Everybody knows that if I take the intersection of a right circular cone with a plane, I get a conic section. My question is, where does the symmetry axis of the cone intersect the plane? Does this point relative to the conic have a name, or a simple description? For example, for an ellipse I first guessed that it was one focus of the ellipse, but that is false.

Following Keenan's suggestion I delete my comment and make it into an answer: Projectively speaking, there is no distinguished point inside a conic because the group of projective transformations that preserves the conic acts transitively on its interior: if someone gives you a circle and an unmarked ruler, you will never be able to construct the center. 

