Answer
$$ r= z\sec \theta \tan \theta .$$
Work Step by Step
Since
$$ x=r\cos \theta,\quad y=r\cos \theta, \quad z=z,$$
then $\dfrac{x^2}{yz}=1$ takes the form
$$\frac{ r^2\cos^2\theta}{z r\sin\theta } =1\Longrightarrow r=\frac{z\sin\theta}{\cos^2\theta}=z\sec \theta \tan \theta .$$