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प्रश्न
If z1, z2, z3 are complex numbers such that |z1| = |z2| = |z3| = `|1/z_1 + 1/z_2 + 1/z_3|` = 1, then |z1 + z2 + z3| is ______.
पर्याय
equal to 1
less than 1
greater than 3
equal to 3
MCQ
रिकाम्या जागा भरा
उत्तर
If z1, z2, z3 are complex numbers such that |z1| = |z2| = |z3| = `|1/z_1 + 1/z_2 + 1/z_3|` = 1, then |z1 + z2 + z3| is equal to 1.
Explanation:
Given: |z1| = |z2| = |z3|
|z1|2 = 1 = `z_1barz_1`
|z2|2 = 1 = `z_2barz_2`
|z3|2 = 1 = `z_3barz_3`
⇒ `|1/z_1 + 1/z_2 + 1/z_3|` = 1
⇒ `|(z_1barz_1)/z_1 + (z_2barz_2)/z_2 + (z_3barz_3)/z_3|` = 1
⇒ `|barz_1 + barz_2 + barz_3|` = 1
⇒ `|bar(z_1 + z_2 + z_3)|` = 1
⇒ |z1 + z2 + z3| = 1
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