If and are different complex numbers with | | = 1 then find

Question

If $\alpha $ and $\beta $ are different complex numbers with $\left| \beta \right| = 1$ then find $\left| {\frac{{\beta - \alpha }}{{1 - \overline \alpha \beta }}} \right|$

✓

Answer

Now ${\left| {\frac{{\beta - \alpha }}{{1 - \overline \alpha \beta }}} \right|^2} = \left[ {\frac{{\beta - \alpha }}{{1 - \overline \alpha \beta }}} \right]\left[ {\frac{{\overline {\beta - \alpha } }}{{1 - \overline \alpha \beta }}} \right]$$\left[ {\because {{\left| z \right|}^2} = z\overline z } \right]$
$ = \left[ {\frac{{\beta - \alpha }}{{1 - \overline \alpha \beta }}} \right]\left[ {\frac{{\overline \beta - \overline \alpha }}{{1 - \alpha \overline \beta }}} \right]$
$ = \frac{{\beta \overline \beta - \beta \overline \alpha - \alpha \overline \beta + \alpha \overline \alpha }}{{1 - \overline \alpha \beta - \alpha \overline \beta + \alpha \overline \alpha \beta \overline \beta }}$$ = \frac{{{{\left| \beta \right|}^2} - \alpha \overline \beta - \alpha \overline \beta + {{\left| \alpha \right|}^2}}}{{1 - \overline \alpha \beta - \alpha \overline \beta + {{\left| \alpha \right|}^2}{{\left| \beta \right|}^2}}}$

$ = \frac{{1 - \overline \alpha \beta - \alpha \overline \beta + {{\left| \alpha \right|}^2}}}{{1 - \overline \alpha \beta - \overline \alpha \beta + {{\left| \alpha \right|}^2}}} = 1$
$\therefore \left| {\frac{{\beta - \alpha }}{{1 - \alpha \beta }}} \right| = 1$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Complex Numbers and Quadratic Equations→Complex Numbers and Quadratic Equations — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Complex Numbers and Quadratic Equations — Maths STD 11 Science — Question

Answer

Need a full question paper?

Explore more

Similar questions