{"id":1955,"date":"2015-11-12T18:30:43","date_gmt":"2015-11-12T18:30:43","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1955"},"modified":"2015-11-12T18:30:43","modified_gmt":"2015-11-12T18:30:43","slug":"finding-a-new-representation-of-the-given-equation-after-rotating-through-a-given-angle","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/finding-a-new-representation-of-the-given-equation-after-rotating-through-a-given-angle\/","title":{"raw":"Finding a New Representation of the Given Equation after Rotating through a Given Angle","rendered":"Finding a New Representation of the Given Equation after Rotating through a Given Angle"},"content":{"raw":"

Until now, we have looked at equations of conic sections without an [latex]xy[\/latex] term, which aligns the graphs with the x<\/em>- and y<\/em>-axes. When we add an [latex]xy[\/latex] term, we are rotating the conic about the origin. If the x<\/em>- and 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 x<\/em>-axis and y<\/em>-axis, and [latex]\\left({x}^{\\prime },{y}^{\\prime }\\right)[\/latex] on the new plane defined by the new, rotated axes, called the x'<\/em>-axis and y'<\/em>-axis.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\"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]{x}^{\\prime }[\/latex] and [latex]{y}^{\\prime }[\/latex] on the new rotated plane.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\"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]\\text{ }\\theta [\/latex].[\/caption]The original coordinate x<\/em>- and y<\/em>-axes have unit vectors [latex]i[\/latex] and [latex]j[\/latex]. The rotated coordinate axes have unit vectors [latex]{i}^{\\prime }[\/latex] and [latex]{j}^{\\prime }[\/latex]. The angle [latex]\\theta [\/latex] is known as the angle of rotation<\/strong>. We may write the new unit vectors in terms of the original ones.\n<\/p>

[latex]\\begin{array}{l}{i}^{\\prime }=\\cos \\text{ }\\theta i+\\sin \\text{ }\\theta j\\hfill \\\\ {j}^{\\prime }=-\\sin \\text{ }\\theta i+\\cos \\text{ }\\theta j\\hfill \\end{array}[\/latex]<\/div>\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\"Figure 5.<\/b> Relationship between the old and new coordinate planes.[\/caption]\n\nConsider a vector [latex]u[\/latex] <\/strong>in the new coordinate plane. It may be represented in terms of its coordinate axes.\n
[latex]\\begin{array}{ll}u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime }\\hfill & \\hfill \\\\ u={x}^{\\prime }\\left(i\\text{ }\\cos \\text{ }\\theta +j\\text{ }\\sin \\text{ }\\theta \\right)+{y}^{\\prime }\\left(-i\\text{ }\\sin \\text{ }\\theta +j\\text{ }\\cos \\text{ }\\theta \\right)\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Substitute}.\\hfill \\\\ u=ix\\text{'}\\text{ }\\cos \\text{ }\\theta +jx\\text{'}\\text{ }\\sin \\text{ }\\theta -iy\\text{'}\\text{ }\\sin \\text{ }\\theta +jy\\text{'}\\text{ }\\cos \\text{ }\\theta \\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Distribute}.\\hfill \\\\ u=ix\\text{'}\\text{ }\\cos \\text{ }\\theta -iy\\text{'}\\text{ }\\sin \\text{ }\\theta +jx\\text{'}\\text{ }\\sin \\text{ }\\theta +jy\\text{'}\\text{ }\\cos \\text{ }\\theta \\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Apply commutative property}.\\hfill \\\\ u=\\left(x\\text{'}\\text{ }\\cos \\text{ }\\theta -y\\text{'}\\text{ }\\sin \\text{ }\\theta \\right)i+\\left(x\\text{'}\\text{ }\\sin \\text{ }\\theta +y\\text{'}\\text{ }\\cos \\text{ }\\theta \\right)j\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Factor by grouping}.\\hfill \\end{array}[\/latex]<\/div>\nBecause [latex]u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime }[\/latex], we have representations of [latex]x[\/latex] and [latex]y[\/latex] in terms of the new coordinate system.\n
[latex]\\begin{array}{c}x={x}^{\\prime }\\cos \\text{ }\\theta -{y}^{\\prime }\\sin \\text{ }\\theta \\\\ \\text{and}\\\\ y={x}^{\\prime }\\sin \\text{ }\\theta +{y}^{\\prime }\\cos \\text{ }\\theta \\end{array}[\/latex]<\/div>\n
\n

