Question

If complex numbers $z_1$ and $z_2$ both satisfy $z+\overline{z}=2|z-1|$ and $\arg (z_1-z_2)=\frac{\pi }{3},$ then the value of $\text{Im}(z_1+z_2)$ is
($\text{Im}\left(z\right)$ denotes the imaginary part of $z$)

• A. $\sin \frac{\pi }{3}$
• B. $\text{cosec}\frac{\pi }{3}$
• C. $\tan \frac{\pi }{3}$
• D. $\cot \frac{\pi }{3}$

Answer 1

The correct option is B $\text{cosec}\frac{\pi }{3}$

Let $z=x+iy$
$\frac{z+\overline{z}}{2}=|z-1|$
$\Rightarrow x^2=(x-1{)}^2+y^2$
$\Rightarrow y^2=2x-1$
Let $z_1(\frac{t_1^2+1}{2},t_1)$ and $z_2(\frac{t_2^2+1}{2},t_2)$
$\arg (z_1-z_2)=\frac{\pi }{3}$
$\Rightarrow \frac{t_2-t_1}{\left(\frac{t_2^2-t_1^2}{2}\right)}=\sqrt{3}$
$\Rightarrow \frac{2}{t_1+t_2}=\sqrt{3}$
$\therefore \text{Im}(z_1+z_2)=t_1+t_2=\frac{2}{\sqrt{3}}$