Question

If $z_1,z_2,z_3$ are three points lying on the circle $|z|=2$ then the minimum value of ${|z_1+z_2|}^2+{|z_2+z_3|}^2+{|z_3+z_1|}^2$ is equal to

• A. $6$
• B. $12$
• C. $15$
• D. $24$

Answer 1

The correct option is B $12$

${|z_1+z_2|}^2+{|z_2+z_3|}^2+{|z_3+z_1|}^2$

$=2({\left|z_1\right|}^2+{\left|z_2\right|}^2+{\left|z_3\right|}^2)+(z_1\overline{z_2}+z_2\overline{z_1}+z_1\overline{z_3}+z_3\overline{z_1}+z_3\overline{z_2}+z_2\overline{z_3})$

$=24+(z_1\overline{z_2}+z_2\overline{z_1}+z_1\overline{z_3}+z_3\overline{z_1}+z_3\overline{z_2}+z_2\overline{z_3})$............1

Also we have, ${|z_1+z_2+z_3|}^2\ge 0$

$\Rightarrow (z_1\overline{z_2}+z_2\overline{z_1}+z_1\overline{z_3}+z_3\overline{z_1}+z_3\overline{z_2}+z_2\overline{z_3})\ge -12$

Thus ${|z_1+z_2|}^2+{|z_2+z_3|}^2+{|z_3+z_1|}^2\ge 12$