Question

If$z_1\ne -z_2$ and $|z_1+z_2|=|\frac{1}{z_1}+\frac{1}{z_2}|$ then

• A. at least one of $z_1,z_2$ is unimodular
• B. both$z_1,z_2$ are unimodular
• C. $z_1.z_2=1$
• D. None of these

Answer 1

The correct option is D None of these

Let
$z_1=x_1+i{y}_1$
$z_2=x_2+i{y}_2$
Now
$z_1+z_2=(x_1+x_2)+i(y_1+y_2)$
$\frac{1}{z_1}+\frac{1}{z_2}$
$=\overline{z_1}+\overline{z_2}$
$=(x_1+x_2)-i(y_1+y_2)$.
Now
$|z_1+z_2|$
$=\sqrt{(x_2+x_1{)}^2+(y_2+y_1{)}^2}$
And
$|\frac{1}{z_1}+\frac{1}{z_2}|$
$=|(x_1+x_2)-i(y_1+y_2)|$
$=\sqrt{(x_2+x_1{)}^2+(y_2+y_1{)}^2}$
Hence
$|z_1+z_2|=|\frac{1}{z_1}+\frac{1}{z_2}|$ for all values of $z_1$ and $z_2$.