Chapter 13 - Vector Geometry - 13.7 Cylindrical and Spherical Coordinates - Exercises - Page 700: 30

$$ r^2= \frac{4}{\cos2\theta} .$$

Since $$ x=r\cos \theta,\quad y=r\cos \theta, \quad z=z,$$ then $ x^2-y^2=4$ takes the form $$ r^2\cos^2\theta-r^2\sin^2\theta =4\Longrightarrow r^2=\frac{4}{\cos^2\theta-\sin^2\theta}=\frac{4}{\cos2\theta} .$$