Question

xy=1 in polar coordinates

Answer 1

$xy=1\iff (r\cos\theta)(r\sin\theta)=r^2\sin\theta\cos\theta=\dfrac12r^2\sin2\theta=1$
$\implies r^2=2\csc2\theta$
$\implies r=\sqrt{2\csc2\theta}$

where we ignore the second root and use the convention that $r>0$.