Circle in the plane of complex numbers

Question

Let $K = \{z \in \mathbb{C}: |z−a|=r \}$ be a circle in $ℂ$.

Show that, for the case that $|a|$ is not equal to r, the image of $K$ under the transformation $z$ $\to$ $\frac {1}{z}$ is a circle too. Also, what's its' radius and center?

I would really appreciate any help.

Answer 1

If $|z-a| = r$ then $z - a = re^{i\theta}$. Then what is $\frac{1}{z}$?