Please, help me with the following problem:

Starting from the figure below,
[![ellipse inscribed in a circle](https://i.stack.imgur.com/S1FoR.png)](https://i.stack.imgur.com/S1FoR.png)

we know:

$FP=OP-OF=a\cos E-ae$

and from the right triangle $OP\_{2}P$, I determined

$P\_{2}P = a\sin E$.

I would like to ask you: how can I prove that $P\_{1}P/P\_{2}P=b/a$ (where a is semi-major axis and b is semi-minor axis of ellipse)?

Can a demonstration be made by means of synthetic geometry or without equation of ellipse?