Parametrize the curve of intersection of x = -y^2 - z^2 and z = y.
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
???
Since the intersection is a parabola, I don't see how those parametric equations can describe it.
To parametrize the curve of intersection of the two given equations x = -y^2 - z^2 and z = y, we can substitute the value of z from the second equation into the first equation.
Replace z with y in the equation x = -y^2 - z^2:
x = -y^2 - (y)^2
x = -2y^2
Now, we have a parametric equation for the curve:
x = -2y^2
y = y
z = y
To summarize, the parametric equations for the curve of intersection are:
x = -2y^2
y = y
z = y