Question

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

• A. $0$
• B. $1$
• C. $\frac{1}{2}$
• D. $-1$

Answer 1

The correct option is B $1$

Let $t=\left|\frac{\beta -\alpha }{1-\overline{\alpha }\beta }\right|$

Multiply the numerator and denominator with $\overline{\beta }$

$\Rightarrow t=\left|\frac{1-\alpha \overline{\beta }}{\overline{\beta }-\overline{\alpha }}\right|=\left|\frac{\overline{1-\alpha \overline{\beta }}}{\overline{\overline{\beta }-\overline{\alpha }}}\right|=\left|\frac{1-\overline{\alpha }\beta }{\beta -\alpha }\right|=\frac{1}{t}$

$\Rightarrow t^2=1$

$\Rightarrow t=1$