Question

Let z and $\omega$ be complex numbers such that $\overline{z}+i\overline{\omega }=0$ and arg $z\omega =\pi$. Then arg z equals

• 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

$\frac{3\pi }{4}$

$\overline{z}+i\overline{\omega }=0\Rightarrow \overline{z}=-i\overline{\omega }\Rightarrow z=i\omega \Rightarrow \omega =-izAlsoarg\left(z\omega \right)=\pi \Rightarrow arg(-i{z}^2)=\pi \Rightarrow arg(-i)+2arg\left(z\right)=\pi \Rightarrow -\frac{\pi }{2}+2arg\left(z\right)=\pi [\because arg(-i)=-\frac{\pi }{2}]\Rightarrow 2arg\left(z\right)=\frac{3\pi }{2}\Rightarrow arg\left(z\right)=\frac{3\pi }{4}$