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Conic sections question

  1. Nov 4, 2006 #1
    Here's the question:
    Consider the equation:
    Ax^2+Cy^2+Dx+Ey+F=0

    Consider the cases AC>0, AC=0 and AC<0 and show that they lead to an ellipse, parabola and hyperbola respectively, except for certain degenerate cases. Discuss these degenerate cases and the curves that arise from them


    Don't really know where to start. I can 'prove' it by doing examples but that is not sufficient. Can somebody get me started on answering this question please?
    Thanks
     
  2. jcsd
  3. Nov 4, 2006 #2

    arildno

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    Complete the squares, and rearrange in the cases where neither A or C are zero.
    Then tackle the cases where either A, C or both are zero.
     
  4. Nov 4, 2006 #3
    Ok... so I complete the square to:
    EDIT: I suck at using LaTeX. Give me a minute...

    Ok here it is:
    [​IMG]
     
    Last edited: Nov 4, 2006
  5. Nov 4, 2006 #4

    arildno

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    In the case of non-zero A and C, it is simpler to do it the following way, in order to not get into the silly trouble of square-rooting negative numbers:
    [tex]A(x+G)^{2}+C(y+H)^{2}=I, G=\frac{D}{2A}, H=\frac{E}{2C}, I=AG^{2}+CH^{2}-F[/tex]
     
    Last edited: Nov 4, 2006
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