Chapter 10 - Analytic Geometry - 10.3 Hyperbolas - 10.3 Exercises - Page 988: 21

$$\eqalign{ & {\text{Center }}\left( {4,7} \right) \cr & {\text{Vertices}}:\left( {4,1} \right){\text{ and }}\left( {4,13} \right) \cr & {\text{Foci: }}\left( {4, - 3} \right){\text{ and }}\left( {4,17} \right) \cr & {\text{Asymptotes: }}y = \pm \frac{3}{4}\left( {x - 4} \right) + 7 \cr & {\text{domain: }}\left( { - \infty , + \infty } \right) \cr & {\text{range: }}\left( { - \infty ,1} \right] \cup \left[ {13,\infty } \right) \cr} $$

$$\eqalign{ & \frac{{{{\left( {y - 7} \right)}^2}}}{{36}} - \frac{{{{\left( {x - 4} \right)}^2}}}{{64}} = 1 \cr & {\text{The equation is written in the form }}\frac{{{{\left( {y - k} \right)}^2}}}{{{a^2}}} - \frac{{{{\left( {x - h} \right)}^2}}}{{{b^2}}} = 1 \cr & \frac{{{{\left( {y - 7} \right)}^2}}}{{36}} - \frac{{{{\left( {x - 4} \right)}^2}}}{{64}} = 1,{\text{ then }}a = 6,\,\,b = 8,\,\,\,h = 4,\,\,\,k = 7 \cr & c = \sqrt {36 + 64} = 10 \cr & \cr & {\text{Characteristics:}} \cr & {\text{Center: }}\left( {h,k} \right) \cr & {\text{Center: }}\left( {4,7} \right) \cr & {\text{Transverse axis: horizontal}} \cr & {\text{vertices: }}\left( {h,k \pm a} \right) \cr & {\text{vertices: }}\left( {4,7 \pm 6} \right) \cr & {\text{vertices: }}\left( {4,1} \right){\text{ and }}\left( {4,13} \right) \cr & {\text{foci:}}\left( {h,k \pm c} \right) \cr & {\text{foci:}}\left( {4,7 \pm 10} \right) \cr & {\text{foci:}}\left( {4, - 3} \right){\text{ and }}\left( {4,17} \right) \cr & {\text{asymptotes: }}y = \pm \frac{a}{b}\left( {x - h} \right) + k \cr & {\text{asymptotes: }}y = \pm \frac{3}{4}\left( {x - 4} \right) + 7 \cr & {\text{domain: }}\left( { - \infty , + \infty } \right) \cr & {\text{The range is: }}\left( { - \infty ,k - a} \right] \cup \left[ {k + a,\infty } \right) \cr & {\text{The range is: }}\left( { - \infty ,1} \right] \cup \left[ {13,\infty } \right) \cr} $$