The following property of an ellipse was told to me by my teachers but was never proved:
"Let E be an ellipse. Let a focal ray R be reflected within the ellipse E. Ultimately after 'infinite reflections', it will travel along the major axis of E."
I am not able to prove this property - mainly because I can't think of any other approach other than the brute force approach to solving it - in which the equations get very tedious.
Brute force approach: Let $$ E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b > 0$$. Let the original ray be $R_0$: $y - b\sin\phi_0 = m_0(x - a\cos\phi_0)$ which is reflected at $P(\phi_0)$ with a slope of $m_0$. [$\phi_0$ is the eccentric angle of the point P on the ellipse.] The idea was to get the sequences $(\phi_n)$ and $m_n$ where $(\phi_n)$ and $m_n$ are the eccentric angle and slope of the ray after the $n^{th}$ reflection; and then show that $m_n \rightarrow 0$. However the problem is that the equations for $m_1$ and $\phi_1$ itself get so tedious, it is very difficult to solve for $m_n$.
How can I prove the property?