Question

Assertion :Let, $z_1$ and $z_2$ be two complex numbers denoted by A and B respectively such that $|z_1-z_2|=|z_1+z_2|$, then the orthocenter $△AOB$ is $\frac{z_1+z_2}{2}$. (where $O$ is origin). Reason: In case of a right-angle triangle, orthocenter is that point at which the triangle is right-angled.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect and Reason is correct

Answer 1

The correct option is D Assertion is incorrect and Reason is correct

$|z_1-z_2|=|z_1+z_2|$ is only possible if they are perpendicular. Thus, in triangle AOB, angle AOB is $\frac{\pi }{2}$.
Here, Ortho-center should be at O. However, $\frac{z_1+z_2}{2}$ represents the midpoint of hypotenuse AB which is the circumcenter.
Hence, Assertion is incorrect while Reason is correct.