{"id":3296,"date":"2025-08-16T01:52:18","date_gmt":"2025-08-16T01:52:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3296"},"modified":"2025-10-23T16:25:18","modified_gmt":"2025-10-23T16:25:18","slug":"rotation-of-axes-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rotation-of-axes-apply-it\/","title":{"raw":"Rotation of Axes: Apply It","rendered":"Rotation of Axes: Apply It"},"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<p data-start=\"33\" data-end=\"338\">A plaza designer tests a tilted spotlight that projects a conic-shaped pool of light onto the ground. With ground axes aligned to nearby paving joints, measurements of the light boundary fit the equation<br data-start=\"246\" data-end=\"249\" \/>[latex]8x^{2}-12xy+17y^{2}=20.[\/latex]<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<ol data-start=\"360\" data-end=\"589\">\r\n \t<li data-start=\"360\" data-end=\"589\">\r\n<p data-start=\"363\" data-end=\"589\">Identify the conic directly from the general form (without rotating axes).<\/p>\r\n<\/li>\r\n \t<li data-start=\"750\" data-end=\"920\">\r\n<p data-start=\"753\" data-end=\"920\">Write the rotated standard form by eliminating the [latex]xy[\/latex] term.<br data-start=\"827\" data-end=\"830\" \/>For rotation angle [latex]\\theta[\/latex], use [latex]\\cot(2\\theta)=\\dfrac{A-C}{B}[\/latex].<\/p>\r\n<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"440928\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"440928\"]\r\n<ol>\r\n \t<li>General form: [latex]Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0[\/latex] with [latex]A=8,;B=-12,;C=17[\/latex].<br data-start=\"536\" data-end=\"539\" \/>Compute the discriminant [latex]B^{2}-4AC[\/latex].[latex]\\\\[\/latex][latex]\r\n\\begin{align}\r\nB^{2}-4AC&amp;=(-12)^{2}-4(8)(17)\\\\\r\n&amp;=144-544\\\\\r\n&amp;=-400&lt;0\r\n\\end{align}\r\n[\/latex]\r\n[latex]\\\\[\/latex]\r\nBecause [latex]B^{2}-4AC&lt;0[\/latex], the curve is an ellipse.\r\n[latex]\\\\[\/latex]<\/li>\r\n \t<li>[latex]\\\\[\/latex][latex]\r\n\\begin{align}\r\n\\cot(2\\theta)&amp;=\\frac{8-17}{-12}=\\frac{9}{12}=\\frac{3}{4}\r\n\\end{align}\r\n[\/latex]This choice yields convenient exact values (from a 3\u20134\u20135 triangle):\r\n[latex]\\\\[\/latex]\r\n<p data-start=\"1092\" data-end=\"1202\">[latex]\r\n\\begin{align}\r\n\\sin\\theta=\\frac{1}{\\sqrt{5}},\\qquad \\cos\\theta=\\frac{2}{\\sqrt{5}}.\r\n\\end{align}\r\n[\/latex]<\/p>\r\n[latex]\\\\[\/latex]\r\n<p data-start=\"1204\" data-end=\"1306\">Rotate axes using<br data-start=\"1221\" data-end=\"1224\" \/>[latex]x=x'\\cos\\theta - y'\\sin\\theta,\\qquad y=x'\\sin\\theta + y'\\cos\\theta.[\/latex]\r\n[latex]\\\\[\/latex]<\/p>\r\n<p data-start=\"1308\" data-end=\"1408\">[latex]\r\n\\begin{align}\r\nx&amp;=\\frac{2x'-y'}{\\sqrt{5}},\\\\\r\ny&amp;=\\frac{x'+2y'}{\\sqrt{5}}.\r\n\\end{align}\r\n[\/latex]<\/p>\r\n[latex]\\\\[\/latex]\r\n<p data-start=\"1410\" data-end=\"1477\">Substitute into [latex]8x^{2}-12xy+17y^{2}=20[\/latex] and simplify.<\/p>\r\n<p data-start=\"1479\" data-end=\"2040\">[latex]\r\n\\begin{align}\r\n8!\\left(\\frac{2x'-y'}{\\sqrt{5}}\\right)^{2}\r\n&amp;-12!\\left(\\frac{2x'-y'}{\\sqrt{5}}\\right)!\\left(\\frac{x'+2y'}{\\sqrt{5}}\\right)\r\n+17!\\left(\\frac{x'+2y'}{\\sqrt{5}}\\right)^{2}=20\\\\\r\n8!\\left(\\frac{4x'^2-4x'y'+y'^2}{5}\\right)\r\n&amp;-12!\\left(\\frac{2x'^2+3x'y'-2y'^2}{5}\\right)\r\n+17!