Answer
$$\frac{{{y^2}}}{{25}} - \frac{{{x^2}}}{3} = 1$$
Work Step by Step
$$\eqalign{
& {\text{vertices at }}\left( {0,5} \right),\left( {0, - 5} \right);{\text{ passing through the point }}\left( { - 3,10} \right) \cr
& {\text{The }}x{\text{ - coordinate in the vertices are the same, then the hyperbola}} \cr
& {\text{has the equation }}\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1 \cr
& {\text{With vertices at }}\left( {0, \pm a} \right) \cr
& {\text{vertices at }}\left( {0, \pm 5} \right) \to a = 5 \cr
& \cr
& \frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1 \cr
& \frac{{{y^2}}}{{{5^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1 \cr
& \frac{{{y^2}}}{{25}} - \frac{{{x^2}}}{{{b^2}}} = 1 \cr
& {\text{passing through the point }}\left( { - 3,10} \right) \cr
& \frac{{100}}{{25}} - \frac{9}{{{b^2}}} = 1 \cr
& {\text{Solving for }}{b^2} \cr
& \frac{9}{{{b^2}}} = 3 \cr
& {b^2} = 3 \cr
& {\text{The equation of the hyperbola is}} \cr
& \frac{{{y^2}}}{{25}} - \frac{{{x^2}}}{3} = 1 \cr} $$
