Question

Let $z$ and $\omega$ are two non -zero complex number such that $|z|=|\omega |$ and arg$z+arg\omega =\pi$. then $z$ equal to ?

• A. $\omega$
• B. $-\omega$
• C. $\overline{\omega }$
• D. $-\overline{\omega }$

Answer 1

Answer: (D)

$-\overline{\omega }$

Let $z={re}^{ia}$

Then, $w={re}^{ib}$ $\{\because |z|=|w|\}$

Given : $\arg \left(z\right)+\arg \left(w\right)=\pi$

$\Rightarrow \alpha +\beta =\pi \iff \alpha =\pi -\beta$

$\therefore z={re}^{i(\pi -\beta )}$

$=\left({re}^{-ib}\right)\left(e^{i\pi }\right)$

$=\left(\varpi \right)(-1)$

$=-\varpi$