Question

Let Z and w be complex numbers. If $Re\left(z\right)=|z-2|,Re\left(w\right)=|w-z|$ and $arg(z-w)=\frac{\pi }{3}$, then the value of $Im(z+w)$, is

• A. $\frac{1}{\sqrt{3}}$
• B. $\frac{2}{\sqrt{3}}$
• C. $\sqrt{3}$
• D. $\frac{4}{\sqrt{3}}$

Answer 1

Answer: (B)

$\frac{2}{\sqrt{3}}$

$Re\left(z\right)=|z-2|$
$Re\left(w\right)=|w-z|$
$arg(z-w)=\frac{\pi }{3}$

Now,
$Re(z+w)=||z-2|+|w-z||$
$=|w-2|=2$
$arg(z-w)=\frac{y}{x}$
$\tan \frac{\pi }{3}=\frac{Im(z+w)}{Re(z+w)}$
$\Rightarrow Im(z+w)=\frac{2}{\sqrt{3}}$
Hence (B) is the correct answer