I have one exercises that i need help.
Let the Möbius transformation $w=T(z)$ defined by
$\frac{w-a}{1-\overline{a}w} = \mu\frac{z-a}{1-\overline{a}z},\mu=e^i\alpha, \lvert a\rvert<1$
1.Put the points whose coordinates are $a$ and $1/a$ in the sphere. Proof that
$\lvert z\rvert<1$ if and only if $\lvert w\rvert<1$
and
$\lvert z\rvert=1$ if and only if $\lvert w\rvert=1$
2.Study how transforms the apolonio circles of focus $a$ and $1 / a$
3.Proof that every Möbius transformation $w=T(z)$ which satisfy the point 1 are of the form
$T(z)=\frac{\alpha z+\beta}{\overline{\beta}z+\overline{\alpha}}, \lvert \alpha\rvert^2=\lvert \beta\rvert^2=1$
Well the first point is excuse me for my english. I have proved that $\phi_a(z)=\frac{z-a}{1-\overline{a}z}$ is a bijection from $\mathbb{D}$ in $\mathbb{D}$ but i cant understand how is the function $w=T(z)$. If someone can help me i will be very gratefull.