If z 1- z 2- 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- isA- Less than 1B- Greater than 3C- Equal to 3D- Equal to 1

1=|1/z_1+1/z_2+1/z_3|= | z_1 z̅_̅1̅/z_1+z_2z̅_̅2̅/z_2+z_3z̅_̅3̅/z_3| (∵ |z_1|^2=1=z_1z̅_1, etc) =|z̅_1+z̅_2+z̅_3|=|z_1+z_2+z_3|=|z_1+z_2+z_3| (∵ |z̅_1|=|z_1| ...

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Byju's Answer

Standard XII

Mathematics

Geometrical Representation of Algebra of Complex Numbers

If z 1, z 2, ...

Question

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

Equal to 1

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Less than 1

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Greater than 3

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Equal to 3

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Solution

The correct option is A Equal to 1
$1 = ∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣ = ∣ ∣ \frac{z_{1}_{1}}{z_{1}} + \frac{z_{2}_{2}}{z_{2}} + \frac{z_{3}_{3}}{z_{3}} ∣ ∣ (∵ | z_{1} |^{2} = 1 = z_{1} {¯ z}_{1}, e t c) = | {¯ z}_{1} + {¯ z}_{2} + {¯ z}_{3} | = | ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ z_{1} + z_{2} + z_{3} | = | z_{1} + z_{2} + z_{3} | (∵ | {¯ z}_{1} | = | z_{1} |)$

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Similar questions

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

Q. If

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are imaginary 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} |

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

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

Q. If

z_{1}, z_{2}

and

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} |

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

The correct option is A Equal to 11=∣∣1z1+1z2+1z3∣∣=∣∣z1¯z1z1+z2¯z2z2+z3¯z3z3∣∣(∵|z1|2=1=z1¯z1, etc)=|¯z1+¯z2+¯z3|=|¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯z1+z2+z3|=|z1+z2+z3|(∵ |¯z1|=|z1|)

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