Let $\displaystyle f:z\mapsto w = {az+b \over cz+d} \ \ \ (ad-bc \ne 0)$ be a Möbius transformation other than the identity map. A point $\displaystyle \alpha$ in $\displaystyle \mathbb{C}$ is said to be a fixed point of $\displaystyle f$ if $\displaystyle f(\alpha ) = \alpha$.

Suppose that $\displaystyle f$ has distinct fixed points, $\displaystyle \alpha$ and $\displaystyle \beta$. Prove that $\displaystyle {w-\alpha \over w-\beta}=k{z-\alpha \over z-\beta}$, where $\displaystyle k={a-c\alpha \over a-c\beta}$.

What is the image under $\displaystyle f$ of

(i) the circline $\displaystyle |{z-\alpha \over z-\beta}|=\lambda$,

(ii) the arc $\displaystyle \arg((z-\alpha )(z-\beta )) = \mu \pmod{2\pi}$?