Question

If the points $A$ and $B$ are represented by the non-zero complex number $z_1$ and $z_2$ on the argand plane such that $|z_1+z_2|=|z_1-z_2|$ and $O$ is the origin, then,

• A. orthocenter of $△OAB$ lies at $O$
• B. circumcenter of $△OAB$ is $\frac{z_1+z_2}{2}$
• C. $arg\left(\frac{z_1}{z_2}\right)=\pm \frac{\pi }{2}$
• D. $△OAB$ is isosceles

Answer 1

Answer: (A), (B), (C)

The correct options are
A circumcenter of $△OAB$ is $\frac{z_1+z_2}{2}$
B $arg\left(\frac{z_1}{z_2}\right)=\pm \frac{\pi }{2}$
C orthocenter of $△OAB$ lies at $O$

We have, $|z_1+z_2|=|z_1-z_2|\Rightarrow (z_1+z_2)(\overline{z_1}+\overline{z_2})=(z_1-z_2)(\overline{z_1}-\overline{z_2})$

$\Rightarrow z_1\overline{z_2}+z_2\overline{z_1}=0$ ...(1)

$\Rightarrow \frac{z_1}{z_2}=-\left(\frac{\overline{z_1}}{z_2}\right)\Rightarrow \frac{z_1}{z_2}$ is purely imaginary

Also, from (1), ${|z_1-z_2|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2$

$\Rightarrow △OAB$ is a right angled triangle, right angled at $O.$

So, orthocentre lies at $O$ and circumcentre $=\frac{z_1+z_2}{2}.$