Confocal ellipses

Question

The foci of an ellipse are S,S’ and P,P’ are two points on the curve on opposite sides of the major axis.SP’ meets S’P at Q’, and S’P’ meets SP at Q. To prove that Q and Q’ lie on another ellipse with foci S,S’. I am uncertain whether the best approach is geometric,or analytic. Any advice would be appreciated.

Answer 1

HINT1:

How to manipulate between ellipses of constant inter-focal distance $SS'= 2c$?

$$\dfrac{x_1^2}{a_1^2}+\dfrac{y_1^2}{a_1^2-c^2}\tag1$$ $$\dfrac{x_2^2}{a_2^2}+\dfrac{y_2^2}{a_2^2-c^2}\tag2$$

HINT2:

Imagine a laser shot in the direction $SQP.$ How does it get reflected/ricocheted at $Q$ and $P$?