Chapter 10 - Analytic Geometry - 10.2 Ellipses - 10.2 Exercises - Page 978: 16

$$\eqalign{ & {\text{domain: }}\left[ { - 6,2} \right] \cr & {\text{range: }}\left[ { - 4,2} \right] \cr & {\text{center: }}\left( { - 2, - 1} \right) \cr & {\text{Vertices: }}\left( {2,1} \right){\text{ and }}\left( { - 6,1} \right) \cr & {\text{Endpoints of the minor axis: }}\left( { - 2, - 3} \right){\text{ and }}\left( { - 2,2} \right) \cr & {\text{Foci }}\left( { - 2 + \sqrt 7 , - 1} \right){\text{ and }}\left( { - 2 - \sqrt 7 , - 1} \right) \cr} $$

$$\eqalign{ & \frac{{{{\left( {x + 2} \right)}^2}}}{{16}} + \frac{{{{\left( {y + 1} \right)}^2}}}{9} = 1 \cr & {\text{The equation of the ellipse is in the form }} \cr & \frac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}} + \frac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1\,\,\,\,\left( {a > b} \right) \cr & a = 4,\,\,b = 3 \cr & h = \, - 2,\,\,k = - 1\, \cr & c = \sqrt {{4^2} - {3^2}} = \sqrt 7 \cr & \cr & {\text{The ellipse with center at }}\left( {h,k} \right) = \left( { - 2, - 1} \right) \cr & {\text{Vertices }}\left( {h \pm a,k} \right) \cr & {\text{Vertices: }}\left( {2,1} \right){\text{ and }}\left( { - 6,1} \right) \cr & \cr & {\text{Foci }}\left( {h \pm c,k} \right) \cr & {\text{Foci: }}\left( { - 2 + \sqrt 7 , - 1} \right){\text{ and }}\left( { - 2 - \sqrt 7 , - 1} \right) \cr & \cr & {\text{The domain of the ellipse is }}\left[ {h - a,h + a} \right] \cr & {\text{domain }}\left[ { - 6,2} \right] \cr & \cr & {\text{The range of the ellipse is }}\left[ {k - b,k + b} \right] \cr & {\text{range }}\left[ { - 4,2} \right] \cr & \cr & {\text{Endpoints of the minor axis: }}\left( {h,k\, \pm b} \right) \cr & {\text{Endpoints of the minor axis: }}\left( { - 2, - 3} \right){\text{ and }}\left( { - 2,2} \right) \cr & \cr & {\text{Therefore,}} \cr & {\text{domain: }}\left[ { - 6,2} \right] \cr & {\text{range: }}\left[ { - 4,2} \right] \cr & {\text{center: }}\left( { - 2, - 1} \right) \cr & {\text{Vertices: }}\left( {2,1} \right){\text{ and }}\left( { - 6,1} \right) \cr & {\text{Endpoints of the minor axis: }}\left( { - 2, - 3} \right){\text{ and }}\left( { - 2,2} \right) \cr & {\text{Foci }}\left( { - 2 + \sqrt 7 , - 1} \right){\text{ and }}\left( { - 2 - \sqrt 7 , - 1} \right) \cr} $$