Find a suitable Möbius transformation that maps $X=\{z|\ |z|<1, |z-q/2|<q/2\}$ onto the annulus $Y=\{z|\ p<|z|<1\}$ for some $p,q\in(0,1)$

Question

Let $p,q\in(0,1)$ define $X=\{z\in\Bbb{C}||z|<1, |z-q/2|<q/2\}$ and $Y=\{z\in\Bbb{C}|\ p<|z|<1\}$. We have to construct a mobious transformation that maps $X$ onto $Y$.

There is a hint suggests $\displaystyle{T(z)=\frac{z-a}{1-\bar{a}z}}$ for suitable $a\in\Bbb{C}$ may do the job. But I don't know how did they get this transformation and what it does.
Can anyone help with the transformation? Thanks for help in advance.