# How to find the minimum value sum of three complex numbers given constrains?

Question) If $|z_1 -1|<1$, $|z_2 -2|<2$, $|z_3 -3|<3$ then $|z_1+z_2+z_3|$ is:

(a) is less than 6
(b) is more than 3
(c) is less than 12
(d) lies between 6 and 12

My Attempt:

(1) Using polygon inequality $|z_1+z_2+z_3|<|z_1|+|z_2|+|z_3|$. I got maximum value =12.

(2)For finding minimum value i plotted three circles with centres at (1,0), (2,0) and (3,0) all passing through origin. Then i considered $z_1,z_2,z_3$ to be lying inside the smallest circle and then arrived at the conclusion that $|z_1+z_2+z_3|\geq0$.

Hence I'm getting option (c) as answer but answer key says option (b) to be the correct answer. Is my way of analysis via graph wrong?