Answer
A hyperbola whose center is at the origin (0,0) with transverse axis $y=x$ and conjugate axis $y=-x$.
Work Step by Step
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 $y=-x$.