What is the equation of the curve of intersection of:
$\displaystyle z = x^2 + y^2$ and
$\displaystyle 2y + z = 0$
(Refer to attachment for graph of both equations).
What have you tried yourself? Do you understand that, because this is a one dimensional curve in three dimensions, you will need to find three equations for x, y, and z in terms of a siingle parameter?
Since $\displaystyle z= x^2+ y^2$, that second equation is $\displaystyle 2y+ z= 2y+ x^2+ y^2= 0$ which is the same as $\displaystyle x^2+ y^2+ 2y+ 1= 1$ or $\displaystyle x^2+ (y+ 1)^2= 1$. That is, the projection of that curve down to the xy-plane is the circle of radius 1 with center at (0, -1).
Do you know how to parameterize such a circle? Use either $\displaystyle z= x^2+ y^2$ or $\displaystyle z= -2y$ to complete the parametric equations for this curve.