If z1, z2, z3 are complex numbers such that |z1|=|z2|=|z3|=|(1/z1)+(1/z2)+(1/z3)|=1, then |z1+z2+z3| is

Question

If z1, z2, z3 are complex numbers such that |z1|=|z2|=|z3|=|(1/z1)+(1/z2)+(1/z3)|=1, then |z1+z2+z3| is (A) equal to 1 (B) less than 1 (C) greater than 3 (D) equal to 3

Tardigrade Admin · Accepted Answer

|z1| =|z2|=|z3|=1 ⇒ z1 barz1 =z2 barz2=z3 barz3=1 ∴|z1+z2+z3| =|(1/ barz1)+(1/ barz2)+(1/ barz3)|=|(1/z1)+(1/z2)+(1/z3)|=1

Q. If z1​,z2​,z3​ are complex numbers such that ∣z1​∣=∣z2​∣=∣z3​∣=∣∣​z1​1​+z2​1​+z3​1​∣∣​=1, then ∣z1​+z2​+z3​∣ is

equal to 1

less than 1

greater than 3

equal to 3

∣z1​∣=∣z2​∣=∣z3​∣=1 ⇒z1​zˉ1​=z2​zˉ2​=z3​zˉ3​=1 ∴∣z1​+z2​+z3​∣=∣∣​zˉ1​1​+zˉ2​1​+zˉ3​1​∣∣​=∣∣​z1​1​+z2​1​+z3​1​∣∣​=1

Q. If $z_{1}, z_{2}, z_{3}$ are complex numbers such that $∣ z_{1} ∣ = ∣ z_{2} ∣ = ∣ z_{3} ∣ = ∣ ∣ \frac{1}{z _{1}} + \frac{1}{z _{2}} + \frac{1}{z _{3}} ∣ ∣ = 1,$ then $∣ z_{1} + z_{2} + z_{3} ∣$ is