Question

If $\alpha$ and $\beta$ are different complex numbers with $\left|\beta \right|=1$, then the value of $\left|\frac{(\beta -\alpha )}{(1-\overline{\alpha }\beta )}\right|.$

Answer 1

Given,
$\left|\beta \right|=1⟹\beta \overline{\beta }=1$ or $\beta =\frac{1}{\overline{\beta }}$ (1)
Now,
$y=\left|\frac{\beta -\alpha }{1-\overline{\alpha }\beta }\right|=\left|\frac{\frac{1}{\overline{\beta }}-\alpha }{1-\overline{\alpha }\beta }\right|$
$\Rightarrow y=\left|\frac{1}{\overline{\beta }}\right|\left|\frac{1-\alpha \overline{\beta }}{1-\overline{\alpha }\beta }\right|$

Using the following conjugate properties:

$\left(i\right)\overline{\overline{z}}=z$,

$\left(ii\right)\overline{z_1-z_2}=\overline{z_1}-\overline{z_2}$
$\Rightarrow y=\frac{1}{\left|\beta \right|}\left|\frac{\overline{1-\overline{\alpha }\beta }}{1-\overline{\alpha }\beta }\right|$ $[\because |\beta |=1]$
$\Rightarrow y=\left|\frac{1-\overline{\alpha }\beta }{1-\overline{\alpha }\beta }\right|=1$
Ans: 1