Let z1 and z2 be two complex numbers such that ∣∣∣z1−2z22−z1¯¯¯¯¯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 A2 ∣∣∣z1−2z22−z1¯¯¯¯¯z2∣∣∣=1⇒|z1−2z2|=|2−z1¯¯¯¯¯z2| Squaring on both sides |z1−2z2|2=|2−z1¯¯¯¯¯z2|2 ⇒(z1−2z2)(¯¯¯¯¯z1−2¯¯¯¯¯z2)=(2−z1¯¯¯¯¯z2)(2−¯¯¯¯¯z1z2) ⇒z1¯¯¯¯¯z1−2z1¯¯¯¯¯z2−2¯¯¯¯¯z1z2+4z2¯¯¯¯¯z2 =4−2z1¯¯¯¯¯z2−2¯¯¯¯¯z1z2+z1¯¯¯¯¯z1z2¯¯¯¯¯z2⇒|z1|2−|z1|2|z2|2+4|z2|2−4=0⇒(|z1|2−4)(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