"Let S denote the elliptical cylinder given by the equation 4y^{2}+ z^{2}=4, and let C be the curve obtained by intersecting S with the plane y=x.

Parameterize C

I am not sure how to go about this. I tried solving the equations for each other

4y^{2}= 4-z

y^{2}= 1- z^{2}/4

y= sqrt(1-z^{2}/4)

and then do i do some sort of parameterization of sqrt(1-z^{2}/4) -x=0 or am i supposed to parameterize the cylinder and plane separately? I know the cylinder parameterizes to

y=cos(theta)

z=2sin(theta)

but i don't know what to do from here. If anyone could hit me with some tips i would greatly appreciate it

EDIT:

would it simply be <cos (t), cos (t), 2 sin (t)> as x=y and y is equal to cos(t)? is it really this simple?