Question

Find a vector function, r(t), that represents the curve of intersection of the two surfaces. the paraboloid z = 7x2 + y2 and the parabolic cylinder y = 3x2

Answer 1

Letting $x=t$, we get $y=3x^2=3t^2$ and $z=7x^2+y^2=7t^2+(3t^2)^2=7t^2+9t^4$, so we can parameterize the intersection by

$\mathbf r(t)=\langle t,3t^2,7t^2+9t^4\rangle$

where $-\infty$.

Image attached.