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.

You can use graphing software to sketch this graph but it is
more of a challenge to see if you can sketch it for yourself and
the steps in this question are designed to help you to do so. You
can download the shareware program Graphmatica for free from
here as
NRICH is an approved distributor of this program. You can find more
information about the program from http://www.graphmatica.com/

(a) Prove that the polar equation of this graph is

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

(b) Deduce that there are 4 points on this graph at distance 2
from the origin and no points closer to the origin. Find the values
of $\theta$ for which there are points on the graph and the values
of $\theta$ for which there are no points on the graph.

(c) Substitute $y=px$ in the Cartesian equation and find an
expression for $x^2$ in terms of $p$. Hence find the values of $p$
for which the lines $y=px$ do not cut the graph in points other
than the origin. For other values of $p$, in how many points do the
lines $y=px$ cut the graph?

(d) Prove that if the point $(a,b)$ lies on the graph then so
do the points $(-a,-b)$, $(-b,a)$ and $(b,-a)$.

(e) What can you say about the symmetries of the graph.