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Math Help - Parametrization/Ellipse

  1. #1
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    Parametrization/Ellipse

    "Let Σ denote that part of the cone x^2 + y^2 = z^2, z > 0 which lies beneath the plane x + 2z = 1.
    Let F(x, y, z) = (0,x,0).
    Show that the projection of ∂Σ vertically to the xy-plane is an ellipse. Parametrise ∂Σ."

    I have no idea how they get an ellipse from this. I've substituted the plane equation into the cone equation and i get a set which isn't an ellipse and hence don't know how to parametrize this. I must be doing something wrong!

    Any help would be greatly appreciated.

    Thanks.
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  2. #2
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    z=\frac{1-x}{2} so plug that into the cone equation to get x^2+y^2=\frac{(1-x)^2}{2}=\frac{1-2x+x^2}{2}=\frac{1}{2}-x+\frac{x^2}{2} so

    \frac{1}{2}x^2+x+y^2=\frac{1}{2} so x^2+2x+2y^2=1 and now complete the square and add 1 to both sides so

    x^2+2x+1+2y^2=\frac{1}{2}+1

    (x+1)^2+2y^2=\frac{3}{2} and this is an ellipse which can be parameterized with sin and cos similar to how you would parameterize a circle
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  3. #3
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    yeah, cool. This is just a horizontal translation of an ellipse in the x-y plane right. thanks.
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