Question

Let $z=1+ib=(a,b)$ be any complex number, $a,b,ϵR$ and $\sqrt{-1}=i.$ Let $z\ne 0+0i,argz={\tan }^{-1}\left(\frac{Imz}{Rez}\right)$ where $-\pi <argz\le \pi$

$arg\left(\overline{z}\right)+arg(-z)=\{\begin{array}{cc}\pi ,ifarg\left(z\right)<0& \\ -\pi ,ifarg\left(z\right)>0& \end{array}$

Let $z$ & $w$ be non-zero complex numbers such that they have equal modulus values and $argz-arg\overline{w}=\pi ,$ then z equals

• A. $-w$
• B. $w$
• C. $-\overline{w}$
• D. $\overline{w}$

Answer 1

The correct option is C $-\overline{w}$

Let $\overline{w}=cos\theta +isin\theta =e^{i\theta }$

Therefore

$arg\left(z\right)=\pi +\theta$

Hence $z=e^{i(\pi +\theta )}$

$=cos(\pi +\theta )+isin(\pi +\theta )$

$=-cos\theta -isin\theta$

$=-\overline{w}$