Intersection of cones tangent to a common sphere

Question

Suppose I have two right circular cones $C1$ and $C2$ (of different size/shape) that are both tangent to a sphere $S$, as shown in the picture. In general, intersection of two conical surfaces is a nasty curve of degree 4. But in our case, because of the common tangency, I suspect that the intersection is actually just a pair of ellipses. Numerical experiments seem to suggest this, anyway. I expect this is a known result from classical geometry, but I'd like to have a proof or a reference, please. All of this is in plain ordinary 3D space.

Edit
I found an answer myself (see answer below). But the proof provided is essentially high-school coordinate geometry. That's fine with me -- I like high-school coordinate geometry. But I wonder if there is some more sophisticated reasoning that makes the result patently obvious without all the algebraic computations. For example, are there tricks of projective geometry that reduce the cone-cone case to the (much easier) cylinder-cylinder case.

Answer 1

There is an answer in this paper. Theorem 8.1 says that the intersection of two cones degenerates into a pair of conics if and only if $ $ the cones have a common inscribed sphere.

Also ...
I asked a more general question on MathOverflow, and received a very satisfying answer.

And finally ...
In fact, there is a far more general result. In Salmon's "Analytic Geometry of Three Dimensions", 4th edition, page 117, we find the following:

Two quadrics having plane contact with the same third quadric intersect each other in plane curves. Proof: Obviously $U-L^2$ and $U-M^2$ have the planes $L-M$ and $L+M$ for their planes of intersection.