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 Math Central Quandaries & Queries
 Question from Pulkit, a student: we get an ellipse on slicing through a cone. Is there a relation between central axis of the cone and this ellipse? Does it pass through the any of the foci of the ellipse?

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.