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:

  1. Since, $|z|=|w|$ and $\arg (z)=\pi-\arg (w)$

$$ \begin{array}{ll} \text { Let } & w=r e^{i \theta}, \text { then } \bar{w}=r e^{-i \theta} \\ \therefore & z=r e^{i(\pi-\theta)}=r e^{i \pi} \cdot e^{-i \theta}=-r e^{-i \theta}=-\bar{w} \end{array} $$



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