KöMaL 2018 B4929

The planes of an ellipse $\mathcal E$ and a hyperbola $\mathcal H$ in the space are perpendicular. The foci of $\mathcal E$ are the endpoints of the real axis of $\mathcal H$, and the foci of $\mathcal H$ are the endpoints of the major axis of $\mathcal E$. Let $A$ be $B$ two fixed points on different branches of hyperbola $\mathcal H$, and let $P$ be an arbitrary point of the ellipse. Prove that the sum of the distances $PA$ and $PB$ is independent of the choice of $P$.