Chapter 10 - Parametric And Polar Curves; Conic Sections - 10.4 Conic Sections - Exercises Set 10.4 - Page 745: 8

$$\eqalign{ & \left( {\bf{a}} \right) \cr & {\text{foci : }}\left( { - \sqrt {21} ,0} \right){\text{ and }}\left( {\sqrt {21} ,0} \right) \cr & {\text{Vertices: }}\left( { - 5,0} \right){\text{ and }}\left( {5,0} \right) \cr & {\text{Ends of the minor axis: }}\left( {0, - 2} \right){\text{ and }}\left( {0,2} \right) \cr & \left( {\bf{b}} \right) \cr & {\text{foci : }}\left( {0, - 3\sqrt 3 } \right){\text{ and }}\left( {0,3\sqrt 3 } \right) \cr & {\text{Vertices: }}\left( {0, - 6} \right){\text{ and }}\left( {0,6} \right) \cr & {\text{Ends of the minor axis: }}\left( { - 3,0} \right){\text{ and }}\left( {3,0} \right) \cr} $$

$$\eqalign{ & \left( {\bf{a}} \right)\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{4} = 1 \cr & {\text{The standard form of the ellipse is }}\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1 \cr & \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1:\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{4} = 1\,\,\,\,\,\,\,\, \to \,\,\,\,\,\,a = 5,\,\,\,\,b = 2 \cr & {c^2} = {a^2} - {b^2} \cr & {c^2} = 25 - 4 \cr & {c^2} = 21 \cr & c = \sqrt {21} \cr & \cr & {\text{With:}} \cr & {\text{foci: }}\left( { - c,0} \right){\text{ and }}\left( {c,0} \right) \to {\text{foci : }}\left( { - \sqrt {21} ,0} \right){\text{ and }}\left( {\sqrt {21} ,0} \right) \cr & {\text{Vertices: }}\left( { - a,0} \right){\text{ and }}\left( {a,0} \right) \to {\text{Vertices: }}\left( { - 5,0} \right){\text{ and }}\left( {5,0} \right) \cr & {\text{Ends of the minor axis: }}\left( {0, - b} \right){\text{ and }}\left( {0,b} \right) \cr & {\text{Ends of the minor axis: }}\left( {0, - 2} \right){\text{ and }}\left( {0,2} \right) \cr & \cr & \left( {\bf{b}} \right)4{x^2} + {y^2} = 36 \cr & {\text{Divide both sides by 36}} \cr & \frac{{4{x^2}}}{{{\text{36}}}} + \frac{{{y^2}}}{{{\text{36}}}} = \frac{{{\text{36}}}}{{{\text{36}}}} \cr & \frac{{{x^2}}}{{\text{9}}} + \frac{{{y^2}}}{{{\text{36}}}} = {\text{1}} \cr & {\text{The standard form of the ellipse is }}\frac{{{x^2}}}{{{b^2}}} + \frac{{{y^2}}}{{{a^2}}} = 1 \cr & \frac{{{x^2}}}{{{b^2}}} + \frac{{{y^2}}}{{{a^2}}} = 1:\frac{{{x^2}}}{{\text{9}}} + \frac{{{y^2}}}{{{\text{36}}}} = {\text{1}}\,\,\,\,\,\,\, \to \,\,\,\,\,\,a = 6,\,\,\,\,b = 3 \cr & {c^2} = {a^2} - {b^2} \cr & {c^2} = 36 - 9 \cr & {c^2} = 27 \cr & c = 3\sqrt 3 \cr & \cr & {\text{With:}} \cr & {\text{foci: }}\left( {0, - c} \right){\text{ and }}\left( {0,c} \right) \to {\text{foci : }}\left( {0, - 3\sqrt 3 } \right){\text{ and }}\left( {0,3\sqrt 3 } \right) \cr & {\text{Vertices: }}\left( {0, - a} \right){\text{ and }}\left( {0,a} \right) \to {\text{Vertices: }}\left( {0, - 6} \right){\text{ and }}\left( {0,6} \right) \cr & {\text{Ends of the minor axis: }}\left( { - b,0} \right){\text{ and }}\left( {b,0} \right) \cr & {\text{Ends of the minor axis: }}\left( { - 3,0} \right){\text{ and }}\left( {3,0} \right) \cr} $$