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
Level curves: PR 3rd ed Cracking Math GRE, Chap 3 q 10

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