Question

If $P$ and $Q$ are respectively by the complex numbers $z_1$ and $z_2$ such that $\left|\frac{1}{z_1}+\frac{1}{z_2}\right|=\left|\frac{1}{z_1}-\frac{1}{z_2}\right|$, then the circumcentre of $\Delta OPQ$ (where $O$ is the origin) is

• A. $\frac{z_1-z_2}{2}$
• B. $\frac{z_1+z_2}{2}$
• C. $\frac{z_1+z_2}{3}$
• D. $z_1+z_2$

Answer 1

Answer: (B)

$\frac{z_1+z_2}{2}$

$z_1=(a,b)$ & $z_2=(c,d)$

$|\frac{1}{z_1}+\frac{1}{z_2}|=|\frac{1}{z_1}-\frac{1}{z_2}|$

$|z_1+z_2|=|z_1-z_2|$

${(a+c)}^2+{(b+d)}^2={(a-c)}^2+{(b-d)}^2$

On solving, we get

$4ac+4bd=0$

$ac+bd=0$

Therefore, OPQ is right angle as $z_1.z_2=0$

In such cases, mid point of hypotenuse is circumcenter, therefore

Circumcenter$=\frac{z_1+z_2}{2}$