Question

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

• A. $\pi /4$
• B. $\pi /2$
• C. $3\pi /4$
• D. $5\pi /4$

Answer 1

The correct option is C $3\pi /4$

$\begin{array}{c}Since\\ z-iw=0\\ \Rightarrow z=iw\\ \Rightarrow w=-iz\\ alsoarg\left(zw\right)=\pi \\ \Rightarrow arg(-i{z}^2)=\pi \\ \Rightarrow -\frac{\pi }{2}+2arg\left(z\right)=\pi \left(\left(-i\right)=-\frac{\pi }{2}\right)\\ \Rightarrow a\left(z\right)=\frac{3\pi }{2}\\ \therefore arg\left(z\right)=\frac{3\pi }{4}\\ Hence,\\ OptionCisthecorrectanswer.\end{array}$