# Ellipses touching a circle

Given a circle and two points $A$, $B$ in the plane, how do I find an ellipse with focal points $A$ and $B$ that touches the circle?

How many such ellipses are there (at least/at most)?

Can I construct the touching point(s) with ruler and compass?

All these questions, as easy as they may sound, turned out to be surprisingly hard at second glance.