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 x<\/em>-axis, then the coordinates of the point with respect to the new axes are [latex]\\left({x}^{\\prime },{y}^{\\prime }\\right)[\/latex]. We can use the following equations of rotation to define the relationship between [latex]\\left(x,y\\right)[\/latex] and [latex]\\left({x}^{\\prime },{y}^{\\prime }\\right):[\/latex]\n
[latex]x={x}^{\\prime }\\cos \\text{ }\\theta -{y}^{\\prime }\\sin \\text{ }\\theta [\/latex]<\/div>\nand\n
[latex]y={x}^{\\prime }\\sin \\text{ }\\theta +{y}^{\\prime }\\cos \\text{ }\\theta [\/latex]<\/div>\n<\/div>\n
\n

How To: Given the equation of a conic, find a new representation after rotating through an angle.\n<\/strong><\/h3>\n
  1. Find [latex]x[\/latex] and [latex]y[\/latex] where [latex]x={x}^{\\prime }\\cos \\text{ }\\theta -{y}^{\\prime }\\sin \\text{ }\\theta [\/latex] and [latex]y={x}^{\\prime }\\sin \\text{ }\\theta +{y}^{\\prime }\\cos \\text{ }\\theta [\/latex].<\/li>\n\t
  2. Substitute the expression for [latex]x[\/latex] and [latex]y[\/latex] into in the given equation, then simplify.<\/li>\n\t
  3. Write the equations with [latex]{x}^{\\prime }[\/latex] and [latex]{y}^{\\prime }[\/latex] in standard form.<\/li>\n<\/ol><\/div>\n
    \n

    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<\/div>\n
    \n

    Solution<\/h3>\nFind [latex]x[\/latex] and [latex]y[\/latex], where [latex]x={x}^{\\prime }\\cos \\text{ }\\theta -{y}^{\\prime }\\sin \\text{ }\\theta [\/latex] and [latex]y={x}^{\\prime }\\sin \\text{ }\\theta +{y}^{\\prime }\\cos \\text{ }\\theta [\/latex].\n\nBecause [latex]\\theta =45^\\circ [\/latex],\n
    [latex]\\begin{array}{l}\\hfill \\\\ x={x}^{\\prime }\\cos \\left(45^\\circ \\right)-{y}^{\\prime }\\sin \\left(45^\\circ \\right)\\hfill \\\\ x={x}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)-{y}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)\\hfill \\\\ x=\\frac{{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{2}}\\hfill \\end{array}[\/latex]<\/div>\nand\n
    [latex]\\begin{array}{l}\\\\ \\begin{array}{l}y={x}^{\\prime }\\sin \\left(45^\\circ \\right)+{y}^{\\prime }\\cos \\left(45^\\circ \\right)\\hfill \\\\ y={x}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)+{y}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)\\hfill \\\\ y=\\frac{{x}^{\\prime }+{y}^{\\prime }}{\\sqrt{2}}\\hfill \\end{array}\\end{array}[\/latex]<\/div>\nSubstitute [latex]x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta [\/latex] and [latex]y={x}^{\\prime }\\sin \\text{ }\\theta +{y}^{\\prime }\\cos \\text{ }\\theta [\/latex] into [latex]2{x}^{2}-xy+2{y}^{2}-30=0[\/latex].\n
    [latex]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[\/latex]<\/div>\nSimplify.\n
    [latex]\\begin{array}{ll}\\overline{)2}\\frac{\\left({x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }-{y}^{\\prime }\\right)}{\\overline{)2}}-\\frac{\\left({x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }+{y}^{\\prime }\\right)}{2}+\\overline{)2}\\frac{\\left({x}^{\\prime }+{y}^{\\prime }\\right)\\left({x}^{\\prime }+{y}^{\\prime }\\right)}{\\overline{)2}}-30=0\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{FOIL method}\\hfill \\\\ \\text{ }{x}^{\\prime }{}^{2}{\\overline{)-2{x}^{\\prime }y}}^{\\prime }+{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}+{x}^{\\prime }{}^{2}\\overline{)+2{x}^{\\prime }{y}^{\\prime }}+{y}^{\\prime }{}^{2}-30=0\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Combine like terms}.\\hfill \\\\ \\text{ }2{x}^{\\prime }{}^{2}+2{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}=30\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Combine like terms}.\\hfill \\\\ \\text{ }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)\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Multiply both sides by 2}.\\hfill \\\\ \\text{ }4{x}^{\\prime }{}^{2}+4{y}^{\\prime }{}^{2}-\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)=60\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Simplify}.\\hfill \\\\ \\text{ }4{x}^{\\prime }{}^{2}+4{y}^{\\prime }{}^{2}-{x}^{\\prime }{}^{2}+{y}^{\\prime }{}^{2}=60\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Distribute}.\\hfill \\\\ \\text{ }\\frac{3{x}^{\\prime }{}^{2}}{60}+\\frac{5{y}^{\\prime }{}^{2}}{60}=\\frac{60}{60}\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Set equal to 1}.\\hfill \\end{array}[\/latex]<\/div>\nWrite the equations with [latex]{x}^{\\prime }[\/latex] and [latex]{y}^{\\prime }[\/latex] in the standard form.\n
    [latex]\\frac{{{x}^{\\prime }}^{2}}{20}+\\frac{{{y}^{\\prime }}^{2}}{12}=1[\/latex]<\/div>\nThis equation is an ellipse. Figure 6\u00a0shows the graph.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\"Figure 6<\/b>[\/caption]\n\n<\/div>","rendered":"

