Show that the projection into the $xy$-plane of the curve of intersection of the parabolic cylinder $z = 1 - 3 y^2$ and the paraboloid $z = x^2 + y^2$ is an ellipse. (a) Find a vector-parametric equation $\vec{r}_1(t) = \langle x(t), y(t), z(t) \rangle$ for the ellipse in the $xy$-plane. Shadow: $\vec{r}_1(t) =$ for $0 \leq t \leq 2\pi$. (b) Find a vector-parametric equation for the curve of intersection of the parabolic cylinder and the paraboloid. Intersection: $\vec{r}_2(t) =$ for $0 \leq t \leq 2\pi$.

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