Help with Planes and Paraboloids

**messianic** · September 17th 2009, 07:42 PM

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

**Chris L T521** · September 17th 2009, 09:06 PM

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?

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