Question 94
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 $A H^{2}+B H^{2}$.
- $\mathcal{P}$ 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 $A H^{2}+B H^{2}$.