Angles and ellipse (proof)

Question

First of all, sorry for my poor English! Can you please help me? I'm trying to prove that, given a point P at an ellipse.

Please help me prove that the angles are equal.

Thanks!

Answer 1

It is not true for any point $P$. See the image:

Answer 2

You haven't stated the problem. You want to prove that a light ray from one focus reflects off the ellipse back to the other focus. You don't tell us what you know. Personally, I would represent the ellipse as a level set of the function $$f(\mathbf x) = \|\mathbf x-F_1\| + \|\mathbf x-F_2\|$$ and use the fact that $\nabla f$ is the normal vector.

You could also write down the equation $$\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$$ and use implicit differentiation and lots of slopes.