Question

Parametrize the curve of intersection of x = -y^2 - z^2 and z = y.

Answer 1

if y = z
x = -y^2 - z^2
= -z^2 - z^2 = -2z^2

let's find any 2 points on the intersection
let z=1
then y = 1
and x = -2 -------> (-2, 1,1)
let z = 2
then y=2
and x = -4 -----> (-4,2,2)
direction vector = (-2,1,1)

a possible set of parametric equations is
x = -2 - 2k
y = 1 + k
z = 1 + k

Answer 2

???
Since the intersection is a parabola, I don't see how those parametric equations can describe it.