Question

(a) Transform the given equation xy=a in Polar coordinates

Answer 1

Let's move on to polar coordinates

$x = r\cos \varphi ,\,y = r\sin \varphi$

We get the equation

$r\cos \varphi \cdot r\sin \varphi = a \Rightarrow {r^2} \cdot \frac{1}{2}\sin 2\varphi = a \Rightarrow {r^2} = \frac{{2a}}{{\sin 2\varphi }}$

Answer: ${r^2} = \frac{{2a}}{{\sin 2\varphi }}$