This question comes from PR 3rd ed Cracking Math GRE, Chap 3 review questions, question 10.

It asks, "which of the following depicts level curve of the surface whose equation in cylindrical coordinates is z = r^2 cos (2 theta)?"

The answer begins by rewriting the equation in cartesian coordinates x,y,z as follows:

y^2 + z^2 = [f(x)]^2

For whatever reason I am not seeing their first, yet critical step. I guess we have the following equations at our disposal.

x^2 + y^2 = r^2

x = r cos theta

y = r sin theta

tan theta = y/x

z = z