Answer
$$\frac{{{y^2}}}{{16}} - \frac{{{x^2}}}{9} = 1$$
Work Step by Step
$$\eqalign{
& {\text{hyperbola; focus at }}\left( {0,5} \right),{\text{ transverse axis with length 8}} \cr
& {\text{The foci of a hyperbola centered at the origin has the equation}} \cr
& \frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1 \cr
& {\text{foci:}}\left( {0, \pm c} \right):\left( {0,5} \right) \to c = 5 \cr
& {\text{transverse axis with length 8, then }} \cr
& 2a = 8 \cr
& a = 4 \cr
& {b^2} = {c^2} - {a^2} \cr
& {b^2} = {5^2} - {4^2} \cr
& {b^2} = 9 \cr
& \cr
& {\text{The equation of the hyperbola is}} \cr
& \frac{{{y^2}}}{{16}} - \frac{{{x^2}}}{9} = 1 \cr} $$