    Until now, we have looked at equations of conic sections without an [latex]xy[\/latex] term, which aligns the graphs with the x<\/em>– and y<\/em>-axes. When we add an [latex]xy[\/latex] term, we are rotating the conic about the origin. If the x<\/em>– and 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 x<\/em>-axis and y<\/em>-axis, and [latex]\\left({x}^{\\prime },{y}^{\\prime }\\right)[\/latex] on the new plane defined by the new, rotated axes, called the x’<\/em>-axis and y’<\/em>-axis.<\/p>\n

    \"\"<\/p>\n

    Figure 3.<\/b> The graph of the rotated ellipse [latex]{x}^{2}+{y}^{2}-xy - 15=0[\/latex]<\/p>\n<\/div>\n

    We will find the relationships between [latex]x[\/latex] and [latex]y[\/latex] on the Cartesian plane with [latex]{x}^{\\prime }[\/latex] and [latex]{y}^{\\prime }[\/latex] on the new rotated plane.<\/p>\n

    \"\"<\/p>\n

    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]\\text{ }\\theta [\/latex].<\/p>\n<\/div>\n

    The original coordinate x<\/em>– and y<\/em>-axes have unit vectors [latex]i[\/latex] and [latex]j[\/latex]. The rotated coordinate axes have unit vectors [latex]{i}^{\\prime }[\/latex] and [latex]{j}^{\\prime }[\/latex]. The angle [latex]\\theta[\/latex] is known as the angle of rotation<\/strong>. We may write the new unit vectors in terms of the original ones.\n<\/p>\n

    [latex]\\begin{array}{l}{i}^{\\prime }=\\cos \\text{ }\\theta i+\\sin \\text{ }\\theta j\\hfill \\\\ {j}^{\\prime }=-\\sin \\text{ }\\theta i+\\cos \\text{ }\\theta j\\hfill \\end{array}[\/latex]<\/div>\n
    \"\"<\/p>\n

    Figure 5.<\/b> Relationship between the old and new coordinate planes.<\/p>\n<\/div>\n

    Consider a vector [latex]u[\/latex] <\/strong>in the new coordinate plane. It may be represented in terms of its coordinate axes.<\/p>\n

