In the classical construction of conic sections, where does the axis of the cone intersect the plane? 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.
 A: 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.
A: This question and accepted answer are almost ten years old, but in case anybody stumbles upon this question, here's some more information on the topic.  (For cone terminology and background, see wikipedia)
First, an edited version of one of the comments:

All [right] circular cones whose section is a given ellipse produce
[different] points. The more narrow [...] the cone, the closer to the
center is the point [of intersection]. In general, [...] the point is
between the foci, at certain distances from them, whose ratio is equal
to the ratio of the radii of the Dandelin spheres. – Pietro Majer

The apex of each cone in this family is on a hyperbola $h$ passing through the foci of the ellipse. The foci of $h$ are the points of intersection of the ellipse and its major axis.  This is easily proven in the same context as the proofs for Dandelin spheres, and uses the property that the difference of distances from a point on the hyperbola to its foci is constant for all points.  See Salmon, Conic Sections, Art. 367.
For a much more detailed treatment see Armstrong, Where is the Cone?.  From the abstract:

Real quadric curves are often referred to as “conic sections,”
implying that they can be realized as plane sections of circular
cones. However, it seems that the details of this equivalence have
been partially forgotten by the mathematical community. The definitive
analytic treatment was given by Otto Staude in the 1880s and a
non-technical description was given in the first chapter of Hilbert
and Cohn-Vossen’s Geometry and the Imagination (1932). [...] Our hope is to revive the
lost knowledge of “conic sections” by providing the slickest possible
modern treatment, by using standard linear algebra that was not
standard in 1932.

