Complex Numbers 3 Question 2
2. If $z$ is a complex number of unit modulus and argument $\theta$, then $\arg \frac{1+z}{1+\bar{z}}$ is equal to
(2013 Main)
(a) $-\theta$
(b) $\frac{\pi}{2}-\theta$
(c) $\theta$
(d) $\pi-\theta$
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Answer:
Correct Answer: 2. (c)
Solution:
- Given, $|z|=1$, $\arg z=\theta \therefore z=e^{i \theta}$
$$ \begin{aligned} & \therefore \quad \bar{z}=e^{-i \theta} \Rightarrow \bar{z}=\frac{1}{z} \\ & \therefore \quad \arg \frac{1+z}{1+\bar{z}}=\arg \frac{1+z}{1+\frac{1}{z}}=\arg (z)=\theta \end{aligned} $$
