{"id":2573,"date":"2025-08-13T18:07:29","date_gmt":"2025-08-13T18:07:29","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2573"},"modified":"2025-08-13T18:07:49","modified_gmt":"2025-08-13T18:07:49","slug":"rotation-of-axes-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rotation-of-axes-learn-it-4\/","title":{"raw":"Rotation of Axes: Learn It 4","rendered":"Rotation of Axes: Learn It 4"},"content":{"raw":"

Identifying Conics without Rotating Axes<\/span><\/h2>\r\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\r\n
[latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\r\nIf we apply the rotation formulas to this equation we get the form\r\n
[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>\r\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> 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.\r\n\r\n
\r\n

the discriminant<\/h3>\r\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].\r\n\r\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.\r\n\r\nIf the discriminant, [latex]{B}^{2}-4AC[\/latex], is\r\n
    \r\n \t
  • [latex]<0[\/latex], the conic section is an ellipse<\/li>\r\n \t
  • [latex]=0[\/latex], the conic section is a parabola<\/li>\r\n \t
  • [latex]>0[\/latex], the conic section is a hyperbola<\/li>\r\n<\/ul>\r\n<\/section>
    Identify the conic for each of the following without rotating axes.\r\n
      \r\n \t
    1. [latex]5{x}^{2}+2\\sqrt{3}xy+2{y}^{2}-5=0[\/latex]<\/li>\r\n \t
    2. [latex]5{x}^{2}+2\\sqrt{3}xy+12{y}^{2}-5=0[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"255593\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"255593\"]\r\n
        \r\n \t
      1. Let\u2019s begin by determining [latex]A,B[\/latex], and [latex]C[\/latex].\r\n
        [latex]\\underset{A}{\\underbrace{5}}{x}^{2}+\\underset{B}{\\underbrace{2\\sqrt{3}}}xy+\\underset{C}{\\underbrace{2}}{y}^{2}-5=0[\/latex]<\/div>\r\nNow, we find the discriminant.\r\n
        [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>\r\nTherefore, [latex]5{x}^{2}+2\\sqrt{3}xy+2{y}^{2}-5=0[\/latex] represents an ellipse.<\/li>\r\n \t
      2. Again, let\u2019s begin by determining [latex]A,B[\/latex], and [latex]C[\/latex].\r\n
        [latex]\\underset{A}{\\underbrace{5}}{x}^{2}+\\underset{B}{\\underbrace{2\\sqrt{3}}}xy+\\underset{C}{\\underbrace{12}}{y}^{2}-5=0[\/latex]<\/div>\r\nNow, we find the discriminant.\r\n
        [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>\r\nTherefore, [latex]5{x}^{2}+2\\sqrt{3}xy+12{y}^{2}-5=0[\/latex] represents an ellipse.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
        Identify the conic for each of the following without rotating axes.\r\n
          \r\n \t
        1. [latex]{x}^{2}-9xy+3{y}^{2}-12=0[\/latex]<\/li>\r\n \t
        2. [latex]10{x}^{2}-9xy+4{y}^{2}-4=0[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"67134\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"67134\"]\r\n
            \r\n \t
          1. hyperbola<\/li>\r\n \t
          2. ellipse<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"

            Identifying Conics without Rotating Axes<\/span><\/h2>\n

            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

            [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex]<\/div>\n

            If we apply the rotation formulas to this equation we get the form<\/p>\n

            [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

            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> The discriminant, [latex]{B}^{2}-4AC[\/latex], is invariant and remains unchanged after rotation. Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section.<\/p>\n

            \n

            the discriminant<\/h3>\n

            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

            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

            If the discriminant, [latex]{B}^{2}-4AC[\/latex], is<\/p>\n

              \n
            • [latex]<0[\/latex], the conic section is an ellipse<\/li>\n
            • [latex]=0[\/latex], the conic section is a parabola<\/li>\n
            • [latex]>0[\/latex], the conic section is a hyperbola<\/li>\n<\/ul>\n<\/section>\n
              Identify the conic for each of the following without rotating axes.<\/p>\n
                \n
              1. [latex]5{x}^{2}+2\\sqrt{3}xy+2{y}^{2}-5=0[\/latex]<\/li>\n
              2. [latex]5{x}^{2}+2\\sqrt{3}xy+12{y}^{2}-5=0[\/latex]<\/li>\n<\/ol>\n