Question

State true or false:

If $z_1,z_2,z_3$ are complex numbers such that $\frac{2}{z_2}=\frac{1}{z_1}+\frac{1}{z_3}$ , then the points $z_1,z_2,z_3$ lie on a circle passing through origin.

Answer 1

$\frac{2}{z_2}=\frac{1}{z_1}+\frac{1}{z_3}$

$⟹z_2=\frac{2z_1{z}_3}{z_1+z_3}$

Now $(\frac{z_1-z_4}{z_2-z_4})(\frac{z_2-z_3}{z_1-z_3})=(\frac{z_1-z_4}{\frac{2z_1{z}_3}{z_1+z_3}-z_4})(\frac{\frac{2z_1{z}_3}{z_1+z_3}-z_3}{z_1-z_3})$

$=\frac{z_1}{\frac{2z_1{z}_3}{z_1+z_3}}(\frac{z_3(z_1-z_3)}{(z_1-z_3)(z_1+z_3)})=\frac{1}{2}$ $($ Putting $z_4=0)$

Hence $z_1,z_2,z_3$ and origin are concyclic points

I.e they lie on a circle