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Math Help - points of intersection between cone and paraboloid

  1. #1
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    points of intersection between cone and paraboloid

    Hey everyone, I need to find the points of intersection between the cone z=\sqrt(x^2+y^2) and z=2-x^2-y^2


    So, what I did was

    2-x^2-y^2=\sqrt(x^2+y^2)
    (2-x^2-y^2)(2-x^2-y^2)=x^2+y^2
    x^4+y^4-5*x^2-5*y^2+2*x^2*y^2+4=0

    The problem is, I don't know what to do for now, can any one give me any tips?

    Best Regards
    Junks
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  2. #2
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    Re: points of intersection between cone and paraboloid

    Quote Originally Posted by junkwisch View Post
    Hey everyone, I need to find the points of intersection between the cone z=\sqrt(x^2+y^2) and z=2-x^2-y^2


    So, what I did was

    2-x^2-y^2=\sqrt(x^2+y^2)
    (2-x^2-y^2)(2-x^2-y^2)=x^2+y^2
    x^4+y^4-5*x^2-5*y^2+2*x^2*y^2+4=0

    The problem is, I don't know what to do for now, can any one give me any tips?

    Best Regards
    Junks
    $z = \sqrt{x^2+y^2}$

    $z = 2 - x^2 - y^2$

    $z = 2 - z^2$

    Spoiler:
    $z^2 + z - 2 =0$

    $(z+2)(z-1) = 0$

    $(z=-2) \vee (z=1)$

    The first equation for $z$ rules out $z=-2$ as a solution so the only solution is $z=1$ i.e.

    $z=\sqrt{x^2+y^2}=1$ and this is a circle of radius 1 centered at (0,0,1) in the plane $z=1$.
    Thanks from junkwisch
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