The problem I am working on requires me to find the family of circles orthogonal to two given circles in the complex plane, namely $\delta \Delta(1)$ and $\delta \Delta (1,2)$ (which are the circles centered at origin with radius 1 and the circle centered at (1,0) with radius 2).
What I have tried to do till now is to send these circles to parallel lines using the transformation z $\rightarrow$ $1/(z+1)$. The images of these circles under this transformation are the vertical lines z = 1/2 and z =1 respectively. So the lines perpendicular to these would be the vertical lines and the preimages of these would give us the family of circles perpendicular to our given circles (as angle would be preserved).
However, I am unable find the preimages of these lines under the given map.
Any help would be greatly appreciated.