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Prove that the intersection of a plane and a object consist of one cone and one upside-down cone where the tip of cone meet is either degenerate conic or conic

Also, idenify in what situation, the intersection is parabola, hyperbola, ellipse and prove it. Thanks in advance.

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See this. – J. M. Dec 6 '11 at 2:09
up vote 2 down vote accepted

Using Dandelin Spheres, as J.M.'s comment suggests, is probably the most straight-forward way to go—it is possible to show that the intersection of the appropriate plane with the cone gives a curve satisfying the locus definition of each conic via geometric properties of the cone, plane, and spheres. Here is a cross-section diagram of the Dandelin Spheres in both the ellipse and hyperbola cases, from

cross-section diagram

That site also has some three-dimensional illustrations and more information.

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