Help with Planes and Paraboloids

• September 17th 2009, 07:42 PM
messianic
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
• September 17th 2009, 09:06 PM
Chris L T521
Quote:

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?