Question

If P and Q are respected by the complex numbers $z_1$ and $z_2$ such that $|\frac{1}{z_1}+\frac{1}{z_2}|=|\frac{1}{z_1}-\frac{1}{z_2}|$ 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)

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

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

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

${|z_2+z_1|}^2={|z_2-z_1|}^2$

$\because {|z_2+z_1|}^2+{|z_2+z_1|}^2=2{\left|z_1\right|}^2+2{\left|z_2\right|}^2$

$\therefore 2{|z_2-z_1|}^2=2{\left|z_1\right|}^2+2{\left|z_2\right|}^2$

$\Rightarrow {|z_2-z_1|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2$

Thus $△OPQ$ is rightangled at $O$.

Hence mid-point of hypotenuse is the circumcentre, i.e., $\frac{z_1+z_2}{2}$.

Hence the circumcentre of $△OPQ$ is $\frac{z_1+z_2}{2}$.