Question

If $z_1,z_2,z_3,$ be 3 complex number in harmonic progression where the points representing $z_1,z_2,z_3,$ are not collinear , then

• A. the origin $z_1{z}_2$ lie on a straight line
• B. the origin $z_2,z_3$ lie on a straight line
• C. $z_3$ lies on the circle through the origin and $z_1{z}_2$
• D. the origin $z_1{z}_3$ are collinear

Answer 1

The correct option is C $z_3$ lies on the circle through the origin and $z_1{z}_2$

$\frac{1}{z_2}-\frac{1}{z_1}=\frac{1}{z_3}-\frac{1}{z_2}$
$\Rightarrow \frac{z_1-z_2}{z_2{z}_1}$=$\frac{z_2-z_3}{z_2{z}_3}$
$\Rightarrow \frac{z_1-z_2}{z_2-z_3}=\frac{z_1}{z_3}$
$\frac{z_1-z_2}{z_3-z_2}=-\left(\frac{z_1}{z_3}\right)$
arg $\left(\frac{z_1-z_2}{z_3-z_2}\right)=\pi -arg\left(\frac{z_1}{z_3}\right)$
$z_3{z}_2{z}_1=\pi -z_{1}O{z}_3$
$\therefore O,z_1,z_2,z_3$ are concyclic.