Prove that any line inside the ellipse not passing through the line segment joining the two foci when reflected off the ellipse would be tangent to the same inner ellipse as the initial line, with the same pair of foci.

Accordingly, if the line were to originally pass through the line segment joining the foci, the reflection on the ellipse would be tangent to the same hyperbola as the initial line, with the same pair of foci as the original ellipse.

The other possibility would be if the line started of from one focus, its reflection then passes through the other focus. (This degenerate case is known as the optical property of the ellipse, whose proof is much simpler.)

I used the analytic approach because I am not familiar with the synthetic approach when working with conics. For example,

Given an ellipse and a point outside it, construct the lines through the point and tangent to the ellipse.

Given an ellipse and a point on it, construct the tangent line to the ellipse at that point.

The same two for a hyperbola.