Locus of complex solutions of $(z+i)^{2011}=z^{2011}$

Question

What do the complex solutions of $(z+i)^{2011}=z^{2011}$ lie on?

1. Hyperbola
2. Ellipse
3. Straight line
4. Circle

I do not know how to approach it.

Answer 1

Hint : This is equivalent (why ?) to $$\left( \frac{z+i}{z}\right)^{2011}=1$$

i.e. to : there exists $k \in \lbrace 0, ..., 2010 \rbrace$ such that $$ \frac{z+i}{z} = \exp\left(\frac{2ik\pi}{2011} \right)$$

Now you can solve that in $z$.