The conversion formulas are $x = \rho \sin \phi \cos \theta$, $y = \rho \sin \phi \sin \theta$, and $z = \rho \cos \phi$. Substituting these into the given equation $xy = z$, we get
\[\rho^2 \sin^2 \phi \cos \theta \sin \theta = \rho \cos \phi.\]
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