Chapter 10 - Parametric And Polar Curves; Conic Sections - 10.4 Conic Sections - Exercises Set 10.4 - Page 745: 7

$$\eqalign{ & \left( {\bf{a}} \right) \cr & {\text{foci : }}\left( { - \sqrt 7 ,0} \right){\text{ and }}\left( {\sqrt 7 ,0} \right) \cr & {\text{Vertices: }}\left( { - 4,0} \right){\text{ and }}\left( {4,0} \right) \cr & {\text{Ends of the minor axis: }}\left( {0, - 3} \right){\text{ and }}\left( {0,3} \right) \cr & \left( {\bf{b}} \right) \cr & {\text{foci : }}\left( {0 - \sqrt 8 } \right){\text{ and }}\left( {0,\sqrt 8 } \right) \cr & {\text{Vertices: }}\left( {0, - 3} \right){\text{ and }}\left( {0,3} \right) \cr & {\text{Ends of the minor axis: }}\left( { - b,0} \right){\text{ and }}\left( {b,0} \right) \cr} $$

$$\eqalign{ & \left( {\bf{a}} \right)\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{9} = 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}}}{{16}} + \frac{{{y^2}}}{9} = 1\,\,\,\,\,\,\,\, \to \,\,\,\,\,\,a = 4,\,\,\,\,b = 3 \cr & {c^2} = {a^2} - {b^2} \cr & {c^2} = 16 - 9 \cr & {c^2} = 7 \cr & c = \sqrt 7 \cr & \cr & {\text{With:}} \cr & {\text{foci: }}\left( { - c,0} \right){\text{ and }}\left( {c,0} \right) \to {\text{foci : }}\left( { - \sqrt 7 ,0} \right){\text{ and }}\left( {\sqrt 7 ,0} \right) \cr & {\text{Vertices: }}\left( { - a,0} \right){\text{ and }}\left( {a,0} \right) \to {\text{Vertices: }}\left( { - 4,0} \right){\text{ and }}\left( {4,0} \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, - 3} \right){\text{ and }}\left( {0,3} \right) \cr & \cr & \left( {\bf{b}} \right)9{x^2} + {y^2} = 9 \cr & {\text{Divide both sides by 9}} \cr & \frac{{9{x^2}}}{9} + \frac{{{y^2}}}{9} = \frac{9}{9} \cr & {x^2} + \frac{{{y^2}}}{9} = 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:{x^2} + \frac{{{y^2}}}{9} = 1\,\,\,\,\,\,\, \to \,\,\,\,\,\,a = 3,\,\,\,\,b = 1 \cr & {c^2} = {a^2} - {b^2} \cr & {c^2} = 9 - 1 \cr & {c^2} = 8 \cr & c = \sqrt 8 \cr & \cr & {\text{With:}} \cr & {\text{foci: }}\left( {0, - c} \right){\text{ and }}\left( {0,c} \right) \to {\text{foci : }}\left( {0 - \sqrt 8 } \right){\text{ and }}\left( {0,\sqrt 8 } \right) \cr & {\text{Vertices: }}\left( {0, - a} \right){\text{ and }}\left( {0,a} \right) \to {\text{Vertices: }}\left( {0, - 3} \right){\text{ and }}\left( {0,3} \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( { - 1,0} \right){\text{ and }}\left( {1,0} \right) \cr & \cr & {\text{Graphs}} \cr} $$