Question

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

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

Answer 1

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

Since $|z|=|\omega |$ and arg$\left(z\right)=\pi -arg\left(\omega \right)$
Let $\omega =r{e}^{i\theta },$ then $\overline{\omega }=r{e}^{-i\theta }\therefore z=r{e}^{i(\pi -\theta )}=r{e}^{i\pi }.e^{-i\theta }=-r{e}^{-i\theta }=-\overline{\omega }$