Focal chords of ellipse

Question

The feet of the perpendiculars from the foci S and S' ,on to the tangent to an ellipse at a point P,are Y and Y', and YQ and Y'Q' are drawn to touch the ellipse in Q and Q'. To show that QS and Q'S' meet the ellipse again at the end of the diameter through P. I know that the feet of the tangents lie on the auxiliary circle of the ellipse, but cannot use this to prove the result.Could we use affine tramsformations here? Any help would be appreciated.

Answer 1

Let $U,V$ be the intersections of the ellipse with $PS$ and, respectively, $QS$ and $X$ be the pole of $UV$. The quadrilater $PQUV$ is inscribed in the ellipse, hence the points \begin{align} PQ\cap UV\\ PU\cap QV\\ PV\cap QU \end{align} are conjugated each other respect to the ellipse. Consequently, the line $(PU\cap QV)(PQ\cap UV)$ is the polar of $PV\cap QU$. Moreover, $S=PU\cap QV$ and $R=PQ\cap UV$ is the pole of $XY$, hence $S$ belongs to $XY$ and the lines $PY$, $VX$ and $SR$ meets at the pole $T$ of $PV$. Since $S$ is a focus for the ellipse, we have $SR\perp SY$, hence $SR$ and $PY$ are parallel, hence $T$ is at infinity. This proves that $PV$ is a diameter of the ellipse.