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Question: If \(\alpha\) and \(\beta\) are different complex numbers with \(|\beta| = 1,\) then \(\left| \frac{...

If α\alpha and β\beta are different complex numbers with ∣β∣=1,|\beta| = 1, then ∣β−α1−αˉβ∣\left| \frac{\beta - \alpha}{1 - \bar{\alpha}\beta} \right| is equal to

A

0

B

½

C

1

D

2

Answer

1

Explanation

Solution

Sol.∣β−α1−αˉβ∣=∣β−αββˉ−αˉβ∣=∣β−αβ(βˉ−αˉ)∣,=1∣β∣∣β−α(β−α‾)∣=1∣β∣=1{∵∣z∣=∣zˉ∣}\left| \frac{\beta - \alpha}{1 - \bar{\alpha}\beta} \right| = \left| \frac{\beta - \alpha}{\beta\bar{\beta} - \bar{\alpha}\beta} \right| = \left| \frac{\beta - \alpha}{\beta(\bar{\beta} - \bar{\alpha})} \right|, = \frac{1}{|\beta|}\left| \frac{\beta - \alpha}{\overline{(\beta - \alpha})} \right| = \frac{1}{|\beta|} = 1\{\because|z| = |\bar{z}|\}