What is the most general mobius transformation that maps $|z-1|=1$ to itself and $|z+1|=1$ to $|w-3|=3$.
I want to find the most general form of such a linear transformation, I'll denote it $T$. My reasoning is as follows:
The origin must be fixed since the origin in the $w$-plane is the only point where the two concentric circles meet. Hence, $T:0 \to 0$. Further, the real line in the $w$-plane intersects both circles at right angles, hence it's pre-image must intersect the circles in the $z$-plane at right angles. The only line or circle that does this is the real line in the $z$-plane. Hence, $T: 2 \to 2$ and $T: -2 \to 6$. This defines three points we can use to arrive at the mobius transformation given by $$ T(z) = w = \frac{3z}{z+1}. $$ However, my goal is to find the most general mobius transformation that does this. Is this form unique?
The reason I think my solution is incorrect is because I was told that there is in fact a family of one-parameter maps that accomplishes this transformation.