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Math Help - Conic Equation

  1. #1
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    Conic Equation

    Can you help me prove if true or false please:

    True or False? Every system of two conic equations whose xy-term is nonexistent has at most 4 points of intersection. graph is not a proof.

    Thanks!
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  2. #2
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    Hello, Anemori!

    I have a rather sloppy proof . . .


    True or False?
    Every system of two conic equations whose xy-term is nonexistent
    has at most 4 points of intersection. . . True!

    We have two conic equations: . \begin{array}{cccc} Ax^2 + By^2 + Cx + Dy + E &=&0 & [1] \\ Px^2 + Qy^2 + Rx + Sy + T &=& 0 & [2] \end{array}


    Solve [1] for y\!:\quad By^2 + Dy + (Ax^2 + Cx + E) \:=\:0

    . . Quadratic Formula: . y \;=\;\frac{-D \pm\sqrt{D^2 - 4B(Ax^2 + Cx + E)}}{2B}


    Substitute into [2]:

    . . Px^2 + Q\left(\frac{-D \pm\sqrt{D^2 - 4B(Ax^2+Cx+E)}}{2B}\right)^2 + Rx \;+ .  S\left(\frac{-D\pm\sqrt{D^2 - 4B(Ax^2+Cx+E)}}{2b}\right) + T \;\;=\;\;0


    This simplifies to a quartic in x . . . which has at most 4 roots.

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