    [latex]\\begin{array}{ll}u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime }\\hfill & \\hfill \\\\ u={x}^{\\prime }\\left(i\\text{ }\\cos \\text{ }\\theta +j\\text{ }\\sin \\text{ }\\theta \\right)+{y}^{\\prime }\\left(-i\\text{ }\\sin \\text{ }\\theta +j\\text{ }\\cos \\text{ }\\theta \\right)\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Substitute}.\\hfill \\\\ u=ix\\text{'}\\text{ }\\cos \\text{ }\\theta +jx\\text{'}\\text{ }\\sin \\text{ }\\theta -iy\\text{'}\\text{ }\\sin \\text{ }\\theta +jy\\text{'}\\text{ }\\cos \\text{ }\\theta \\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Distribute}.\\hfill \\\\ u=ix\\text{'}\\text{ }\\cos \\text{ }\\theta -iy\\text{'}\\text{ }\\sin \\text{ }\\theta +jx\\text{'}\\text{ }\\sin \\text{ }\\theta +jy\\text{'}\\text{ }\\cos \\text{ }\\theta \\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Apply commutative property}.\\hfill \\\\ u=\\left(x\\text{'}\\text{ }\\cos \\text{ }\\theta -y\\text{'}\\text{ }\\sin \\text{ }\\theta \\right)i+\\left(x\\text{'}\\text{ }\\sin \\text{ }\\theta +y\\text{'}\\text{ }\\cos \\text{ }\\theta \\right)j\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Factor by grouping}.\\hfill \\end{array}[\/latex]<\/div>\n

    Because [latex]u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime }[\/latex], we have representations of [latex]x[\/latex] and [latex]y[\/latex] in terms of the new coordinate system.<\/p>\n

    [latex]\\begin{array}{c}x={x}^{\\prime }\\cos \\text{ }\\theta -{y}^{\\prime }\\sin \\text{ }\\theta \\\\ \\text{and}\\\\ y={x}^{\\prime }\\sin \\text{ }\\theta +{y}^{\\prime }\\cos \\text{ }\\theta \\end{array}[\/latex]<\/div>\n
    \n

    A General Note: Equations of Rotation<\/h3>\n

    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 x<\/em>-axis, then the coordinates of the point with respect to the new axes are [latex]\\left({x}^{\\prime },{y}^{\\prime }\\right)[\/latex]. We can use the following equations of rotation to define the relationship between [latex]\\left(x,y\\right)[\/latex] and [latex]\\left({x}^{\\prime },{y}^{\\prime }\\right):[\/latex]<\/p>\n

    [latex]x={x}^{\\prime }\\cos \\text{ }\\theta -{y}^{\\prime }\\sin \\text{ }\\theta[\/latex]<\/div>\n

    and<\/p>\n

    [latex]y={x}^{\\prime }\\sin \\text{ }\\theta +{y}^{\\prime }\\cos \\text{ }\\theta[\/latex]<\/div>\n<\/div>\n
    \n

    How To: Given the equation of a conic, find a new representation after rotating through an angle.
    \n<\/strong><\/h3>\n
      \n
    1. Find [latex]x[\/latex] and [latex]y[\/latex] where [latex]x={x}^{\\prime }\\cos \\text{ }\\theta -{y}^{\\prime }\\sin \\text{ }\\theta[\/latex] and [latex]y={x}^{\\prime }\\sin \\text{ }\\theta +{y}^{\\prime }\\cos \\text{ }\\theta[\/latex].<\/li>\n
    2. Substitute the expression for [latex]x[\/latex] and [latex]y[\/latex] into in the given equation, then simplify.<\/li>\n
    3. Write the equations with [latex]{x}^{\\prime }[\/latex] and [latex]{y}^{\\prime }[\/latex] in standard form.<\/li>\n<\/ol>\n<\/div>\n
      \n

      Example 2: Finding a New Representation of an Equation after Rotating through a Given Angle<\/h3>\n

