(1) Does the axis of a cone pass through the foci of any its conic sections? Consider the image below:
Is the intersection of the axis of cone and the ellipse the same as the focus of the ellipse? Also for the parabola? If not, what is the significance of this point where the conic section intersects the axis of the cone? Does the diameter of the base of the cone pass through the focus of the hyperbola?
(2) It is easy to see that the axis of the cone passes through the center of the circle, and that a plane passing through the vertex of a double cone would pass through the center point of a hyperbola. But how does one compute the center of the ellipse from this picture?
(3) The center of a conic section is equidistant from its foci. One can think of a circle as having one focus at its center, where an ellipse's foci would coincide if its major and minor axes were equal in length. But a parabola also has one focus, so what is the "center" of a parabola?
I am guessing all of these have something to do with Dandelin spheres as described here, but not sure that this answers my questions or not.