Stage: 5 Challenge Level: 
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.
Long problems. Families of Graphs. Symmetry. Conversion from cartesian to polar forms. Asymptotes. Functions and their inverses. Rotations. Graph sketching. Turning points. Polar coordinates.
Published March 2003.
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