Answer
$$\frac{{{y^2}}}{{144}} - \frac{{{x^2}}}{{81}} = 1$$
Work Step by Step
$$\eqalign{
& y{\text{ - intercepts }}\left( {0, \pm 12} \right);{\text{ foci }}\left( {0, - 15} \right),\left( {0,15} \right) \cr
& {\text{The }}x{\text{ coordinate in the foci are the same, then the hyperbola}} \cr
& {\text{has the equation }}\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1 \cr
& {\text{With }}y{\text{ - intercepts at }}\left( {0, \pm a} \right){\text{ and foci }}\left( {0, \pm c} \right) \cr
& {\text{,Then}} \cr
& {\text{foci }}\left( {0, - 15} \right),\left( {0,15} \right) \to \,\,\,c = 15 \cr
& y{\text{ - intercepts at }}\left( {0, \pm 12} \right) \to \,\,a = 12 \cr
& {b^2} = {c^2} - {a^2} \cr
& {b^2} = {15^2} - {12^2} \cr
& {b^2} = 81 \cr
& {\text{The equation of the hyperbola is}} \cr
& \frac{{{y^2}}}{{144}} - \frac{{{x^2}}}{{81}} = 1 \cr} $$
