Chapter 10 - Parametric And Polar Curves; Conic Sections - 10.4 Conic Sections - Exercises Set 10.4 - Page 745: 10

$$\eqalign{ & \left( {\bf{a}} \right) \cr & {\text{foci : }}\left( {1, - 3 - \sqrt 5 } \right){\text{ and }}\left( {1, - 3 + \sqrt 5 } \right) \cr & {\text{Vertices: }}\left( {1, - 6} \right){\text{ and }}\left( {1,0} \right) \cr & {\text{Ends of the minor axis: }}\left( { - 1, - 3} \right){\text{ and }}\left( {3, - 3} \right) \cr & \left( {\bf{b}} \right) \cr & {\text{foci : }}\left( { - 2,1} \right){\text{ and }}\left( { - 2,5} \right) \cr & {\text{Vertices: }}\left( { - 2,0} \right){\text{ and }}\left( { - 2,6} \right) \cr & {\text{Ends of the minor axis: }}\left( { - 2 - \sqrt 5 ,3} \right){\text{ and }}\left( { - 2 + \sqrt 5 ,3} \right) \cr} $$

$$\eqalign{ & \left( {\bf{a}} \right)9{x^2} + 4{y^2} - 18x + 24y + 9 = 0 \cr & \left( {9{x^2} - 18x} \right) + \left( {4{y^2} + 24y} \right) = - 9 \cr & 9\left( {{x^2} - 2x} \right) + 4\left( {{y^2} + 6y} \right) = - 9 \cr & {\text{Complete the square}} \cr & 9\left( {{x^2} - 2x + 1} \right) + 4\left( {{y^2} + 6y + 9} \right) = - 9 + 9\left( 1 \right) + 4\left( 9 \right) \cr & 9{\left( {x - 1} \right)^2} + 4{\left( {y + 3} \right)^2} = 36 \cr & {\text{Divide by 36}} \cr & \frac{{{{\left( {x - 1} \right)}^2}}}{4} + \frac{{{{\left( {y + 3} \right)}^2}}}{9} = 1 \cr & {\text{The standard form of the ellipse is }}\frac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}} + \frac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1 \cr & \frac{{{{\left( {x - h} \right)}^2}}}{{{b^2}}} + \frac{{{{\left( {y - k} \right)}^2}}}{{{a^2}}} = 1\,\,\,:\,\,\,\,\frac{{{{\left( {x - 1} \right)}^2}}}{4} + \frac{{{{\left( {y + 3} \right)}^2}}}{9} = 1 \cr & \,\, \to \,\,a = 3,\,\,\,\,b = 2,\,\,\,\,\,h = 1,\,\,\,\,\,k = - 3 \cr & {c^2} = {a^2} - {b^2} \cr & {c^2} = 9 - 4 \cr & c = \sqrt 5 \cr & \cr & {\text{With:}} \cr & {\text{foci: }}\left( {h,k - c} \right){\text{ and }}\left( {h,k + c} \right) \cr & {\text{foci : }}\left( {1, - 3 - \sqrt 5 } \right){\text{ and }}\left( {1, - 3 + \sqrt 5 } \right) \cr & {\text{Vertices: }}\left( {h,k - a} \right){\text{ and }}\left( {h,k + a} \right) \cr & {\text{Vertices: }}\left( {1, - 6} \right){\text{ and }}\left( {1,0} \right) \cr & {\text{Ends of the minor axis: }}\left( {h - b,k} \right){\text{ and }}\left( {h + b,k} \right) \cr & {\text{Ends of the minor axis: }}\left( { - 1, - 3} \right){\text{ and }}\left( {3, - 3} \right) \cr & \cr & \left( {\bf{b}} \right)5{x^2} + 9{y^2} + 20x - 54y = - 56 \cr & \left( {5{x^2} + 20x} \right) + \left( {9{y^2} - 54y} \right) = - 56 \cr & 5\left( {{x^2} + 4x} \right) + 9\left( {{y^2} - 6y} \right) = - 56 \cr & {\text{Complete the square}} \cr & 5\left( {{x^2} + 4x + 4} \right) + 9\left( {{y^2} - 6y + 9} \right) = - 56 + 5\left( 4 \right) + 9\left( 9 \right) \cr & 5{\left( {x + 2} \right)^2} + 9{\left( {y - 3} \right)^2} = 45 \cr & {\text{Divide by 45}} \cr & \frac{{{{\left( {x + 2} \right)}^2}}}{9} + \frac{{{{\left( {y - 3} \right)}^2}}}{5} = 1 \cr & {\text{The standard form of the ellipse is }}\frac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}} + \frac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1 \cr & \frac{{{{\left( {x - h} \right)}^2}}}{{{b^2}}} + \frac{{{{\left( {y - k} \right)}^2}}}{{{a^2}}} = 1\,\,\,:\,\,\,\,\frac{{{{\left( {x + 2} \right)}^2}}}{9} + \frac{{{{\left( {y - 3} \right)}^2}}}{5} = 1 \cr & \,\, \to \,\,a = 3,\,\,\,\,b = \sqrt 5 ,\,\,\,\,\,h = - 2,\,\,\,\,\,k = 3 \cr & {c^2} = {a^2} - {b^2} \cr & {c^2} = 9 - 5 \cr & c = 2 \cr & \cr & {\text{With:}} \cr & {\text{foci: }}\left( {h,k - c} \right){\text{ and }}\left( {h,k + c} \right) \cr & {\text{foci : }}\left( { - 2,1} \right){\text{ and }}\left( { - 2,5} \right) \cr & {\text{Vertices: }}\left( {h,k - a} \right){\text{ and }}\left( {h,k + a} \right) \cr & {\text{Vertices: }}\left( { - 2,0} \right){\text{ and }}\left( { - 2,6} \right) \cr & {\text{Ends of the minor axis: }}\left( {h - b,k} \right){\text{ and }}\left( {h + b,k} \right) \cr & {\text{Ends of the minor axis: }}\left( { - 2 - \sqrt 5 ,3} \right){\text{ and }}\left( { - 2 + \sqrt 5 ,3} \right) \cr} $$