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Question

Let z1 and z2 be two complex numbers such that z12z22z1¯¯¯¯¯z2=1 and |z2|1. Then the value of |z1| is

A
2
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B
4
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C
1
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D
8
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Solution

The correct option is A 2
z12z22z1¯¯¯¯¯z2=1|z12z2|=|2z1¯¯¯¯¯z2|
Squaring on both sides
|z12z2|2=|2z1¯¯¯¯¯z2|2
(z12z2)(¯¯¯¯¯z12¯¯¯¯¯z2)=(2z1¯¯¯¯¯z2)(2¯¯¯¯¯z1z2)
z1¯¯¯¯¯z12z1¯¯¯¯¯z22¯¯¯¯¯z1z2+4z2¯¯¯¯¯z2
=42z1¯¯¯¯¯z22¯¯¯¯¯z1z2+z1¯¯¯¯¯z1z2¯¯¯¯¯z2|z1|2|z1|2|z2|2+4|z2|24=0(|z1|24)(1|z2|2)=0|z1|2=4 (|z2|1)

Hence, |z1|=2

Alternate solution:
Put z2=0 (or any other specific value whose modulus is not 1)
z12=1|z1|=2

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