Question

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

• A. $\frac{1}{2}$
• B. $1$
• C. $\frac{1}{3}$
• D. $2$

Answer 1

The correct option is B $1$

$|\beta -\alpha |=\sqrt{{|\beta |}^2+{|\alpha |}^2-2Re\left(\overline{\alpha }\beta \right)}$

So $|\frac{\beta -\alpha }{1-\overline{\alpha }\beta }|=\frac{|\beta -\alpha |}{|1-\overline{\alpha }\beta |}$

$=\sqrt { \dfrac { { |\beta | }^{ 2 }+{ |\alpha | }^{ 2 }-2Re(\bar { \alpha } \beta ) }{ {

|1| }^{ 2 }+{ |\bar { \alpha } \beta | }^{ 2 }-2Re(\bar { 1 } \cdot \bar { \alpha } \beta ) } } $

$=\sqrt{\frac{1+{|\alpha |}^2-2Re\left(\overline{\alpha }\beta \right)}{1+{|\alpha |}^2-2Re\left(\overline{\alpha }\beta \right)}}$

$=1$ ($\because |\beta |=1$)