      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>\n

      \n

      Solution<\/h3>\n

      Find [latex]x[\/latex] and [latex]y[\/latex], where [latex]x={x}^{\\prime }\\cos \\text{ }\\theta -{y}^{\\prime }\\sin \\text{ }\\theta[\/latex] and [latex]y={x}^{\\prime }\\sin \\text{ }\\theta +{y}^{\\prime }\\cos \\text{ }\\theta[\/latex].<\/p>\n

      Because [latex]\\theta =45^\\circ[\/latex],<\/p>\n

      [latex]\\begin{array}{l}\\hfill \\\\ x={x}^{\\prime }\\cos \\left(45^\\circ \\right)-{y}^{\\prime }\\sin \\left(45^\\circ \\right)\\hfill \\\\ x={x}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)-{y}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)\\hfill \\\\ x=\\frac{{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{2}}\\hfill \\end{array}[\/latex]<\/div>\n

      and<\/p>\n

      [latex]\\begin{array}{l}\\\\ \\begin{array}{l}y={x}^{\\prime }\\sin \\left(45^\\circ \\right)+{y}^{\\prime }\\cos \\left(45^\\circ \\right)\\hfill \\\\ y={x}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)+{y}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)\\hfill \\\\ y=\\frac{{x}^{\\prime }+{y}^{\\prime }}{\\sqrt{2}}\\hfill \\end{array}\\end{array}[\/latex]<\/div>\n

      Substitute [latex]x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta[\/latex] and [latex]y={x}^{\\prime }\\sin \\text{ }\\theta +{y}^{\\prime }\\cos \\text{ }\\theta[\/latex] into [latex]2{x}^{2}-xy+2{y}^{2}-30=0[\/latex].<\/p>\n

      [latex]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[\/latex]<\/div>\n

      Simplify.<\/p>\n

      [latex]\\begin{array}{ll}\\overline{)2}\\frac{\\left({x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }-{y}^{\\prime }\\right)}{\\overline{)2}}-\\frac{\\left({x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }+{y}^{\\prime }\\right)}{2}+\\overline{)2}\\frac{\\left({x}^{\\prime }+{y}^{\\prime }\\right)\\left({x}^{\\prime }+{y}^{\\prime }\\right)}{\\overline{)2}}-30=0\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{FOIL method}\\hfill \\\\ \\text{ }{x}^{\\prime }{}^{2}{\\overline{)-2{x}^{\\prime }y}}^{\\prime }+{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}+{x}^{\\prime }{}^{2}\\overline{)+2{x}^{\\prime }{y}^{\\prime }}+{y}^{\\prime }{}^{2}-30=0\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Combine like terms}.\\hfill \\\\ \\text{ }2{x}^{\\prime }{}^{2}+2{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}=30\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Combine like terms}.\\hfill \\\\ \\text{ }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)\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Multiply both sides by 2}.\\hfill \\\\ \\text{ }4{x}^{\\prime }{}^{2}+4{y}^{\\prime }{}^{2}-\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)=60\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Simplify}.\\hfill \\\\ \\text{ }4{x}^{\\prime }{}^{2}+4{y}^{\\prime }{}^{2}-{x}^{\\prime }{}^{2}+{y}^{\\prime }{}^{2}=60\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Distribute}.\\hfill \\\\ \\text{ }\\frac{3{x}^{\\prime }{}^{2}}{60}+\\frac{5{y}^{\\prime }{}^{2}}{60}=\\frac{60}{60}\\hfill & \\begin{array}{cccc}& & & \\end{array}\\text{Set equal to 1}.\\hfill \\end{array}[\/latex]<\/div>\n

      Write the equations with [latex]{x}^{\\prime }[\/latex] and [latex]{y}^{\\prime }[\/latex] in the standard form.<\/p>\n

      [latex]\\frac{{{x}^{\\prime }}^{2}}{20}+\\frac{{{y}^{\\prime }}^{2}}{12}=1[\/latex]<\/div>\n

      This equation is an ellipse. Figure 6\u00a0shows the graph.<\/p>\n

      \"\"<\/p>\n

      Figure 6<\/b><\/p>\n<\/div>\n<\/div>\n\n\t\t\t

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