Answer
$\frac{x^2}{5} + \frac{y^2}{16}=1$
Work Step by Step
Step 1. Identify the given values: use the diagram given in the Exercise, we can identify one vertex at $(0,4)$ so that $a=4$, and one focal point as $(0,3)$ so that $c=3$.
Step 2. Write a general equation: as the vertex and focus are on the y-axis, we can write an equation as $\frac{x^2}{b^2} + \frac{y^2}{a^2}=1$ where $a\gt b\gt0$.
Step 3. Find the unknowns: with $a=4, c=3$, use the relationship $b^2=a^2-c^2$. we have $b=\sqrt {4^2-3^2}=\sqrt 5$
Step 4. Conclusion: the equation for the graph can be written as $\frac{x^2}{5} + \frac{y^2}{16}=1$
