Let z₁, z₂, z₃ be three complex number such that |z₁ |= |z₂ |= |z₃ |=1 and z₁^2 z₂ z₃ + z₂^2 z₁ z₃ + z₃^2 z₁ z₂ +1=0 then |z₁+z₂+z₃ | can take the value equal to

Step-by-step solution

Given |z₁ |= |z₂ |= |z₃ |=1 z₁= 1 z ₁ etc. also z₁^2 z₂ z₃ + z₂^2 z₁ z₃ + z₃^2 z₁ z₂ +1=0 aligned & (z₁ )^3+ (z₂ )^3+ (z₃ )^3+z₁ z₂ z₃=0 & (z₁ )^3+ (z₂ )^3+ (z₃ )^3-3 z₁ z₂ z₃=-4 z₁ z₂ z₃ aligned aligned & (z₁+z₂+z₃ ) [ (z₁ )^2+ (z₂ )^2+ (z₃ )^2- z₁ z₂ ]=-4 z₁ z₂ z₃ & z₁ [ ( z₁ )^2-3 z₁ z₂ ]=-4 z₁ z₂ z₃ aligned aligned & let z₁+z₂+z₃=z & z ₁+ z ₂+ z ₃= z aligned aligned & z [ z ^2-3 z ₁ z ₂ ]=-4 z ₁ z ₂ z ₃ & z ^3=3 z z ₁ z ₂-4 z ₁ z ₂ z ₃ aligned z^3=z₁ z₂ z₃ [3 z ( 1 z₁ + 1 z₂ + 1 z₃ )-4 ]=z₁ z₂ z₃ [3 z ( z ₁+ z