Mathematics - Complex Number Question with Solution | TestHub

$\frac{2}{z_1}=\frac{1}{z_2}+\frac{1}{z_3}\Rightarrow \arg \left(\frac{z_2-z_1}{z_3-z_1}\right)=\arg \left(-\frac{z_2}{z_3}\right)=\arg \left(\frac{z_2}{z_3}\right)\pm \pi$

$$\begin{array}{rlr}& \Rightarrow \arg \left(\frac{z_2-z_1}{z_3-z_1}\right)=\arg \left(\frac{z_2-0}{z_3-0}\right)\pm \pi & \Rightarrow \arg \left(\frac{z_2-z_1}{z_3-z_1}\right)-\arg \left(\frac{z_2-0}{z_3-0}\right)=\pm \pi \end{array}$$

Sum of angles at A and origin is $\pm \pi$. Hence points $\mathrm{O},\mathrm{B},\mathrm{A},\mathrm{C}$ are concyclic.