If z1, z2, z3 are complex numbers such that |z1|=|z2|=|z3|= | (1/z1) + (1/z2) + (1/z3) | =1 then |z1+z2+z3|=?? How to solve these types of sums??

Question

Arun · Answer

Given : | z₁ | = | z₂ | = | z₃ | = 1 ............. (1)

. . . . . . | 1/z₁ + 1/z₂ + 1/z₃ | = 1 ............ (2)
__________________________

To Find : | z₁ + z₂ + z₃ |.
__________________________

We note that

(i) | z | = | z' |, ... z' is conjugate of z

(ii) z·z' = | z |² = | z' |²

(iii) Σ(z') = [ Σ(z) ]'.
__________________________

If | z | = 1, then : z·z' = 1, i.e., (1/z) = z'...... (3)

∴ | z₁ + z₂ + z₃ |

= | ( z₁ + z₂ + z₃ )' |

= | (z₁)' + (z₂)' + (z₃)' |

= | 1/z₁ + 1/z₂ + 1/z₃ | ........ from (3)

= 1 ...................................... from (2)
__________________

∴ | z₁ + z₂ + z₃ | = 1

ankit singh · Answer

Given : | z₁ | = | z₂ | = | z₃ | = 1 ............. (1)

. . . . . . | 1/z₁ + 1/z₂ + 1/z₃ | = 1 ............ (2)
__________________________

To Find : | z₁ + z₂ + z₃ |.
__________________________

We note that

(i) | z | = | z' |, ... z' is conjugate of z

(ii) z·z' = | z |² = | z' |²

(iii) Σ(z') = [ Σ(z) ]'.
__________________________

If | z | = 1, then : z·z' = 1, i.e., (1/z) = z'...... (3)

∴ | z₁ + z₂ + z₃ |

= | ( z₁ + z₂ + z₃ )' |

= | (z₁)' + (z₂)' + (z₃)' |

= | 1/z₁ + 1/z₂ + 1/z₃ | ........ from (3)

= 1 ...................................... from (2)
__________________

∴ | z₁ + z₂ + z₃ | = 1

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If z1, z2, z3 are complex numbers such that |z1|=|z2|=|z3|= | (1/z1) + (1/z2) + (1/z3) | =1 then |z1+z2+z3|=?? How to solve these types of sums??

If z1, z2, z3 are complex numbers such that |z1|=|z2|=|z3|= | (1/z1) + (1/z2) + (1/z3) | =1 then |z1+z2+z3|=?? How to solve these types of sums??

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If z1, z2, z3 are complex numbers such that |z1|=|z2|=|z3|= | (1/z1) + (1/z2) + (1/z3) | =1 then |z1+z2+z3|=?? How to solve these types of sums??

If z1, z2, z3 are complex numbers such that |z1|=|z2|=|z3|= | (1/z1) + (1/z2) + (1/z3) | =1 then |z1+z2+z3|=?? How to solve these types of sums??

2 Answers

Think You Can Provide A Better Answer ?

Other Related Questions on Algebra

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Answer and earn gift vouchers

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