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  4. If α and β are different complex numbers with | β | = 1, then (β-α/1- overline α β) is equal to

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Q. If $\alpha $ and $\beta$ are different complex numbers with $ | \beta | = 1$, then $\frac {\beta-\alpha}{1-\overline {\alpha} \beta}$ is equal to

KCETKCET 2012Complex Numbers and Quadratic Equations

Solution:

Since, $|\beta|=1$
$ \therefore |\beta|^{2}=\beta \bar{\beta}=1 $
$\therefore \left|\frac{\beta-\alpha}{1-\bar{\alpha} \beta}\right| =\left|\frac{\beta-\alpha}{\beta \bar{\beta}-\bar{\alpha} \beta}\right|$
$=\frac{|\beta-\alpha|}{|\beta||\bar{\beta}-\bar{\alpha}|}$
$=\frac{|\beta-\alpha|}{1 \cdot|\overline{\beta-\alpha}|} $
$=1$