Intersection of focal radii and tangents on an ellipse

Question

If SP and S’P’ are focal radii of an ellipse drawn in the same direction, and the tangents at P and P’meet S’P’,SP in Q’ and Q respectively, is the line Q’Q parallel to P’P? I have calculated the gradient of PP’ as $$-[x’[b^2]/[y’[a^2]$$ where (x’,y’) are the coordinates of the intersection of PQ’ and P’Q, but can’t seem to show QQ’ has equal gradient. Any help would be appreciated.

Answer 1

The proof is quite simple if we employ a very useful property of the ellipse:

the line joining the intersection point $T$ of the tangents at $P$ and $P'$, with the midpoint $M$ of $PP'$, passes through the center $O$ of the ellipse.

As $O$ is the midpoint of $SS'$, then line $TO$ is parallel to $QS$ and $Q'S'$. Hence $T$ is the midpoint of both $PQ'$ and $P'Q$, and consequently $PP'Q'Q$ is a parallelogram.