Chapter 10 - Analytic Geometry - 10.2 Ellipses - 10.2 Exercises - Page 978: 11

$$\eqalign{ & {\text{domain: }}\left[ { - 3,3} \right] \cr & {\text{range: }}\left[ { - 9,9} \right] \cr & {\text{center: }}\left( {0,0} \right) \cr & {\text{Vertices: }}\left( {0, \pm 9} \right) \cr & {\text{Endpoints of the minor axis:}}\left( {0, \pm 3} \right) \cr & {\text{Foci }}\left( {0, \pm 6\sqrt 2 } \right) \cr} $$

$$\eqalign{ & 9{x^2} + {y^2} = 81 \cr & {\text{Divide both sides by 81}} \cr & \frac{{9{x^2}}}{{81}} + \frac{{{y^2}}}{{81}} = \frac{{81}}{{81}} \cr & \frac{{{x^2}}}{9} + \frac{{{y^2}}}{{81}} = 1 \cr & {\text{The equation of the ellipse is in the form }}\frac{{{x^2}}}{{{b^2}}} + \frac{{{y^2}}}{{{a^2}}} = 1\,\,\left( {a > b} \right) \cr & a = 9,\,\,b = 3 \cr & c = \sqrt {{9^2} - {3^2}} = 6\sqrt 2 \cr & \cr & {\text{The ellipse with center at the origin}} \cr & {\text{Vertices }}\left( {0, \pm a} \right) \cr & {\text{Vertices }}\left( {0, \pm 9} \right) \cr & \cr & {\text{Foci }}\left( {0, \pm c} \right) \cr & {\text{Foci }}\left( {0, \pm 6\sqrt 2 } \right) \cr & \cr & {\text{The domain of the ellipse is }}\left[ { - b,b} \right] \cr & {\text{domain }}\left[ { - 3,3} \right] \cr & \cr & {\text{The range of the ellipse is }}\left[ { - a,a} \right] \cr & {\text{range }}\left[ { - 9,9} \right] \cr & \cr & {\text{Endpoints of the minor axis: }}\left( {0,\, \pm b} \right) = \left( {0, \pm 3} \right) \cr & \cr & {\text{Therefore,}} \cr & {\text{domain: }}\left[ { - 3,3} \right] \cr & {\text{range: }}\left[ { - 9,9} \right] \cr & {\text{center: }}\left( {0,0} \right) \cr & {\text{Vertices: }}\left( {0, \pm 9} \right) \cr & {\text{Endpoints of the minor axis:}}\left( {0, \pm 3} \right) \cr & {\text{Foci }}\left( {0, \pm 6\sqrt 2 } \right) \cr} $$