Chapter 10 - Analytic Geometry - 10.3 Hyperbolas - 10.3 Exercises - Page 988: 17

$$\eqalign{ & {\text{Center }}\left( {0,0} \right) \cr & {\text{Vertices}}:\left( {0, \pm 5} \right) \cr & {\text{Foci: }}\left( {0, \pm \sqrt {29} } \right) \cr & {\text{Asymptotes: }}y = \pm \frac{5}{2}x \cr & {\text{domain: }}\left( { - \infty , + \infty } \right) \cr & {\text{range: }}\left( { - \infty , - 5} \right] \cup \left[ {5,\infty } \right) \cr} $$

$$\eqalign{ & 4{y^2} - 25{x^2} = 100 \cr & {\text{Divide both sides by 100}} \cr & \frac{{4{y^2}}}{{100}} - \frac{{25{x^2}}}{{100}} = \frac{{100}}{{100}} \cr & \frac{{{y^2}}}{{25}} - \frac{{{x^2}}}{4} = 1 \cr & {\text{The equation is written in the form }}\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1 \cr & \frac{{{y^2}}}{{25}} - \frac{{{x^2}}}{4} = 1,{\text{ then }}a = 5,\,\,b = 2 \cr & c = \sqrt {{a^2} + {b^2}} = \sqrt {25 + 4} = \sqrt {29} \cr & \cr & {\text{Therefore,}} \cr & {\text{Center }}\left( {0,0} \right) \cr & {\text{Vertices: }}\left( {0, \pm a} \right):\left( {0, \pm 5} \right) \cr & {\text{Foci: }}\left( {0, \pm c} \right):\left( {0, \pm \sqrt {29} } \right) \cr & {\text{Asymptotes: }}y = \pm \frac{a}{b}x \cr & {\text{Asymptotes: }}y = \pm \frac{5}{2}x \cr & {\text{The domain of the hyperbola is }}\left( { - \infty , + \infty } \right) \cr & {\text{The range of the hyperbola is: }}\left( { - \infty ,a} \right] \cup \left[ {a,\infty } \right) \cr & {\text{range: }}\left( { - \infty , - 5} \right] \cup \left[ {5,\infty } \right) \cr & \cr & {\text{Graph}} \cr} $$