Question

Let $z,\omega$ be complex numbers such that $\overrightarrow{z}+i\overrightarrow{\omega }=0$ and $Arg\left(z\omega \right)=\pi$ then $Arg\left(z\right)=$.

• A. $\frac{\pi }{4}$
• B. $\frac{5\pi }{4}$
• C. $\frac{3\pi }{4}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is C $\frac{3\pi }{4}$

Since $z+iw=0$

$\Rightarrow z=-iw$

$\Rightarrow w=-iz$

Also $arg\left(zw\right)=\pi$

$\Rightarrow arg(-i{z}^2)=\pi$

$\Rightarrow arg(-i)+2arg\left(z\right)=\pi$

$\Rightarrow \frac{-\pi }{2}+2arg\left(z\right)=\pi$ ....... $[\because arg(-i)=\frac{-\pi }{2}]$

$\Rightarrow 2arg\left(z\right)=\frac{3\pi }{2}$

$\Rightarrow arg\left(z\right)=\frac{3\pi }{4}$

Hence, the answer is $\frac{3\pi }{4}$.