Find a specific Möbius transformation Question: Find a Möbius transformation mapping the circle $|z - 2i| = 1$ onto the circle $|w| = 1$, and, at the same time, mapping the circle $|z| = 1$ onto a line parallel with the imaginary axis.
Comments: Clearly the best way to proceed is to use the cross ratio formula, you try to use the information given to find three points (mapping $i$ to $1$ seems to be a candidate, etc) and so on, but I don't remember exactly what to do.
This is a question from an exam that I took last August, I think I only got 3 credits out of 5 for the question. At any rate I passed the exam so I'm asking here on behalf of a classmate who is retaking this exam (because I could not give him a proper full solution and there might be a similar question again). So it would be very nice if someone could show us how this is to be solved.
 A: First step:
Find a transformation mapping $|z - 2 i| = 1$ to $|z| = 1.$ ($z->z-2i$ leaps to mind). Call this transformation $\phi.$
Second step. Recall that the set of transformations preserving the unit circle have the form 
$$e^{2\pi i \theta} \frac{z-a}{1-\overline{a} z}.$$ Call this general transformation $\psi(a, \theta).$ (remember $a$ is inside the disk).
Third step: look at $\psi(a, \theta) \circ \phi.$ What does it do to the unit circle? It SHOULD map exactly one point thereof to $\infty,$ that should restrict your $a,$ and the angle with the real axis should constrain the $\theta.$
A: You could also first choose a Möbius transformation mapping the unit circle to the imaginary axis, $\dfrac{z-1}{z+1}$ for example. 
Now, you have to check what the center and radius of the other circle has become (for example by looking at the images of three points of the circle), and apply a translation of the midpoint to the origin and then scale the radius to one (both transformations will not change the direction of the straight line).
