Question

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

Answer 1

Answers : (1)

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

. . . . . . | 1/z₁ + 1/z₂ + 1/z₃ | = 1 ............ (2) 
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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) ]'. 
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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) 
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∴ | z₁ + z₂ + z₃ | = 1 

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Answer 2

Step-by-step explanation:

we have

|z1| = |z2|=|z3|= |1/z1 +1/z2+1/z3|= 1

NOW

|1/z1 + 1/z2 + 1/z3| = 1

that replies |z1+ z2 + z3 /z1z2z3| = 1

that replies |z1+z2+z3|/|z1z2z3|=1. {since |z1/z2| = |z1|/|z2|

that replies |z1+z2+z3| = |z1z2z3|

that replies |z1+z2+z3|= |z1|.|z2|.|z3|{since |z1.z2.z3| = |z1|.|z2|.|z3|

that replies |z1+z2+ z3| = 1.1.1

that replies |z1+z2+z3|= 1.Ans.....

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