# Thread: Help with Planes and Paraboloids

1. ## Help with Planes and Paraboloids

I need some help on this problem:

Prove that the projection into the xy plane of the intersection of the plane z = y and the paraboloid z = x^2 + y^2 is a circle

2. Originally Posted by messianic
I need some help on this problem:

Prove that the projection into the xy plane of the intersection of the plane z = y and the paraboloid z = x^2 + y^2 is a circle
Substitute $z=y$ into the $z=x^2+y^2$ to get $y=x^2+y^2\implies x^2+y^2-y=0$.

Completing the square gives $x^2+\left(y-\tfrac{1}{2}\right)^2=\tfrac{1}{4}$, which is a circle.

Does this make sense?