Thread: Conics

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.

Originally Posted by Ivan

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.

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

Hello, Ivan!

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).

A parabola is formed by the intersection of a double-napped cone
. . 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.