{"id":1961,"date":"2015-11-12T18:30:43","date_gmt":"2015-11-12T18:30:43","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1961"},"modified":"2015-11-12T18:30:43","modified_gmt":"2015-11-12T18:30:43","slug":"identifying-conics-without-rotating-axes","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/identifying-conics-without-rotating-axes\/","title":{"raw":"Identifying Conics without Rotating Axes","rendered":"Identifying Conics without Rotating Axes"},"content":{"raw":"<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\n<\/p><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]{A}^{\\prime }{{x}^{\\prime }}^{2}+{B}^{\\prime }{x}^{\\prime }{y}^{\\prime }+{C}^{\\prime }{{y}^{\\prime }}^{2}+{D}^{\\prime }{x}^{\\prime }+{E}^{\\prime }{y}^{\\prime }+{F}^{\\prime }=0[\/latex]<\/div>\nIt may be shown that [latex]{B}^{2}-4AC={{B}^{\\prime }}^{2}-4{A}^{\\prime }{C}^{\\prime }[\/latex]. The expression does not vary after rotation, so we call the expression invariant<strong>.<\/strong> The discriminant, [latex]{B}^{2}-4AC[\/latex], is invariant and remains unchanged after rotation. Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section.\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]{A}^{\\prime }{{x}^{\\prime }}^{2}+{B}^{\\prime }{x}^{\\prime }{y}^{\\prime }+{C}^{\\prime }{{y}^{\\prime }}^{2}+{D}^{\\prime }{x}^{\\prime }+{E}^{\\prime }{y}^{\\prime }+{F}^{\\prime }=0[\/latex], then [latex]{B}^{2}-4AC={{B}^{\\prime }}^{2}-4{A}^{\\prime }{C}^{\\prime }[\/latex].\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><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><\/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><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><\/div>\n<div class=\"textbox shaded\">\n<h3>Solution<\/h3>\n<ol><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{array}{l}{B}^{2}-4AC={\\left(2\\sqrt{3}\\right)}^{2}-4\\left(5\\right)\\left(2\\right)\\hfill \\\\ \\text{ }=4\\left(3\\right)-40\\hfill \\\\ \\text{ }=12 - 40\\hfill \\\\ \\text{ }=-28&lt;0\\hfill \\end{array}[\/latex]<\/div>\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{array}{l}{B}^{2}-4AC={\\left(2\\sqrt{3}\\right)}^{2}-4\\left(5\\right)\\left(12\\right)\\hfill \\\\ \\text{ }=4\\left(3\\right)-240\\hfill \\\\ \\text{ }=12 - 240\\hfill \\\\ \\text{ }=-228&lt;0\\hfill \\end{array}[\/latex]<\/div>\nTherefore, [latex]5{x}^{2}+2\\sqrt{3}xy+12{y}^{2}-5=0[\/latex] represents an ellipse.<\/li>\n<\/ol><\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 3<\/h3>\nIdentify the conic for each of the following without rotating axes.\n<ol><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><a href=\"https:\/\/courses.candelalearning.com\/precalctwo1xmaster\/chapter\/solutions-26\/\" target=\"_blank\">Solution<\/a>\n\n<\/div>","rendered":"<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\n<\/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]{A}^{\\prime }{{x}^{\\prime }}^{2}+{B}^{\\prime }{x}^{\\prime }{y}^{\\prime }+{C}^{\\prime }{{y}^{\\prime }}^{2}+{D}^{\\prime }{x}^{\\prime }+{E}^{\\prime }{y}^{\\prime }+{F}^{\\prime }=0[\/latex]<\/div>\n<p>It may be shown that [latex]{B}^{2}-4AC={{B}^{\\prime }}^{2}-4{A}^{\\prime }{C}^{\\prime }[\/latex]. The expression does not vary after rotation, so we call the expression invariant<strong>.<\/strong> The discriminant, [latex]{B}^{2}-4AC[\/latex], is invariant and remains unchanged after rotation. Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section.<\/p>\n<div 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]{A}^{\\prime }{{x}^{\\prime }}^{2}+{B}^{\\prime }{x}^{\\prime }{y}^{\\prime }+{C}^{\\prime }{{y}^{\\prime }}^{2}+{D}^{\\prime }{x}^{\\prime }+{E}^{\\prime }{y}^{\\prime }+{F}^{\\prime }=0[\/latex], then [latex]{B}^{2}-4AC={{B}^{\\prime }}^{2}-4{A}^{\\prime }{C}^{\\prime }[\/latex].<\/p>\n<p>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>\n<div class=\"textbox shaded\">\n<h3>Solution<\/h3>\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{array}{l}{B}^{2}-4AC={\\left(2\\sqrt{3}\\right)}^{2}-4\\left(5\\right)\\left(2\\right)\\hfill \\\\ \\text{ }=4\\left(3\\right)-40\\hfill \\\\ \\text{ }=12 - 40\\hfill \\\\ \\text{ }=-28<0\\hfill \\end{array}[\/latex]<\/div>\n<p>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{array}{l}{B}^{2}-4AC={\\left(2\\sqrt{3}\\right)}^{2}-4\\left(5\\right)\\left(12\\right)\\hfill \\\\ \\text{ }=4\\left(3\\right)-240\\hfill \\\\ \\text{ }=12 - 240\\hfill \\\\ \\text{ }=-228<0\\hfill \\end{array}[\/latex]<\/div>\n<p>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 class=\"bcc-box bcc-success\">\n<h3>Try It 3<\/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<p><a href=\"https:\/\/courses.candelalearning.com\/precalctwo1xmaster\/chapter\/solutions-26\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\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-1961\">\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":276,"menu_order":5,"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-1961","chapter","type-chapter","status-publish","hentry"],"part":1942,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1961","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1961\/revisions"}],"predecessor-version":[{"id":2194,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1961\/revisions\/2194"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1942"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1961\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/media?parent=1961"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1961"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1961"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/license?post=1961"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}