First slide

Algebra of complex numbers

Question

If $| z | = 1$ and $w = \frac{z - 1}{z + 1} ($ where $z \neq - 1)$ , then $R e (w)$ equals

Moderate

A

0
B

$- \frac{1}{| z + 1 |^{2}}$
C

$|\frac{z}{z + 1}| \frac{1}{| z + 1 |^{2}}$
D

$\frac{\sqrt{2}}{| z + |^{2}}$

Solution

$| z | = 1 \Rightarrow z z = 1$
$2 Re (w) = w + \bar{w} = \frac{z - 1}{z + 1} + \frac{\bar{z} - 1}{z + 1}$
$\begin{aligned} = \frac{z - 1}{z + 1} + \frac{(1 / z) - 1}{(1 / z) + 1} \\ = \frac{z - 1}{z + 1} + \frac{1 - z}{1 + z} = 0 \end{aligned}$
$\Rightarrow Re (w) = 0$

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