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.

(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.