Which degenerate form or forms of the parabola cannot be obtained from the intersection of a plane and a double-napped cone? Describe how to obtain this (or these) form(s).

I will greatly appreciate your help.

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- Sep 17th 2007, 11:59 PMIvanConics
Which degenerate form or forms of the parabola cannot be obtained from the intersection of a plane and a double-napped cone? Describe how to obtain this (or these) form(s).

I will greatly appreciate your help. - Sep 18th 2007, 05:48 AMtopsquark
Given this definition of a degenerate conic section, which I

*think*is standard, I can't see how to make a parabola from a degenerate form. Unless the answer is supposed to be a pair of intersecting lines.

Edit: No, upon more reflection, the degenerate parabola should just be a single line passing through the apex.

-Dan - Sep 18th 2007, 06:02 AMSoroban
Hello, Ivan!

Quote:

Which degenerate form or forms of the parabola cannot be obtained

from the intersection of a plane and a double-napped cone?

Describe how to obtain this (these) form(s).

. . and a plane parallel to its "slant".

One degenerate parabola, a line, is obtained by passing the plane

. . through the vertex of the cone.

The other degenerate parabola, a point, cannot be obtained.

It can be formed with a plane through the vertex

. . which is*not*parallel to the cone's slant.