Question

find the curve of intersection of the elliptic cyclinder x^2 +4z^2 = 16 and the plane x+3y-2z = 3

Answer 1

You have to impose equality:
x^2 + 4*z^2 - 16 = x + 3*y - 2*z - 3
So, you have
y = 1/3*(x^2 + 4*z^2 - x + 2*z - 13)
which is an ellipse