Answer
$\displaystyle \frac{x^{2}}{9}+\frac{y^{2}}{25}=1$
Work Step by Step
Foci on the y-axis: $\quad\Rightarrow \quad \displaystyle \frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1 \quad (a\gt b)$
Foci: $\quad(0, \pm c) \quad\Rightarrow \quad c^{2}=16$
Vertices: $\quad (0, \pm a)\quad\Rightarrow \quad a^{2}=25$
Center-to-focus distance: $\quad c=\sqrt{a^{2}-b^{2}} \quad\Rightarrow b^{2}=a^{2}-c^{2}$
$b^{2}= a^{2}-c^{2}$
$b^{2}=25-16=9$
So, the equation is
$\displaystyle \frac{x^{2}}{9}+\frac{y^{2}}{25}=1$
