Answer
$$\frac{{{x^2}}}{{72}} - \frac{{{y^2}}}{{72}} = 1$$
Work Step by Step
$$\eqalign{
& {\text{hyperbola with foci at }}\left( {0, \pm 12} \right){\text{ and asymptotes }}y = \pm x \cr
& {\text{The coordinates of the foci are }}\left( {0, \pm c} \right),{\text{ then}} \cr
& {\text{The equation of the ellipse is of the form }}\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1 \cr
& {\text{foci}}\left( {0, \pm c} \right),\,\,\left( {0, \pm 12} \right) \to c = 12 \cr
& {\text{Asymptotes }}y = \pm \frac{b}{a}x \cr
& \frac{b}{a} = 1,\,\,\,a = b \cr
& {c^2} = {a^2} + {b^2} \cr
& {c^2} = 2{a^2} \cr
& {12^2} = 2{a^2} \cr
& {a^2} = 72 = {b^2} \cr
& \cr
& {\text{The equation of the hyperbola is}} \cr
& \frac{{{x^2}}}{{72}} - \frac{{{y^2}}}{{72}} = 1 \cr} $$
