Chapter 10 - Conics, Parametric Equations, and Polar Coordinates - 10.1 Exercises - Page 692: 26

$$\eqalign{ & {\text{Vertex }}\left( { - 5, - 6} \right){\text{ and }}\left( { - 3, - 6} \right) \cr & {\text{Center}}\left( { - 4, - 6} \right) \cr & {\text{Foci }}\left( { - 4 - \frac{{\sqrt {15} }}{4}, - 6} \right){\text{ and }}\left( { - 4 + \frac{{\sqrt {15} }}{4}, - 6} \right) \cr} $$

$$\eqalign{ & {\left( {x + 4} \right)^2} + \frac{{{{\left( {y + 6} \right)}^2}}}{{1/4}} = 1 \cr & {\text{The equation has the standard form}} \cr & \frac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}} + \frac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1,{\text{ }}a > b \cr & {\text{Comparing}} \cr & {\left( {x + 4} \right)^2} + \frac{{{{\left( {y + 6} \right)}^2}}}{{{{\left( {1/2} \right)}^2}}} = 1 \cr & \to h = - 4,{\text{ }}k = - 6,{\text{ }}a = 1,{\text{ }}b = 1/2,{\text{ }}c = \frac{{\sqrt {15} }}{4} \cr & {\text{With}} \cr & {\text{Orientation of Major Axis Horizontal along the }}x{\text{ - axis}} \cr & \underbrace {{\text{Vertex }}\left( {h - a,k} \right){\text{ and }}\left( {h + a,k} \right)}_ \Downarrow \cr & {\text{Vertex }}\left( { - 4 - 1, - 6} \right){\text{ and }}\left( { - 4 + 1, - 6} \right) \cr & {\text{Vertex }}\left( { - 5, - 6} \right){\text{ and }}\left( { - 3, - 6} \right) \cr & \underbrace {{\text{Foci }}\left( {h - c,k} \right){\text{ and }}\left( {h + c,k} \right)}_ \Downarrow \cr & {\text{Foci }}\left( { - 4 - \frac{{\sqrt {15} }}{4}, - 6} \right){\text{ and }}\left( { - 4 + \frac{{\sqrt {15} }}{4}, - 6} \right) \cr & {\text{Excentricity }}e = \frac{{\frac{{\sqrt {15} }}{4}}}{1} = \frac{{\sqrt {15} }}{4} \cr & {\text{Center}}\left( {h,k} \right) \to {\text{Center}}\left( { - 4, - 6} \right) \cr & {\text{Graph:}} \cr & \cr & {\text{Vertex }}\left( { - 5, - 6} \right){\text{ and }}\left( { - 3, - 6} \right) \cr & {\text{Center}}\left( { - 4, - 6} \right) \cr & {\text{Foci }}\left( { - 4 - \frac{{\sqrt {15} }}{4}, - 6} \right){\text{ and }}\left( { - 4 + \frac{{\sqrt {15} }}{4}, - 6} \right) \cr} $$