If z1, z2, z3 are any three complex numbers such that |z1|=|z2|=|z3|=|(1/z1)+(1/z2)+(1/z3)|=1, then find the value of |z1+z2+z3|.

Question

If z1, z2, z3 are any three complex numbers such that |z1|=|z2|=|z3|=|(1/z1)+(1/z2)+(1/z3)|=1, then find the value of |z1+z2+z3|. (A) 1 (B) 2 (C) 3 (D) 4

Tardigrade Admin · Accepted Answer

|z1|=|z2|=|z3|=1 ⇒ |z1|2=|z2|2=|z3|2=1 ⇒ z1 barz1 =z2 barz2=z3 barz3=1 ⇒ barz1=(1/z1), barz2=(1/z2), barz3=(1/z3) Given that, |(1/z1)+(1/z2)+(1/z3)|=1 ⇒ | barz1+ barz2+ barz3|=1 ⇒ | overlinez1+z2+z3|=1 ⇒ |z1+z2+z3|=1

Q. If $z_{1}$ , $z_{2}$ , $z_{3}$ are any three complex numbers such that $∣ z_{1} ∣ = ∣ z_{2} ∣ = ∣ z_{3} ∣ = ∣ ∣ \frac{1}{z _{1}} + \frac{1}{z _{2}} + \frac{1}{z _{3}} ∣ ∣ = 1$ , then find the value of $∣ z_{1} + z_{2} + z_{3} ∣$ .

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Q. If z1​, z2​, z3​ are any three complex numbers such that ∣z1​∣=∣z2​∣=∣z3​∣=∣∣​z1​1​+z2​1​+z3​1​∣∣​=1, then find the value of ∣z1​+z2​+z3​∣.

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Q. If $z_{1}$ , $z_{2}$ , $z_{3}$ are any three complex numbers such that $∣ z_{1} ∣ = ∣ z_{2} ∣ = ∣ z_{3} ∣ = ∣ ∣ \frac{1}{z _{1}} + \frac{1}{z _{2}} + \frac{1}{z _{3}} ∣ ∣ = 1$ , then find the value of $∣ z_{1} + z_{2} + z_{3} ∣$ .

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