Question

Let $z$ and $w$ be complex numbers such that $\overline{z}+i\overline{w}=0$ and $argzw=\pi$. 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 A $\frac{5\pi }{4}$

Let $z=x+iy,w=a+bi$

$z+iw=(x+iy)+i(a+bi)$

$=(x-b)+i(y+a)$

Conjugate A $(z+iw)=0$

$\therefore (x-b)-i(y+a)=0$

$\therefore x-b=0,y+a=0$

$x=b$ & $y=-a$

$\therefore z=b-ai$ & $w=a+bi$

$zw=(b-ai)(a+bi)$

$=ab+b^{2}i-a^{2}i+ab$

$zw=2ab+(b^2-a^2)i$

$\therefore arg\left(zw\right)=\pi$

$ta{n}^{-1}\left(\frac{b^2-a^2}{2ab}\right)=\pi$

$\frac{b^2-a^2}{2-b}=tan\pi =0$

$b^2-a^2=0$

$b^2=a^2$

$\therefore b=-a$

$\therefore z=-a-ai$

$\therefore arg\left(2x\right)=ta{n}^{-1}\left(\frac{-a}{-a}\right)$

$=ta{n}^{-1}\left(1\right)$

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

$Arg\left(z\right)=\frac{5\pi }{4}$