Question

If $P$ and $Q$ are represented by the complex numbers $z_1$ and $z_2$, such that $\left|\frac{1}{z_2}+\frac{1}{z_1}\right|=\left|\frac{1}{z_2}-\frac{1}{z_1}\right|$, then

• A. $△OPQ$ (where o is the origin ) is equilateral
• B. $△OPQ$ is right angled
• C. the circumcenter of $△OPQ$ is $\frac{1}{2}(z_1+z_2)$
• D. the circiucenter of $△OPQ$ is $\frac{1}{3}(z_1+z_2)$

Answer 1

Answer: (B), (C)

the circumcenter of $△OPQ$ is $\frac{1}{2}(z_1+z_2)$
$△OPQ$ is right angled

$|\frac{1}{z_2}+\frac{1}{z_1}|=\frac{1}{z_2}-\frac{1}{z_1}|$
Thus, $\frac{|z_1+z_2|}{|z_1||z_2|}=\frac{|z_1-z_2|}{|z_1||z_2|}=>|z_1+z_2|=|z_1-z_2|$
This is only possible if $z_1$ and $z_2$ make a right angle at the origin.
Thus, the circum-center is the midpoint of the hypotenuse i.e. $\frac{z_1+z_2}{2}$
Hence, (b), (c) are correct.