Challenge Level

You are told that the graph of points $(x,y)$ satisfying the equation

$$xy(x^2 - y^2) = x^2 + y^2$$

consists of four curves together with a single point at the origin.

To help you to get started, think of a point $P$ on the graph, at a distance $r$ from the origin, having coordinates $(x,y)$, where the line $OP$ joining $P$ to the origin makes an angle $\theta$ with the $x$-axis. Now, to find the polar equation, make the substitution

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

and don't be put off by the long expression you get. Use the trig identities for $\sin^2\theta $ and $\cos^2\theta$ and you should be able to simplify the expression to prove that the polar equation of this graph is

$$r^4 \sin 4\theta = 4r^2.$$

Clearly $r=0$ satisfies this equation. Is it possible to have $r< 2$?

What are the valuesof $\theta$ when $r=2$?

One more hint and all should be plain sailing: notice that $\sin 4\theta$ is never negative.