Chapter 10 - Conics, Parametric Equations, and Polar Coordinates - 10.1 Exercises - Page 692: 36

$$\eqalign{ & {\text{Vertices}}\left( {5, - 18} \right){\text{ and }}\left( {5,12} \right){\text{ }} \cr & {\text{Foci}}\left( {5, - 20} \right){\text{ and }}\left( {5,14} \right) \cr & {\text{Center}}\left( {5, - 3} \right) \cr & {\text{Asymptotes}}:{\text{ }}y = \pm \frac{{15}}{8}\left( {x - 5} \right) - 3 \cr} $$

$$\eqalign{ & \frac{{{{\left( {y + 3} \right)}^2}}}{{225}} - \frac{{{{\left( {x - 5} \right)}^2}}}{{64}} = 1 \cr & {\text{This equation is written in the standard form}} \cr & \frac{{{{\left( {y - k} \right)}^2}}}{{{a^2}}} - \frac{{{{\left( {x - h} \right)}^2}}}{{{b^2}}} = 1 \cr & {\text{Therefore}}{\text{,}} \cr & \frac{{{{\left( {y + 3} \right)}^2}}}{{225}} - \frac{{{{\left( {x - 5} \right)}^2}}}{{64}} = 1 \Rightarrow \frac{{{{\left( {y + 3} \right)}^2}}}{{{{15}^2}}} - \frac{{{{\left( {x - 5} \right)}^2}}}{{{8^2}}} = 1 \cr & \underbrace {\frac{{{{\left( {y + 3} \right)}^2}}}{{{{15}^2}}} - \frac{{{{\left( {x - 5} \right)}^2}}}{{{8^2}}} = 1}_{\frac{{{{\left( {y - k} \right)}^2}}}{{{a^2}}} - \frac{{{{\left( {x - h} \right)}^2}}}{{{b^2}}} = 1} \cr & h = 5,{\text{ }}k = - 3,{\text{ }}a = 15,{\text{ }}b = 8 \cr & c = \sqrt {{a^2} + {b^2}} = \sqrt {225 + 64} \Rightarrow c = 17 \cr & {\text{Characteristics of the hyperbola }} \cr & {\text{Orientation}}:{\text{ Vertical transverse axis}} \cr & {\text{Center}}\left( {h,k} \right) \to {\text{Center}}\left( {5, - 3} \right) \cr & \underbrace {{\text{Vertices}}\left( {h,k - a} \right){\text{ and }}\left( {h,k + a} \right)}_ \Downarrow \cr & {\text{Vertices}}\left( {5, - 3 - 15} \right){\text{ and }}\left( {5, - 3 + 15} \right){\text{ }} \cr & {\text{Vertices}}\left( {5, - 18} \right){\text{ and }}\left( {5,12} \right){\text{ }} \cr & \underbrace {{\text{Foci}}\left( {h,k - c} \right){\text{ and }}\left( {h,k + c} \right)}_ \Downarrow \cr & {\text{Foci}}\left( {5, - 3 - 17} \right){\text{ and }}\left( {5, - 3 + 17} \right) \cr & {\text{Foci}}\left( {5, - 20} \right){\text{ and }}\left( {5,14} \right) \cr & \underbrace {{\text{Asymptotes}}:{\text{ }}y = \pm \frac{a}{b}\left( {x - h} \right) + k}_ \Downarrow \cr & {\text{Asymptotes}}:{\text{ }}y = \pm \frac{{15}}{8}\left( {x - 5} \right) - 3 \cr & {\text{Graph:}} \cr & \cr} $$