Q98 / 225 · JEE Main 2002 · Mathematics

$z$ and $w$ are two non zero complex no.s such that $|z|=|w|$ and $\operatorname{Arg} z+\operatorname{Arg} w=\pi$ then $z$ equals

  • A
    $\overline{\mathrm{W}}$
  • Correct answer
    B
    -$\overline{\mathrm{W}}$
  • C
    $\mathrm{W}$
  • D
    -$\mathrm{W}$

Solution

Let $|\mathrm{z}|=|\omega|=\mathrm{r} \quad \therefore \mathrm{z}=\mathrm{re}^{\mathrm{i} \theta}, \omega=\mathrm{re}^{\mathrm{i} \phi}$ where $\theta+\phi=\pi \quad \therefore \bar{\omega}=\mathrm{re}^{-\mathrm{i \phi}}$ $\therefore z=\mathrm{re}^{i(\pi-\phi)}=\mathrm{re}^{\mathrm{i} \pi} \cdot \mathrm{e}^{-i \phi}=-\mathrm{re}^{-i \phi}=-\bar{\omega}$

Hence, the correct answer is option: 'B'.

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