Question

z and $\omega$ are two non zero complex numbers, such that $|z|=|\omega |$ and arg z +arg $\omega =\pi$ then z equals

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

Answer 1

The correct option is B $-\overline{\omega }$

Let
$w=r(cosA+isinA)$
Therefore
$arg\left(z\right)=\pi -arg\left(w\right)$
$=\pi -A$
Hence $z$ will be
$z=|z|\left(cos\right(\pi -A)+isin(\pi -A\left)\right)$
$=|z|(-cosA+isinA)$
$=-|z|(cosA-isinA)$
Now $|z|=|w|=r$
Therefore
$z=-|w|(cosA-isinA)$
$=-\overline{w}$
Hence, option 'B' is correct.