Question

Let z, w be complex numbers such that $\overline{z}+i\overline{w}=0$ and $argzw=\pi$. 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)$

$\because \overline{z}+i\overline{w}=0$ ......[Given]

By comparing real and imaginry part of equation we have,

Now $cosA+sinB=0$ and $cosB-sinA=0$

$\therefore A+B=\pi$

$\Rightarrow A=\pi -B$

$\therefore cos\left(B\right)=sin(\pi -B)$

$\Rightarrow cos\left(B\right)=sin\left(B\right)$

$\therefore arg\left(Z\right)$ $=\pi -\frac{\pi }{4}$

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