The number of complex numbers z, satisfying , is: [2025]

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Solution

Q.
The number of complex numbers z, satisfying $| z |$ $= 1 and$ $| \frac{z}{\bar{z}} + \frac{\bar{z}}{z} |$ $= 1$ , is: [2025]

1 8

2 10

3 6

4 4

Ans.
(1)

Let $z = e^{i θ}$ , then $\bar{z} = e^{- i θ} \Rightarrow \frac{z}{\bar{z}} = e^{i 2 θ}$

Also, |z| = 1

$∵$ $| \frac{z}{\bar{z}} + \frac{\bar{z}}{z} |$ $= 1 \Rightarrow$ $| e^{i 2 θ} + e^{- i 2 θ} |$ $= 1 \Rightarrow$ $| \cos 2 θ |$ $= \frac{1}{2}$

$∴$ Number of solutions = 8.

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