If A and B represent the complex numbers `z_(1)` and `z_(2)` such that `|z_(1)+z_(2)|=|z_(1)-z_(2)|`, then the circumcenter of `triangleOAB`, where O is the origin, is

Options

A. `(z_(1)+z_(2))/(3)`
B. `(z_(1)+z_(2))/(2)`
C. `(z_(1)-z_(2))/(2)`
D. none of these

Answer

B

Explanation

We have,
`|z_(1)+z_(2)|=|z_(1)-z_(2)|`
`rArr |z_(1)+z_(2)|^(2)=|z_(1)-z_(2)|^(2)`
`rArr 4|z_(1)||z_(2)|^(2)=|z_(1)-z_(2)|^(2)`
(wher `theta_(1)="arg"(z_(1))` and `theta_(2)="arg"(z_(2)`]
`rArr theta_(2)-theta_(1)=pi/2 rArr angleAOB=pi/2`

So, the circumcenter of `triangleAOB` is the mid-point of hypotenuse AB and its affix is `(z_(1)+z_(2))/(2)`.