Question

If $z_{1}, z_{2}, z_{3}$ are three points lying on the circle $|z|=2$ then the minimum value of $\left|z_{1}+z_{2}\right|^{2}+\left|z_{2}+z_{3}\right|^{2}+$ $\left|z_{3}+z_{1}\right|^{2}$ is equal to

• A. 6
• B. 12
• C. 15
• D. 24

Answer 1

Answer: (B)

$\left|z_{1}+z_{2}\right|^{2}+\left|z_{2}+z_{3}\right|^{2}+\left|z_{3}+z_{1}\right|^{2}$
$=2\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)+\left(z_{1} \bar{z}_{2}+\bar{z}_{1} z_{2}\right.\left.+z_{2} \bar{z}_{3}+\bar{z}_{2} z_{3}+z_{3} \bar{z}_{1}+z_{1} \bar{z}_{3}\right)$
$=24+\left(z_{1} \bar{z}_{2}+\bar{z}_{1} z_{2}+z_{2} \bar{z}_{3}+\bar{z}_{2} z_{3}+z_{3} \bar{z}_{1}+z_{3} \bar{z}_{1}\right)\,\,\,$(1)
Also,
$\left|z_{1}+z_{2}+z_{3}\right|^{2} \geq 0 $
$\Rightarrow z_{1} \bar{z}_{2}+\bar{z}_{1} z_{2}+z_{2} \bar{z}_{3}+\bar{z}_{2} z_{3}+z_{3} \bar{z}_{1}+\bar{z}_{3} z_{1} \geq-12$
$\therefore \left|z_{1}+z_{2}\right|^{2}+\left|z_{2}+z_{3}\right|^{2}+\left|z_{3}+z_{1}\right|^{2} \geq 12$