Question

If $\alpha, \beta$ be any two complex numbers such that $\left|\frac{\alpha-\beta}{1-\bar{\alpha} \beta}\right|=1$, then :

• A. $|\alpha|=1$
• B. $|\beta|=1$
• C. $\alpha= e ^{ i \theta}, \theta \in R -2 n \pi, n \in I$
• D. $\beta= e ^{ i \theta}, \theta \in R-2 n \pi, n \in I$

Answer 1

Answer: (A), (B), (C), (D)

$|\alpha-\beta|=|1-\bar{\alpha} \beta| \Rightarrow(\alpha-\beta)(\bar{\alpha}-\bar{\beta})=(1-\bar{\alpha} \beta)(1-\alpha \bar{\beta})$
$\Rightarrow|\alpha|^2+|\beta|^2-\alpha \bar{\beta}-\bar{\alpha} \beta=1+|\alpha|^2 \cdot|\beta|^2-\bar{\alpha} \beta-\alpha \bar{\beta} $
$\Rightarrow|\alpha|^2+|\beta|^2-|\alpha|^2 \cdot|\beta|^2-1=0 \Rightarrow\left(|\alpha|^2-1\right)\left(|\beta|^2-1\right)=0 $
$\Rightarrow|\alpha|=1 \text { or }|\beta|=1 \Rightarrow \alpha= e ^{ i \theta}, \theta \in R \text { or } \beta= e ^{ i \theta}, \theta \in R $