If z1, z2, z3 are three points lying on the circle |z|=2 then the minimum value of |z1+z2|2+|z2+z3|2+ |z3+z1|2 is equal to

Question

If z1, z2, z3 are three points lying on the circle |z|=2 then the minimum value of |z1+z2|2+|z2+z3|2+ |z3+z1|2 is equal to (A) 6 (B) 12 (C) 15 (D) 24

Tardigrade Admin · Accepted Answer

|z1+z2|2+|z2+z3|2+|z3+z1|2 =2(|z1|2+|z2|2+|z3|2)+(z1 barz2+ barz1 z2..+z2 barz3+ barz2 z3+z3 barz1+z1 barz3) =24+(z1 barz2+ barz1 z2+z2 barz3+ barz2 z3+z3 barz1+z3 barz1) (1) Also, |z1+z2+z3|2 ≥ 0 ⇒ z1 barz2+ barz1 z2+z2 barz3+ barz2 z3+z3 barz1+ barz3 z1 ≥-12 ∴ |z1+z2|2+|z2+z3|2+|z3+z1|2 ≥ 12

Q. If z1​,z2​,z3​ are three points lying on the circle ∣z∣=2 then the minimum value of ∣z1​+z2​∣2+∣z2​+z3​∣2+ ∣z3​+z1​∣2 is equal to

6

12

15

24

Q. 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