Question

If P and Q are represented by the numbers $z_1$ and $z_2$ such that $|\frac{1}{z_2}+\frac{1}{z_1}|$ = $|\frac{1}{z_2}-\frac{1}{z_1}|$, then the circumcentre of $△OPQ$,(where O is the origin) is

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

$|\frac{1}{z_2}+\frac{1}{z_1}|$ = $|\frac{1}{z_2}-\frac{1}{z_1}|$

$\Rightarrow |z_1+z_2|$ = $|z_1-z_2|$

$\Rightarrow z_1{\overline{z}}_2+z_2{\overline{z}}_1$ = 0

$\Rightarrow \frac{z_1}{z_2}$ is purely imaginary

$\Rightarrow$ arg $⟮\frac{z_1}{z_2}$ = $\pm \frac{\pi }{2}$

$\Rightarrow \angle POQ$ = $\frac{\pi }{2}$

Circumcentre of $△POQ$ is the midpoint of PQ i.e. $\frac{z_1+z_2}{2}$