is a complex number such that z≠1,|z|=1. The real part of z−1z+1 is

A
$\frac{1}{| z + 1 |^{2}}$
B
$\frac{- 1}{| z + 1 |^{2}}$
C
$\frac{\sqrt{2}}{| z + 1 |^{2}}$
D
0

Solution:

$| z | = 1 \Rightarrow z \bar{z} = 1, \bar{z} = \frac{1}{z}$ and $Re z = \frac{1}{2} (z + \bar{z})$

$\begin{array}{l} ∴ Re (\frac{z - 1}{z + 1}) = \frac{1}{2} (\frac{z - 1}{z + 1} + (\frac{z - 1}{z + 1})) = \frac{1}{2} (\frac{z - 1}{z + 1} + \frac{\bar{z} - 1}{\bar{z} + 1}) \\ = \frac{1}{2} (\frac{z - 1}{z + 1} + \frac{\frac{1}{z} - 1}{\frac{1}{z} + 1}) = \frac{1}{2} (\frac{z - 1}{z + 1} + \frac{1 - z}{z + 1}) = 0 \end{array}$

is a complex number such that $z \neq 1, | z | = 1$ . The real part of $\frac{z - 1}{z + 1}$ is

Solution:

Related content