Question

If $\alpha$ and $\beta$ are different complex numbers with $|\beta |=1,$ then the value of $\left|\frac{\beta -\alpha }{1-\overline{\alpha }\beta }\right|$ is

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

Answer 1

The correct option is B $1$

Given, $|\beta |=1\Rightarrow |\beta {|}^2=1$
$\Rightarrow \beta \overline{\beta }=1$
$\Rightarrow \beta =\frac{1}{\overline{\beta }}\cdots \left(1\right)$

Now, $\left|\frac{\beta -\alpha }{1-\overline{\alpha }\beta }\right|=\left|\frac{\frac{1}{\overline{\beta }}-\alpha }{1-\overline{\alpha }\beta }\right|$
$=|\beta |\left|\frac{1-\alpha \overline{\beta }}{1-\overline{\alpha }\beta }\right|\left[\text{From}\right(1\left)\right]=\frac{|\overline{1-\alpha \overline{\beta }}|}{|1-\overline{\alpha }\beta |}(\because |\beta |=1\text{and}|\overline{z}|=|z|)=\frac{|1-\overline{\alpha }\beta |}{|1-\overline{\alpha }\beta |}=1$