Graph of a curve

Question

Today in my test, there was a question which had contour C: $|z+\dfrac{1}{z}| = 2$.
What does the curve represent? Is it a discrete set of points or really a curve?

Answer 1

We have $$ |z+z^{-1}|=\frac{|z^2+1|}{|z|}=2 \iff |z^2+1|=2|z| \iff |z^2+1|^2=4|z|^2. $$ Setting $$ z=re^{i\theta}, $$ we get: $$ 4r^2=|r^2e^{i2\theta}+1|^2=(r^2e^{2i\theta}+1)(r^2e^{-2i\theta}+1)=r^4+2r^2\cos2\theta+1, $$ i.e. $$ 0=r^4-2(2-\cos2\theta)r^2+1=r^4-2(1+2\sin^2\theta)r^2+1. $$ It follows that $$ r^2=1+2\sin^2\theta\pm\sqrt{(1+2\sin^2\theta)^2-1}. $$ Hence $C$ is the union of the two curves $$ C_+: \ r^2=1+2\sin^2\theta+\sqrt{(1+2\sin^2\theta)^2-1},\ \theta \in [0,2\pi] $$ and $$ C_-: \ r^2=1+2\sin^2\theta-\sqrt{(1+2\sin^2\theta)^2-1},\ \theta \in [0,2\pi]. $$ It's clear that $C_\pm$ is not a circle!

Answer 2

From this diagram,

we can apply the Law of Cosines to get $$ 4 = r^2 + 1/r^2 + 2\cos 2\theta $$

which is a simple enough equation but I do not have enough experience in polar coordinates to recognize curve shapes. Solving that equation seems to get dirty rather quickly.