Question

If $z_1,z_2,z_3$ non-zero, non-collinear complex numbers such that $\frac{2}{z_1}=\frac{1}{z_2}+\frac{1}{z_3},$ then the points $z_1,z_2,z_3$ lie

• A. in the interior of a circle
• B. on a circle passing through origin
• C. in the exterior of a circle
• D. None of these

Answer 1

The correct option is B on a circle passing through origin

We have,

$\frac{2}{z_1}=\frac{1}{z_2}+\frac{1}{z_3}=\frac{z_3+z_2}{z_2{z}_3}$

$\Rightarrow z_1=\frac{2z_2{z}_3}{z_2+z_3}$

Now, $\left(\frac{z_2-z_4}{z_1-z_4}\right)\left(\frac{z_1-z_3}{z_2-z_3}\right)$

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

$=\frac{z_2}{\frac{2z_2{z}_3}{z_2+z_3}}\left(\frac{z_3(z_2-z_3)}{(z_2-z_3)(z_2-z_3)}\right)$ [taking $z_4=0]$

$=\frac{1}{2}$ (a real number).

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

and therefore, $z_1,z_2,z_3$ lie on a circle passing through the origin.