Calc 3

Show that the projection into the xy-plane of the curve of intersection of the parabolic cylinder z=1−2y^2 and the paraboloid z=x^2+y^2 is an ellipse.

Find a vector-parametric equation r→1(t)=⟨x(t),y(t),z(t)⟩ for the ellipse in the xy-plane.
Shadow: r→1(t)=____ for 0<t<pi/2

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  1. well, the intersection is where
    1-2y^2 = x^2+y^2
    x^2 + 3y^2 = 1
    or
    x^2 + y^2/(1/3) = 1
    x = r cost
    y = 1/√3 r sint

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