Stage: 5 Challenge Level:
You are told that the graph of points $(x,y)$ satisfying the
$$xy(x^2 - y^2) = x^2 + y^2$$
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
$$x = r \cos \theta,\ y = r \sin \theta$$
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
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.