [latex]\\Delta = 0[\/latex] \u2192 parabola.<\/p>\n<\/li>\n<li data-start=\"991\" data-end=\"1036\">\n<p data-start=\"993\" data-end=\"1036\">If [latex]\\Delta > 0[\/latex] \u2192 hyperbola.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1038\" data-end=\"1175\">\n<p data-start=\"1041\" data-end=\"1067\"><strong data-start=\"1041\" data-end=\"1065\">Special Case: Circle<\/strong><\/p>\n<ul data-start=\"1682\" data-end=\"1806\">\n<li data-start=\"1071\" data-end=\"1175\">\n<p data-start=\"1073\" data-end=\"1175\">If [latex]\\Delta < 0[\/latex] and [latex]A=C[\/latex], with [latex]B=0[\/latex], the conic is a circle.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Use the discriminant to identify each conic section.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">a) [latex]9x^2 + 16y^2 + 24x - 32y - 36 = 0[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">b) [latex]x^2 - 6x + 4y + 9 = 0[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">c) [latex]4x^2 - 9y^2 + 8xy = 12[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qdisc001\">Show Solution<\/button> <\/p>\n<div id=\"qdisc001\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">The discriminant is [latex]\\Delta = B^2 - 4AC[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">a) [latex]9x^2 + 16y^2 + 24x - 32y - 36 = 0[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">Identify coefficients: [latex]A = 9, \\quad B = 0, \\quad C = 16[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">Calculate the discriminant: [latex]\\begin{aligned} \\Delta &= B^2 - 4AC \\ &= 0^2 - 4(9)(16) \\ &= 0 - 576 \\ &= -576 \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Since [latex]\\Delta < 0[\/latex], this is an ellipse.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">b) [latex]x^2 - 6x + 4y + 9 = 0[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Rewrite: [latex]x^2 + 0xy + 0y^2 - 6x + 4y + 9 = 0[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">[latex]A = 1, \\quad B = 0, \\quad C = 0[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">Calculate the discriminant: [latex]\\begin{aligned} \\Delta &= 0^2 - 4(1)(0) \\ &= 0 \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Since [latex]\\Delta = 0[\/latex], this is a parabola.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">c) [latex]4x^2 - 9y^2 + 8xy = 12[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Rewrite: [latex]4x^2 + 8xy - 9y^2 - 12 = 0[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">[latex]A = 4, \\quad B = 8, \\quad C = -9[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">Calculate the discriminant: [latex]\\begin{aligned} \\Delta &= B^2 - 4AC \\ &= 8^2 - 4(4)(-9) \\ &= 64 - (-144) \\ &= 64 + 144 \\ &= 208 \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Since [latex]\\Delta > 0[\/latex], this is a hyperbola.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-haahfhfa-sVWuWSPRHX8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/sVWuWSPRHX8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-haahfhfa-sVWuWSPRHX8\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661456&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-haahfhfa-sVWuWSPRHX8&#38;vembed=0&#38;video_id=sVWuWSPRHX8&#38;video_target=tpm-plugin-haahfhfa-sVWuWSPRHX8\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Using+the+Discriminant+and+Coefficients+to+Identify+a+Conic_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cUsing the Discriminant and Coefficients to Identify a Conic\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":27,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Determining What Type of Conic Section from General Form\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/auD46ZWZxQo\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Rotation of Conics\",\"author\":\"\",\"organization\":\"Symplit Math\",\"url\":\"https:\/\/youtu.be\/L3phw2wFWS4\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Using the Discriminant and Coefficients to Identify a Conic\",\"author\":\"\",\"organization\":\"ThinkwellVids\",\"url\":\"https:\/\/youtu.be\/sVWuWSPRHX8\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":522,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Determining What Type of Conic Section from General Form","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/auD46ZWZxQo","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Rotation of Conics","author":"","organization":"Symplit Math","url":"https:\/\/youtu.be\/L3phw2wFWS4","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Using the Discriminant and Coefficients to Identify a Conic","author":"","organization":"ThinkwellVids","url":"https:\/\/youtu.be\/sVWuWSPRHX8","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661454&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bbhacebc-auD46ZWZxQo&vembed=0&video_id=auD46ZWZxQo&video_target=tpm-plugin-bbhacebc-auD46ZWZxQo'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661455&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-egbhfbda-L3phw2wFWS4&vembed=0&video_id=L3phw2wFWS4&video_target=tpm-plugin-egbhfbda-L3phw2wFWS4'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661456&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-haahfhfa-sVWuWSPRHX8&vembed=0&video_id=sVWuWSPRHX8&video_target=tpm-plugin-haahfhfa-sVWuWSPRHX8'><\/script>\n","media_targets":["tpm-plugin-bbhacebc-auD46ZWZxQo","tpm-plugin-egbhfbda-L3phw2wFWS4","tpm-plugin-haahfhfa-sVWuWSPRHX8"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1600"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1600\/revisions"}],"predecessor-version":[{"id":5828,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1600\/revisions\/5828"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/522"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1600\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1600"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1600"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1600"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1600"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}