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.

Question

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

Answer 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