Dandelin spheres

Remarks

When $C$ is an ellipse there are exactly two Dandelin spheres, one on each side of $\pi$. For a hyperbola there are two, one in each nappe, and for a parabola there is one. Check of the focal property for an ellipse: let $\Sigma_1, \Sigma_2$ touch $\pi$ at $F_1, F_2$ and touch $K$ along circles $c_1, c_2$. For $P \in C$, the generator line $VP$ meets $c_1$ and $c_2$ at points $Q_1, Q_2$. Two tangent segments from $P$ to the same sphere have the same length, so $PF_1 = PQ_1$ and $PF_2 = PQ_2$. Hence $PF_1 + PF_2 = Q_1 Q_2$, which is the distance between the parallel circles $c_1$ and $c_2$ measured along a generator, the same for all $P$; so $C$ is an ellipse with foci $F_1, F_2$. When $C$ is not a circle, the plane of each circle $c_i$ meets $\pi$ in a line, which is the directrix belonging to the focus $F_i$. Non-example: take a small sphere inscribed in $K$ near the vertex, on the vertex side of $\pi$ and far from it. It is tangent to $K$ along a circle but does not touch $\pi$, so clause 2 fails.