Question

Let $z$ and $\omega$ be two non zero complex numbers such that $\left|z\right|=\left|\omega \right|$ and $Argz+Arg\omega =\pi$, then $z$ equals:

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

Answer 1

The correct option is D

$-\overline{\omega }$

Explanation for the correct option:

Finding the value of $z$:

Given that,

$\left|z\right|=\left|\omega \right|...\left(i\right)$

We know that for a complex number $z$

$z=\left|z\right|(\cos \theta +i\sin \theta )...\left(ii\right)$

Where, $\theta =argz$

Considering ${\theta }_1=argw$

Given, $Argz+Arg\omega =\pi$

Therefore,

$$\begin{gathered} \theta +{\theta }_1=\pi \\ \Rightarrow \theta =\pi -{\theta }_1...\left(iii\right) \end{gathered}$$

From equation $\left(i\right),\left(ii\right)\&\left(iii\right)$ we have:

$$\begin{gathered} z=\left|\omega \right|\left(\right(\cos (\pi -{\theta }_1)+i\sin (\pi -{\theta }_1)) \\ =|\omega |((-\cos ({\theta }_1)+i\sin ({\theta }_1\left)\right) \\ =-\left|\omega \right|\left(\right(\cos \left({\theta }_1\right)-i\sin \left({\theta }_1\right)) \\ =–\overline{\omega } \end{gathered}$$
Hence, the correct answer is option (D).