Question

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

Answer 1

Find the value of $\left|\frac{(\beta -\alpha )}{(1-\overline{\alpha }\beta }\right|$:

Given $\left|\beta \right|=1$

$$\begin{gathered} \therefore |\beta {|}^2=1 \\ \Rightarrow \beta .\overline{\beta }=1 \end{gathered}$$

Now, $\left|\frac{(\beta -\alpha )}{(1-\overline{\alpha }\beta )}\right|$

$$\begin{gathered} =\left|\frac{\beta -\alpha }{\beta \overline{\beta }-\overline{\alpha }\beta }\right| \\ =\left|\frac{\beta -\alpha }{\beta (\overline{\beta }-\overline{\alpha })}\right| \\ =\frac{(\beta -\alpha )}{\left|\beta \right|\overline{|\beta -\alpha |}} \\ =\frac{1}{\left|\beta \right|} \\ =1 \end{gathered}$$

Hence, the value of $\left|\frac{(\beta -\alpha )}{(1-\overline{\alpha }\beta }\right|$is $\mathbf{1}$.