Solveeit Logo
Login

Question

Question: If a and b are different complex numbers with \|b\| = 1, then \(\left| \frac{\beta - \alpha}{1 - \ba...

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

A

0

B

1/2

C

1

D

2

Answer

1

Explanation

Solution

Sol. |b| = 1 Q |b|2 = 1

bβˉ\bar{\beta} =1

∣β−α1−αˉβ∣\left| \frac{\beta - \alpha}{1 - \bar{\alpha}\beta} \right|= ∣β−αββˉ−αˉβ∣\left| \frac{\beta - \alpha}{\beta\bar{\beta} - \bar{\alpha}\beta} \right|= ∣β−α∣∣β∣∣βˉ−αˉ∣\frac{|\beta - \alpha|}{|\beta||\bar{\beta} - \bar{\alpha}|}= ∣β−α∣∣β−α‾∣\frac{|\beta - \alpha|}{|\overline{\beta - \alpha}|}= 1

(Q ∣zˉ∣|\bar{z}|= |z|)