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Math Help - Degenerated Conic

  1. #1
    Senior Member pankaj's Avatar
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    Degenerated Conic

    If the equation of the curve is

     <br />
ax^2+2hxy+by^2+2gx+2fy+c=0<br />

    and this curve is such a conic that the focus lies on the directrix.

    Can anybody describe the possible situations i.e what happens if h^2-ab is >,< or = to 0(zero)

    Or

    Can anybody provide a link where such a situation has been described.

    Thanks in advance.
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by pankaj View Post
    If the equation of the curve is

     <br />
ax^2+2hxy+by^2+2gx+2fy+c=0<br />

    and this curve is such a conic that the focus lies on the directrix.

    Can anybody describe the possible situations i.e what happens if h^2-ab is >,\ < or = to 0(zero)
    h^2-ab>0: degenerate hyperbola = two straight lines, e.g. x^2-y^2=0.
    h^2-ab=0: degenerate parabola = single straight line, e.g. x^2=0.
    h^2-ab<0: degenerate ellipse = single point or nothing at all, e.g. x^2+y^2=0 or x^2+y^2=-1.
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  3. #3
    Senior Member pankaj's Avatar
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    Quote Originally Posted by Opalg View Post
    h^2-ab>0: degenerate hyperbola = two straight lines, e.g. x^2-y^2=0.
    h^2-ab=0: degenerate parabola = single straight line, e.g. x^2=0.
    h^2-ab<0: degenerate ellipse = single point or nothing at all, e.g. x^2+y^2=0 or x^2+y^2=-1.
    Is their proof available
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