\\left(\\frac{x'^2+4x'y'+4y'^2}{5}\\right)=20\\\\\r\n\\frac{1}{5}\\Big(32x'^2-32x'y'+8y'^2\r\n&amp;-24x'^2-36x'y'+24y'^2\r\n+17x'^2+68x'y'+68y'^2\\Big)=20\\\\\r\n\\frac{1}{5}\\Big(25x'^2+100y'^2\\Big)&amp;=20\\\\\r\n25x'^2+100y'^2&amp;=100\\\\\r\n\\frac{x'^2}{4}+\\frac{y'^2}{1}&amp;=1\r\n\\end{align}\r\n[\/latex]<\/p>\r\n<p data-start=\"2042\" data-end=\"2345\">Standard form relative to the rotated axes is<br data-start=\"2087\" data-end=\"2090\" \/>[latex]\\dfrac{x'^2}{4}+\\dfrac{y'^2}{1}=1,[\/latex]<br data-start=\"2139\" data-end=\"2142\" \/>an ellipse with semi-axes [latex]a=2[\/latex] and [latex]b=1[\/latex] (meters, in this context). The rotation angle satisfies [latex]\\cot(2\\theta)=3\/4[\/latex], so [latex]\\theta\\approx 18.435^\\circ[\/latex].<\/p>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">Use the discriminant [latex]B^{2}-4AC[\/latex] to classify without rotating.<br data-start=\"2440\" data-end=\"2443\" \/>\u2022 To remove [latex]xy[\/latex], rotate by [latex]\\cot(2\\theta)=\\dfrac{A-C}{B}[\/latex] and substitute [latex]x=x'\\cos\\theta-y'\\sin\\theta,;y=x'\\sin\\theta+y'\\cos\\theta[\/latex].<br data-start=\"2616\" data-end=\"2619\" \/>\u2022 Write the result in standard form in the [latex]x',y'[\/latex]\u00a0system.<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p data-start=\"2711\" data-end=\"2854\">A survey drone maps a reflective sculpture; a cross-section relative to chosen ground axes is modeled by<br data-start=\"2815\" data-end=\"2818\" \/>[latex]5x^{2}+6xy+5y^{2}=50.[\/latex]<\/p>\r\n<p data-start=\"2856\" data-end=\"2900\">a) Identify the conic without rotating axes.<\/p>\r\n<p data-start=\"2902\" data-end=\"3020\">b) Find a rotation that eliminates [latex]xy[\/latex] and write the standard form in the [latex]x',y'[\/latex]\u00a0system.<\/p>\r\n\r\n<\/section>","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<p data-start=\"33\" data-end=\"338\">A plaza designer tests a tilted spotlight that projects a conic-shaped pool of light onto the ground. With ground axes aligned to nearby paving joints, measurements of the light boundary fit the equation<br data-start=\"246\" data-end=\"249\" \/>[latex]8x^{2}-12xy+17y^{2}=20.[\/latex]<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<ol data-start=\"360\" data-end=\"589\">\n<li data-start=\"360\" data-end=\"589\">\n<p data-start=\"363\" data-end=\"589\">Identify the conic directly from the general form (without rotating axes).<\/p>\n<\/li>\n<li data-start=\"750\" data-end=\"920\">\n<p data-start=\"753\" data-end=\"920\">Write the rotated standard form by eliminating the [latex]xy[\/latex] term.<br data-start=\"827\" data-end=\"830\" \/>For rotation angle [latex]\\theta[\/latex], use [latex]\\cot(2\\theta)=\\dfrac{A-C}{B}[\/latex].<\/p>\n<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q440928\">Show Solution<\/button><\/p>\n<div id=\"q440928\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>General form: [latex]Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0[\/latex] with [latex]A=8,;B=-12,;C=17[\/latex].<br data-start=\"536\" data-end=\"539\" \/>Compute the discriminant [latex]B^{2}-4AC[\/latex].[latex]\\\\[\/latex][latex]\\begin{align}  B^{2}-4AC&=(-12)^{2}-4(8)(17)\\\\  &=144-544\\\\  &=-400<0  \\end{align}[\/latex]\n[latex]\\\\[\/latex]\nBecause [latex]B^{2}-4AC<0[\/latex], the curve is an ellipse.\n[latex]\\\\[\/latex]<\/li>\n<li>[latex]\\\\[\/latex][latex]\\begin{align}  \\cot(2\\theta)&=\\frac{8-17}{-12}=\\frac{9}{12}=\\frac{3}{4}  \\end{align}[\/latex]This choice yields convenient exact values (from a 3\u20134\u20135 triangle):<br \/>\n[latex]\\\\[\/latex]<\/p>\n<p data-start=\"1092\" data-end=\"1202\">[latex]\\begin{align}  \\sin\\theta=\\frac{1}{\\sqrt{5}},\\qquad \\cos\\theta=\\frac{2}{\\sqrt{5}}.  \\end{align}[\/latex]<\/p>\n<p>[latex]\\\\[\/latex]<\/p>\n<p data-start=\"1204\" data-end=\"1306\">Rotate axes using<br data-start=\"1221\" data-end=\"1224\" \/>[latex]x=x'\\cos\\theta - y'\\sin\\theta,\\qquad y=x'\\sin\\theta + y'\\cos\\theta.[\/latex]<br \/>\n[latex]\\\\[\/latex]<\/p>\n<p data-start=\"1308\" data-end=\"1408\">[latex]\\begin{align}  x&=\\frac{2x'-y'}{\\sqrt{5}},\\\\  y&=\\frac{x'+2y'}{\\sqrt{5}}.  \\end{align}[\/latex]<\/p>\n<p>[latex]\\\\[\/latex]<\/p>\n<p data-start=\"1410\" data-end=\"1477\">Substitute into [latex]8x^{2}-12xy+17y^{2}=20[\/latex] and simplify.<\/p>\n<p data-start=\"1479\" data-end=\"2040\">[latex]\\begin{align}  8!\\left(\\frac{2x'-y'}{\\sqrt{5}}\\right)^{2}  &-12!\\left(\\frac{2x'-y'}{\\sqrt{5}}\\right)!\\left(\\frac{x'+2y'}{\\sqrt{5}}\\right)  +17!\\left(\\frac{x'+2y'}{\\sqrt{5}}\\right)^{2}=20\\\\  8!\\left(\\frac{4x'^2-4x'y'+y'^2}{5}\\right)  &-12!\\left(\\frac{2x'^2+3x'y'-2y'^2}{5}\\right)  +17!\\left(\\frac{x'^2+4x'y'+4y'^2}{5}\\right)=20\\\\  \\frac{1}{5}\\Big(32x'^2-32x'y'+8y'^2  &-24x'^2-36x'y'+24y'^2  +17x'^2+68x'y'+68y'^2\\Big)=20\\\\  \\frac{1}{5}\\Big(25x'^2+100y'^2\\Big)&=20\\\\  25x'^2+100y'^2&=100\\\\  \\frac{x'^2}{4}+\\frac{y'^2}{1}&=1  \\end{align}[\/latex]<\/p>\n<p data-start=\"2042\" data-end=\"2345\">Standard form relative to the rotated axes is<br data-start=\"2087\" data-end=\"2090\" \/>[latex]\\dfrac{x'^2}{4}+\\dfrac{y'^2}{1}=1,[\/latex]<br data-start=\"2139\" data-end=\"2142\" \/>an ellipse with semi-axes [latex]a=2[\/latex] and [latex]b=1[\/latex] (meters, in this context). The rotation angle satisfies [latex]\\cot(2\\theta)=3\/4[\/latex], so [latex]\\theta\\approx 18.435^\\circ[\/latex].<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Use the discriminant [latex]B^{2}-4AC[\/latex] to classify without rotating.<br data-start=\"2440\" data-end=\"2443\" \/>\u2022 To remove [latex]xy[\/latex], rotate by [latex]\\cot(2\\theta)=\\dfrac{A-C}{B}[\/latex] and substitute [latex]x=x'\\cos\\theta-y'\\sin\\theta,;y=x'\\sin\\theta+y'\\cos\\theta[\/latex].<br data-start=\"2616\" data-end=\"2619\" \/>\u2022 Write the result in standard form in the [latex]x',y'[\/latex]\u00a0system.<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p data-start=\"2711\" data-end=\"2854\">A survey drone maps a reflective sculpture; a cross-section relative to chosen ground axes is modeled by<br data-start=\"2815\" data-end=\"2818\" \/>[latex]5x^{2}+6xy+5y^{2}=50.[\/latex]<\/p>\n<p data-start=\"2856\" data-end=\"2900\">a) Identify the conic without rotating axes.<\/p>\n<p data-start=\"2902\" data-end=\"3020\">b) Find a rotation that eliminates [latex]xy[\/latex] and write the standard form in the [latex]x',y'[\/latex]\u00a0system.<\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":26,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":522,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3296"}],"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":14,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3296\/revisions"}],"predecessor-version":[{"id":4857,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3296\/revisions\/4857"}],"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\/3296\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3296"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3296"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3296"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3296"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}