How many cones pass through a given conic section? Given a conic section in the $xy$-plane, how many cones (infinite double cone) in the surrounding 3D space intersect the $xy$-plane at that conic? Is the family continuous, with a nice parametization?
At least one must exist, and I expect symmetry in the conic to give a few such cones by reflection, but are there more than that?
Edit:
Following Peter Smith's answer, it seems possible that a continuum of such cones exist. If that were to be the case, what is the locus of the apexes of those cones?

To take the simplest case, take the circle to be centred at $(0, 0, 0)$ in the $xy$-plane; and now take any point $(0, 0, z)$. Then plainly there is a double cone of rays which pass through $(0, 0, z)$ and some point on the circle (and this is a right circular cone). So there are continuum-many distinct such cones (i.e. as many as there are are points $(0, 0, z)$) which have the given circle as section in the $xy$-plane. [This observation generalizes, mutatis mutandis, to other sorts of conic section: you'll get continuum-many possibilities.]
