A problem about finding shortest path and a property of the ellipse

Today we will look at two problems that seem to be unrelated. The first one is a beautiful geometry problem about finding shortest path and the other one is about a property of an ellipse.

But first, let us introduce the ellipse. An ellipse is drawn below.

for any point $P$ on the ellipse, $PF_1 + PF_2 = \ell$

Solve for special cases first!

I would like to share with you a lesson that I have learnt. That is when facing a problem and we do not know what to do, the first thing we can do is to look at special cases of that problem. Investigating special cases can help us gain a greater understanding of the problem. To illustrate the point, let us solve some problems.