Let’s say you have a ellipse $\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1 , |a|>|b|$ and you have a ray of light starting on a line $y=mx+c$ such that it intersects the ellipse twice and does not pass between the foci then after infinite reflections you will get a second ellipse like the following…
$\textbf{My Attempt}$
We can obviously solve the line equation and ellipse equation and we get
$$x = \frac{-a^2mc \pm ab\alpha}{a^2m^2+b^2}$$ $$y = \frac{b^2c \pm abm\alpha}{a^2m^2+b^2}$$
where $\alpha = \sqrt{a^2m^2+b^2-c^2}$
and then through a little bit of calculus and taking the starting point of ray as the negative signed solution of line and ellipse (another question here btw, is it necessary to check with positive signed solution too), we can find the angle of incidence
$$i = \arctan\left(\frac{ab^2mc+a^2m^2b\alpha -b^3\alpha+ab^2mc}{ab^2c+a^2bm\alpha+mb^3\alpha-ab^2m^2c}\right)$$
Now I am not getting how to proceed further, like how to enforce the conditions of not passing between the foci and even after finding that how to navigate through this mess of a equation and find the region of no light…