Movement of $1/z$ in complex plane How is moving $\frac{1}{z}$ in complex plane if $z$ is described by a circle which has radius $r$ and center $a+b*i$
I've just started complex algebra and still having some trouble imagining it.
How one does even solve this kind of problems, I don't want the complete solution I just want some appropriate ways for working with complex numbers to solve this kind of problems. 
Thank you.)

Multiply it by $z^*/z^*$, where $z = r[\cos\theta + i \sin\theta]$ and $z^* = r[\cos\theta - i \sin\theta]$, that should get you
$$
\frac{1}{z} = \frac{1}{z}\frac{z^*}{z^*} = \frac{z^*}{|z|^2} = \frac{r[\cos\theta - i \sin\theta]}{r^2} = \frac{1}{r}[\cos\theta - i\sin\theta]
$$
and
$$
\theta = arctan(b/a)
$$
Can you take it from there?
