Answer to Question #190717 in Complex Analysis for Vishal

Question

The inverse point of 1+i with respect to the circle |z-1|=2 is

Answer 1

Formula for symmetric point with respect to the circle-

$|z-z_o|=r$

So, $z'=\dfrac{r^2}{\bar{z}-z_o}+z_o~~~~~~~~~-(1)$

Given circle is-

$|z-1|=2$  

$\text{ So, } z_o=1,r=2 \text{ and point }z=1+i$

Subsititue above values in (1)-

$z'=\dfrac{2^2}{\overline{1+i}-1}=\dfrac{4}{1-i-1}=\dfrac{4}{-i}=4i$

Hence The inverse is 4i.