Complex Numbers 3 Question 5
5. Let $z$ and $w$ be two non-zero complex numbers such that $|z|=|w|$ and $\arg (z)+\arg (w)=\pi$, then $z$ equals
(1995, 2M)
(a) $w$
(b) $-w$
(c) $\bar{w}$
(d) $-\bar{w}$
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Answer:
Correct Answer: 5. (d)
Solution:
- Since, $|z|=|w|$ and $\arg (z)=\pi-\arg (w)$
$$ \begin{array}{ll} \text { Let } & w=r e^{i \theta}, \text { then } \bar{w}=r e^{-i \theta} \\ \therefore & z=r e^{i(\pi-\theta)}=r e^{i \pi} \cdot e^{-i \theta}=-r e^{-i \theta}=-\bar{w} \end{array} $$
