If z1 z2 z3 are complex numbers such that
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If z1,z2 and z3 are complex numbers such that |z1| = |z2| = |z3| = |1/z1 + 1/z2 + 1/z3| = 1, then |z1 + z2 + z3| is
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Answers : (1)
Given : | z₁ | = | z₂ | = | z₃ | = 1 ............. (1)
. . . . . . | 1/z₁ + 1/z₂ + 1/z₃ | = 1 ............ (2)
__________________________
To Find : | z₁ + z₂ + z₃ |.
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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
Plz mark this answer as brainlist
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