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Math Help - Conics

  1. #1
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    Question 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.
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  2. #2
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    Quote Originally Posted by Ivan View Post
    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
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  3. #3
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    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.

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