Let $z_1,z_2,z_3$ be complex number such that $|z_1|=|z_2|=|z_3|=$ I and$$\frac{z^2_1}{z_2z_3}+\frac{z^2_2}{z_3z_1}+\frac{z^2_3}{z_1z_2}=-1,$$ then value of $|z_1+z_2+z_3|$ can be:
a) 2
b) 3
c) 4
d) 1
Solution 1: Let $\omega = \frac{1+i\sqrt{3}}{2}$. Then let $z_{1} = 1, z_{2} = \omega, z_{3} = \omega^{5}$. Then it satisfies, so the answer is $2$
Solution 2: $z_1 = -1, z_2 = z_3 = 1$ also satisfies the requirement and yields $|z_1 +z_2 + z_3| = 1$
Which of the two solutions is correct?