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
Conversion of polar coordinates and Cartesian coordinates are as follows:
a)$r^2=x^2+y^2 \implies r=\sqrt {x^2+y^2}$
b) $\tan \theta =\dfrac{y}{x} \implies \theta =\tan^{-1} (\dfrac{y}{x})$
c) $x=r \cos \theta$ d) $y=r \sin \theta$
Here, $r^2 \sin 2 \theta=2$ can be written as $r^2 (2 \sin \theta \cos \theta)=2$
Thereforeg, our Cartesian equation is $xy=1 \implies y=\dfrac{1}{x}$
This shows a hyperbola whose center is at the origin (0,0) with transverse axis $y=x$ and conjugate axis $y=-x$.
