complex number substitution within a function

Question

I'm trying to make a simple substitution:

$a \in C$ and $0<|a|<1$

$f_{a}(z)=\frac{z-a}{1-\bar{a}z}$

Demonstrate that if |z|=1 then |$f_a(z)|=1$ .

I used. $z=|z|e^{I\theta}$ but that did not get me anywhere.

Answer 1

Hint: $\displaystyle\frac{z-a}{1-\overline az}=\frac{z-a}{\overline zz-\overline az}=\frac{z-a}{\overline z-\overline a}\cdot\frac1z.$

Answer 2

Hint. Realise the denominator multiplying by $$\frac{1-a\bar z}{1-a\bar z}.$$

Answer 3

I did not notice that $|z-a|=|\bar z-\bar a|$ which leads to the answer.