Let $P(z)=az-b$ and $Q(z)=cz-d$, where $a,b,c,d$ are nonzero complex numbers such that $bc\neq ad$.
Suppose that $|\frac{P(z)}{Q(z)}|=1$ whenever $|z|=1$. Show that
$$b\bar{d}=a\bar{c}\hspace{3mm}\text{and} \hspace{3mm} a\bar{b}=c\bar{d}$$
and deduce that
$$\frac{P(z)}{Q(z)}=\omega \frac{z-\alpha}{1-\bar{\alpha}z}$$
where $\alpha=\frac{b}{a}$ and $\omega$ is a complex number with modulus $1$.
$\textbf{My attempt so far:}$
Since $|\frac{P(z)}{Q(z)}|=1$ whenever $|z|=1$, then
$$|az-b|^2=|cz-d|^2 \rightarrow (az-b)\overline{(az-b)}=(cz-d)\overline{(cz-d)}$$
$$\rightarrow (az-b)(\bar{a}\bar{z}-\bar{b})=(cz-d)(\bar{c}\bar{z}-\bar{d})$$
$$\rightarrow a\bar{a}|z|^2-a\bar{b}z-\bar{a}b\bar{z}+b\bar{b}=c\bar{c}|z|^2-c\bar{d}z-\bar{c}d\bar{z}+d\bar{d}$$
$$\rightarrow |a|^2 -(a\bar{b}-c\bar{d})z+|b|^2=|c|^2-(\bar{c}d-\bar{a}b)\bar{z}+|d|^2$$
I'm stuck at this point. Am I on the right track, or is my method of approach wrong on a whole?