The surfaces intersect in a space curve C. Determine the projection of C onto the xy-plane.

The paraboloids x^2+y^2 + z=4 and x^2 + 3y^2=z

As the paraboloids is given with the following equation

x^2+y^2 + z=4 and x^2 + 3y^2=z

So,

x^2+y^2 + z-4 = x^2 + 3y^2 - z

x^2+y^2 + z-4 - x^2 - 3y^2 + z = 0

= - 2y^2 + 2z - 4

which is the projection of C onto the xy-Plane and it will be an eclipse.