# If $\alpha$ and $\beta$ are distinct complex numbers with $\mid \beta\mid=1$,then value of $\big|\large\frac{\beta-\alpha}{1-\overline{\alpha}\beta}\big|$ equals

$\begin{array}{1 1}(A)\;1&(B)\;2\\(C)\;\mid \alpha\mid&(D)\;\text{None of these}\end{array}$

$\mid \beta\mid=1\Rightarrow \beta\overline{\beta}=1$
$\therefore \big|\large\frac{\beta-\alpha}{1-\overline{\alpha}\beta}\big|=\big|\large\frac{\beta-\alpha}{1-\Large\frac{\overline{\alpha}}{\overline{\beta}}}\big|$
$\Rightarrow \big|\large\frac{\overline{\beta}(\beta-\alpha)}{\overline{\beta}-\overline{\alpha}}\big|$
$\Rightarrow |\beta|\big|\large\frac{\beta-\alpha}{\overline{\beta}-\overline{\alpha}}\big|$
$\Rightarrow |\beta|(1)=\mid \beta\mid=1$
Hence (A) is the correct answer.
answered Apr 10, 2014