



 
Nice question  it's something I've never thought about. I would conjecture that the axis of a right circular cone passes through the center of the conic (and therefore through its coincident foci) precisely when the conic is a circle (in a plane perpendicular to the axis). All I know for certain about the plane sections of a cone can be found in the Wikipedia article on Dandelin spheres, although I find the picture there a bit confusing. To see how the axis of the cone intersects a plane section, imagine rotating a plane about a line that intersects the axis at a right angle, starting with a plane perpendicular to the axis  the ellipse of intersection becomes more and more elongated until in the limit it becomes a pair of intersecting lines. It seems to me that one focus moves slowly away from the axis while the other moves quickly. Chris  


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 