Chapter 13 - Vector Geometry - 13.7 Cylindrical and Spherical Coordinates - Exercises - Page 700: 32

$$ r^2= \frac{2z}{3 \sin2\theta} .$$

Since $$ x=r\cos \theta,\quad y=r\cos \theta, \quad z=z,$$ then $ z=3xy $ takes the form $$ z=3r^2\cos \theta \sin\theta \Longrightarrow r^2=\frac{z}{3\cos \theta \sin\theta}=\frac{2z}{3 \sin2\theta} .$$