I have a circle A which passes through the points $10$, $0$ and $-7+7i$. This is mapped onto a circle $m(A)=B$ by the Möbius transformation $m(z)=\frac{7}{7+z}$.
The question asks for the equations of $A$ and $B$ as well as their centres and radii, to show that the Euclidean centre of A doesn't map to the Euclidean centre of B and finally whether there exists a Möbius transformation which takes $A$ to $B$ and the Euclidean centre of $A$ to the Euclidean centre of $B$.
I believe that I have done all of the question apart from the last part. I think that the equation for A is: $z \bar z + (-5+12i)z + (-5-12i)\bar z=0$
And the equation for B is: $z \bar z + (-\frac{12}{17}-\frac{12}{17}i)z + (-\frac{12}{17}+\frac{12}{17}i)\bar z=- \frac{7}{17}$.
So now I need to find a Möbius transformation which takes the centre of $A$ ($-5+12i)$ to the centre of $B$ ($\frac{12}{17} -\frac{12}{17}i$) as well as taking $A$ to $B$ or prove that it does not exist but I have no idea how I would go about doing this!
The question as you quote it says you have to take $A$ to $B$ but not neccessarily using the same transformation $m$. So think about some special Möbius transformations. In particular, which real affine transformations would be Möbius transformations as well? Translation, rotation and isotropic scaling are all special cases of Möbius transformations as well. Shearing or non-isotropic scaling on the other hand are not. So do you think you can satisfy your requirements with the first class of affine transformations? If so, build a suitable transformation one step at a time, then translate all these steps into Möbius transformations and combine them in the right order. If not, this would be less useful since it only demonstrates one way of how not to solve the question, but I can give you the hint that it will be possible.