Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 1A / a / i / Solution

For $f(z)=(az+b)/(z+1)$, the finite fixed points are the roots of
$$
z^2+(1-a)z-b=0.
$$
Thus a repeated fixed point or two prescribed fixed points determine $a$ and $b$ by the usual sum and product of roots.


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 Incoming links: Prescribed fixed points of a Möbius transformation