Question

Let $z_1,z_2,z_3$​ be three distinct complex numbers lying on a circle whose centre is at the origin. If $z_i+z_j{z}_k,$ ​where $i,j,k\in \{1,2,3\}$ and $i\ne j\ne k$ are real numbers, then the value of $4(z_1\times z_2\times z_3)$ is

Answer 1

As $z_1,z_2,z_3$ ​lie on the circle centered at origin.
$\therefore |z_1|=|z_2|=|z_3|=r$
(where $r$ is the radius of the circle)
Let $z_1=r{e}^{i{\theta }_1}$
$z_2=r{e}^{i{\theta }_2}$
$z_3=r{e}^{i{\theta }_3}$
​It is given that $z_1+z_2{z}_3\in R$
$\therefore z_1=\overline{z_2{z}_3}\Rightarrow r{e}^{i{\theta }_1}=r{e}^{-i{\theta }_2}\cdot r{e}^{-i{\theta }_3}\Rightarrow r=r^2{e}^{-i({\theta }_1+{\theta }_2+{\theta }_3)}\Rightarrow r{e}^{-i({\theta }_1+{\theta }_2+{\theta }_3)}=1(\because r\ne 0)$
Taking mod both sides, we get
$r=1$

Hence, $4(z_1\times z_2\times z_3)$
$=4(\overline{z_2{z}_3}\times z_2{z}_3)=4(|z_2|\times |z_3|{)}^2=4(\because |z_2|=|z_3|=1)$