Chapter 10 - Analytic Geometry - 10.3 Hyperbolas - 10.3 Exercises - Page 988: 26

$$\eqalign{ & {\text{Center }}\left( { - 9, - 6} \right) \cr & {\text{Vertices}}:\left( { - 14, - 6} \right){\text{ and }}\left( { - 4, - 6} \right) \cr & {\text{Foci: }}\left( { - 9 \pm \sqrt {29} , - 6} \right) \cr & {\text{Asymptotes: }}y = \pm \frac{2}{5}\left( {x + 9} \right) - 6 \cr & {\text{domain: }}\left( { - \infty , - 14} \right] \cup \left[ { - 4,\infty } \right) \cr & {\text{range: }}\left( { - \infty , + \infty } \right) \cr} $$

$$\eqalign{ & 4{\left( {x + 9} \right)^2} - 25{\left( {y + 6} \right)^2} = 100 \cr & {\text{Divide both sides by 100}} \cr & \frac{{4{{\left( {x + 9} \right)}^2}}}{{100}} - \frac{{25{{\left( {y + 6} \right)}^2}}}{{100}} = \frac{{100}}{{100}} \cr & \frac{{{{\left( {x + 9} \right)}^2}}}{{25}} - \frac{{{{\left( {y + 6} \right)}^2}}}{4} = 1 \cr & {\text{The equation is written in the form }}\frac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}} - \frac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1 \cr & \frac{{{{\left( {x + 9} \right)}^2}}}{{25}} - \frac{{{{\left( {y + 6} \right)}^2}}}{4} = 1,{\text{ then }}a = 5,\,\,b = 2,\,\,\,h = - 9,\,\,\,k = - 6 \cr & c = \sqrt {25 + 4} = \sqrt {29} \cr & \cr & {\text{Characteristics:}} \cr & {\text{Center: }}\left( {h,k} \right) \cr & {\text{Center: }}\left( { - 9, - 6} \right) \cr & {\text{Transverse axis: horizontal}} \cr & {\text{vertices: }}\left( {h \pm a,k} \right) \cr & {\text{vertices: }}\left( { - 9 \pm 5, - 6} \right) \cr & {\text{vertices: }}\left( { - 14, - 6} \right){\text{ and }}\left( { - 4, - 6} \right) \cr & {\text{foci:}}\left( {h \pm c,k} \right) \cr & {\text{foci:}}\left( { - 9 \pm \sqrt {29} , - 6} \right) \cr & {\text{asymptotes: }}y = \pm \frac{b}{a}\left( {x - h} \right) + k \cr & {\text{asymptotes: }}y = \pm \frac{2}{5}\left( {x + 9} \right) - 6 \cr & {\text{The domain is: }}\left( { - \infty ,h - a} \right] \cup \left[ {h + a,\infty } \right) \cr & {\text{domain: }}\left( { - \infty , - 14} \right] \cup \left[ { - 4,\infty } \right) \cr & {\text{The range is: }}\left( { - \infty , + \infty } \right) \cr} $$