Circle Inversion Mapping: Proof of w = 1/z Transforming |z-1| = 1 to x = 1/2

[hadroneater]

Homework Statement

Show that the inversion mapping w = f(z) = 1/z maps the circle |z - 1| = 1 onto the vertical line x = 1/2.

Homework Equations

The Attempt at a Solution

z = a + ib
w = x + yi = a^2/(a^2 + b^2) + ib^2/(a^2 + b^2)
|z - 1| = |(a -1) + ib | = 1
(a - 1)^2 + b^2 = 1
a^2 + b^2 = 2ab

Not sure what to do afterwards.

 

[A. Bahat]

Hint: The last equation is wrong. It should read $a^2+b^2=2a$, which makes your job a whole lot easier.

 

[hadroneater]

Thanks! That was really easy.