## Thomas' Calculus 13th Edition

A hyperbola whose center is at the origin (0,0) with transverse axis $y=x$ and conjugate axis $y=-x$.
The conversion of polar coordinates and Cartesian coordinates are described as follows: 1. $r^2=x^2+y^2$ and $r=\sqrt {x^2+y^2}$ 2. $\tan \theta =\dfrac{y}{x} \implies \theta =\tan^{-1} (\dfrac{y}{x})$ 3. $x=r \cos \theta$ and 4. $y=r \sin \theta$ Given: $r^2 \sin 2 \theta=2$ This can be rewritten as: $r^2 (2 \sin \theta \cos \theta)=2$ Now, the cartesian equation is $xy=1$ and $y=\dfrac{1}{x}$ Hence, this shows the hyperbola whose center is at the origin (0,0) with transverse axis $y=x$ and conjugate axis $y=-x$.