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Question
If z1, z2, z3 are complex numbers such that `|z_1| = |z_2| = |z_3| = |1/z_1 + 1/z_2 + 1/z_3|` = 1, then find the value of |z1 + z2 + z3|.
Solution
`|z_1| = |z_2| = |z_3|` = 1
⇒ `|z_1|^2 = |z_2|^2 = |z_3|^2` = 1
⇒ `z_1 barz_1 = z_2 barz_2 = z_3 barz_3` = 1
⇒ `barz_1 = 1/barz_1, barz_2 = 1/barz_2, barz_3 = 1/z_3`
Given that `|1/z_1 + 1/z_2 + 1/z_3|` = 1
⇒ `|barz_1 + barz_2 + barz_3|` = 1, i.e., `|bar(z_1 + z_2 + z_3)|` = 1
⇒ |z1 + z2 + z3| = 1
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