[math-fun] Dandelin spheres, ellipses & projections of circles

Let's consider the version of Dandelin spheres internally tangent to a circular cylinder instead of a cone. The usual ellipse has a semi-minor axis the same as the radius of the cylinder. Is there a version where an ellipse is created whose semi-MAJOR axis the same as the radius of the cylinder? I'm trying to understand the meaning of the foci of the ellipse formed by the projection of a arbitrarily oriented circle in 3D orthographically projected onto the XY plane.