Question

Let $z$ and $w$ be two non-zero complex numbers such that $|z|=|w|$ and $arg\left(z\right)+arg\left(w\right)=\pi$, then $z$ equals

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

Answer 1

The correct option is D $-\overline{w}$

$z=|z|(\cos \theta +i\sin \theta )$

where $\theta =argz$

if ${\theta }_1=argw$ then $\theta =\pi -{\theta }_1$

$\therefore z=|w|\left\{\cos \right(\pi -{\theta }_1)+i\sin (\pi -{\theta }_1\left)\right\}$

$=|w|(-\cos {\theta }_1+i\sin {\theta }_1)$

$=-|w|(-\cos {\theta }_1+i\sin {\theta }_1)$

$=-|w|(\cos {\theta }_1-i\sin {\theta }_1)$

$z=-|\overline{w}|$