Question

If P and Q are represented by the complex 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

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

Answer 1

Answer: (B), (C)

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

We have
$|z_1{|}^2+|z_2{|}^2+2(z_1{\overline{z}}_2+{\overline{z}}_1{z}_2)=|z_1{|}^2+|z_2{|}^2-2(z_1{\overline{z}}_2+{\overline{z}}_1{z}_2)$
$\Rightarrow 4(z_1{\overline{z}}_2+{\overline{z}}_1{z}_2)=0$
$\Rightarrow \frac{z_1}{z_2}+\frac{{\overline{z}}_1}{{\overline{z}}_2}=0$
$\Rightarrow arg\left(\frac{z_1}{z_2}\right)=\frac{\pi }{2}=arg\left(\frac{z_1-0}{z_2-0}\right)$
The angle between $z_2$, O, and $z_1$ is a right angle, taken in order, as shown in the above diagram. Now, the circumcenter of the above diagram will lie on the line PQ as diameter and is represented by C which is the center of PQ, such that $z=(z_1+z_2)/2$ where z is the affix of circumcenter.