TITLE: Intersection of two quadrics that have a common inscribed sphere
QUESTION [9 upvotes]: This is related to a question I asked here on math.stackexchange. It didn't receive an answer there (except for my answer), and my question here is a generalization of that one, anyway.
Suppose I have two quadric surfaces $Q1$ and $Q2$ in plain ordinary 3D space, $\mathbb{R}^3$, and they have a common inscribed tangent sphere $S$, as shown in the picture. 

In general, the intersection of two quadric surfaces is a nasty curve of degree 4. But in our case, because of the common tangency, I suspect that the intersection of $Q1$ and $Q2$ is just a pair of ellipses. Numerical experiments seem to suggest this, anyway. My question: is my conjecture true, and, if so, what is the proof?
My tags might well be wrong, so please feel free to edit.

REPLY [3 votes]: 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.

This is essentially the same proof given in Ivan's answer.