Chapter 10 - Analytic Geometry - 10.2 Ellipses - 10.2 Exercises - Page 978: 13

$$\eqalign{ & {\text{domain: }}\left[ { - 5,5} \right] \cr & {\text{range: }}\left[ { - 2,2} \right] \cr & {\text{center: }}\left( {0,0} \right) \cr & {\text{Vertices: }}\left( { \pm 5,0} \right) \cr & {\text{Endpoints of the minor axis:}}\left( {0, \pm 2} \right) \cr & {\text{Foci }}\left( { \pm \sqrt {21} ,0} \right) \cr} $$

$$\eqalign{ & 4{x^2} = 100 - 25{y^2} \cr & 4{x^2} + 25{y^2} = 100 \cr & {\text{Divide both sides by }}100 \cr & \frac{{4{x^2}}}{{100}} + \frac{{25{y^2}}}{{100}} = \frac{{100}}{{100}} \cr & \frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{4} = 1 \cr & {\text{The equation of the ellipse is in the form }}\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\,\,\left( {a > b} \right) \cr & a = 5,\,\,b = 2 \cr & c = \sqrt {{5^2} - {2^2}} = \sqrt {21} \cr & \cr & {\text{The ellipse with center at the origin}} \cr & {\text{Vertices }}\left( { \pm a,0} \right) \cr & {\text{Vertices }}\left( { \pm 5,0} \right) \cr & \cr & {\text{Foci }}\left( { \pm c,0} \right) \cr & {\text{Foci }}\left( { \pm \sqrt {21} ,0} \right) \cr & \cr & {\text{The domain of the ellipse is }}\left[ { - a,a} \right] \cr & {\text{domain }}\left[ { - 5,5} \right] \cr & \cr & {\text{The range of the ellipse is }}\left[ { - b,b} \right] \cr & {\text{range }}\left[ { - 2,2} \right] \cr & \cr & {\text{Endpoints of the minor axis: }}\left( {0,\, \pm b} \right) = \left( {0, \pm 2} \right) \cr & \cr & {\text{Therefore,}} \cr & {\text{domain: }}\left[ { - 5,5} \right] \cr & {\text{range: }}\left[ { - 2,2} \right] \cr & {\text{center: }}\left( {0,0} \right) \cr & {\text{Vertices: }}\left( { \pm 5,0} \right) \cr & {\text{Endpoints of the minor axis:}}\left( {0, \pm 2} \right) \cr & {\text{Foci }}\left( { \pm \sqrt {21} ,0} \right) \cr} $$