fractional linear maps what is the fractional linear map maps the circles $|z-5|=3$ and $|z+5|=3$ to concentric circles? What is the general method to find such maps? What is the image of $|z+2|=2$ under the map: $z \to 1/\bar{z}$? Thank you very much.

Per Numth's comment, let's send the negative circle to the unit circle with the positive circle to be concentric inside it. So we need to send:
$$\begin{array}{rcl}
-5-3 = -8 &\to& +1\\
-5+3 = -2 &\to& -1\\
+5-3 = +2 &\to& -x\\
+5+3 = +8 &\to& +x\end{array}$$
With a rational transformation given by $f(z) = (Az+B)/(z+C)$ we have four equations in the four unknowns $x,A,B,C$:
$$\begin{array}{rcl}
(-8A+B)/(-8+C) &=& +1\\
(-2A+B)/(-2+C) &=& -1\\
(+2A+B)/(+2+C) &=& -x\\
(+8A+B)/(+8+C) &=& +x\end{array}$$
All the circles are on the real axis, so all the variables are real.
