A billiard trajectory inside an ellipsoid in $n$-dimensional space is tangent to $n-1$ quadrics confocal with this ellipsoid.

In your case, the disk bounded by $C$ is a limit of ellipsoids confocal to $E$, lines intersecting $C$ play the role of tangents, so a line starting at $C$ returns to $C$ after one reflection.

The proof of the general result above can be found in

<cite authors="Tabachnikov, Serge">_Tabachnikov, Serge_, Geometry and billiards, Student Mathematical Library 30. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3919-5/pbk). xi, 176&nbsp;p. (2005). [ZBL1119.37001](https://zbmath.org/?q=an:1119.37001).</cite>

The idea is to use ellipsoidal coordinates as Jacobi did when studying geodesics on ellipsoids.
Billiard trajectory inside an ellipsoid is the limit case of a geodesic on an ellipsoid one dimension higher (as the higher-dimensional ellipsoid flattens, a geodesic going over the "edge" becomes billiard trajectory).