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.
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.