Question

If $z_1,z_2,z_3$ be complex number such that
$|z_1|=|z_2|=|z_3|=|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|$ = 1 then $|z_1+z_2+z_3|$ is

• A. Greater than Equal to 1
• B. Less than 1
• C. Greater than 3
• D. Equal to 3

Answer 1

The correct option is A Greater than Equal to 1

$|z|$ is a real positive number hence

$AM\ge HM$

$\frac{|z_1|+|z_2|+|z_3|}{3}\ge \frac{3}{|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|}$

$\frac{3|z_1|}{3}(\frac{1}{|z_1|}+\frac{1}{|z_2|}+\frac{1}{|z_3|})\ge 3$

$⟹|z_1+z_2+z_3|\le |z_1|+|z_2|+|z_3|$

$⟹|z_1+z_2+z_3|\le 3$

$|(z_1+z_2)+z_3|\ge ||z_1+z_2|-z_3|$

$|z_1+z_2+z_3|\ge ||z_1+z_2|-1|$

Now, $|z_1+z_2|\ge |z_1|-|z_2|,|z_1+z_2|\ge 0$

$|z_1+z_2+z_3|\ge |0-1|$

$|z_1+z_2+z_3|\ge 1$