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

Answer 1

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

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