If z 1 , z 2 a n d z 3 are complex numbers such that | z 1 |=| z 2 |=| z 3 |= ∣ ∣ 1 z 1 + 1 z 2 + 1 z 3 ∣ ∣ = 1 , then | z 1 + z 2 + z 3 | Options: A . &nbsp Equal to 1 B . &nbsp Less than 1 C . &nbsp Greater than 3 D . &nbsp Equal to 3

Answer: Option A : A 1= ∣ ∣ 1 z 1 + 1 z 2 + 1 z 3 ∣ ∣ = ∣ ∣ ∣ z 1 ∗ ¯ z 1 z 1 + z 2 ∗ ¯ z 2 z 2 + z 3 ∗ ¯ z 3 z 3 ∣ ∣ ∣ (hence, | z 1 | 2 =1= z 1 ¯ ¯¯¯ ¯ z 1 , etc) ⇒ | z 1 + z 2 + z 3 |=| ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ z 1 + z 2 + z 3 |= | z 1 + z 2 + z 3 | (hence,| ¯ ¯¯¯ ¯ z 1 |=| z 1 |)

If z1,z2 and z3 are complex numbers such that |z1|=|z2|=|z3|=Ȣ

Discussion Forum : Complex Numbers

Question -

If $z_{1}, z_{2} a n d z_{3}$ 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 | $z_{1}$ + $z_{2}$ + $z_{3}$ |

Options:

A . Equal to 1

B . Less than 1

C . Greater than 3

D . Equal to 3

Answer: Option A
:
A

1= $∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣$ = $∣ ∣ ∣ \frac{z_{1} * {¯ z}_{1}}{z_{1}} + \frac{z_{2} * {¯ z}_{2}}{z_{2}} + \frac{z_{3} * {¯ z}_{3}}{z_{3}} ∣ ∣ ∣$

(hence, ${| z_{1} |}^{2}$ =1= $z_{1}$ $_{1}$ , etc)

$\Rightarrow$ | $z_{1}$ + $z_{2}$ + $z_{3}$ |=| $¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ z_{1} + z_{2} + z_{3}$ |= | $z_{1}$ + $z_{2}$ + $z_{3}$ |

(hence,| $_{1}$ |=| $z_{1}$ |)

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