Question

Let z,w be complex numbers such that $z+iw=0$ and arg $zw=\pi$ then arg z equals

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

Answer 1

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

Given that arg $zw=\pi$ .....(i)

$\overline{z}+i\overline{w}=0$$\Rightarrow \overline{z}=-i\overline{w}\Rightarrow z=iw\Rightarrow w=-iz$
From (i),arg $(-i{z}^2)=\pi$
arg$(-i)+2arg\left(z\right)=\pi ;\frac{-\pi }{2}+2arg\left(z\right)=\pi$
$2arg\left(z\right)=\frac{3\pi }{2};arg\left(z\right)=\frac{3\pi }{4}$