HMMT 2022 G8

Let $\mathcal{E}$ be an ellipse with foci $A$ and $B$. Suppose there exists a parabola $\mathcal{P}$ such that

• $\mathcal{P}$ passes through $A$ and $B$,
• the focus $F$ of $\mathcal{P}$ lies on $\mathcal{E}$,
• the orthocenter $H$ of $\triangle F A B$ lies on the directrix of $\mathcal{P}$.

If the major and minor axes of $\mathcal{E}$ have lengths 50 and 14 , respectively, compute $A H^{2}+B H^{2}$.