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Properties of Conjugate of a Complex Number

If z1,z2,z3...

Question

If $z_{1}, z_{2}, z_{3}$ be the complex numbers such that
$z_{1} + z_{2} + z_{3} = 0$ and $| z_{1} | = | z_{2} | = | z_{3} | = 1$ , then $\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} = 0$

True

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False

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Solution

The correct option is A True
True.

|z|=| $¯ z$ |

$| z | = 1 \Rightarrow z \cdot ¯ z = 1$

$(\frac{1}{z})$ = $¯ z$

$z_{1} + z_{2} + z_{3} = 0$

$\Rightarrow_{1} + {¯ z}_{2} + {¯ z}_{3} = 0$

$\Rightarrow \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} = 0$

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Similar questions

z_{1}, z_{2}, z_{3}

| z_{1} | = | z_{2} | = | z_{3} | = ∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣

| z_{1} + z_{2} + z_{3} |

z_{1}, z_{2}, z_{3}

| z_{1} | = | z_{2} | = | z_{3} | = ∣ ∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣ ∣ = 1

| z_{1} + z_{2} + z_{3} |

z_{1,} z_{_{2},} z_{3}

| z_{1} | = | z_{2} | =

| z_{3} |

∣ ∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣ ∣ = 1,

| z_{1 +} z_{2} + z_{3} |

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 | $z_{1}$ + $z_{2}$ + $z_{3}$ |

z - 1, z_{2}

z_{3}

| z_{1} | = | z_{2} | = | z_{3} | = | (1 / z_{1}) + (1 / z_{2}) + (1 / z_{2}) | = 1

| z_{1} + z_{2} + z_{3} |

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