Question

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

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

Answer 1

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

Let
$z=cosA+isinA$ and $w=cosB+isinB$
$\overline{z}=cosA-isinA$ and $\overline{w}=cosB-isinB$
$zw=cos(A+B)+isin(A+B)$
$\overline{z}+i\overline{w}$
$=cosA+sinB+i(cosB-sinA)$
Now $cosA+sinB=0$ and $cosB-sinA=0$
And $A+B=\pi$
$A=\pi -B$
$cos\left(B\right)=sin(\pi -B)$
$cos\left(B\right)=sin\left(B\right)$
Therefore $arg\left(Z\right)$
$=\pi -\frac{\pi }{4}$
$=\frac{3\pi }{4}$