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← JEE Main 2015 · 4 Apr · Shift 1
MathematicsSingle correct+4 / -1Question 62 of 90

JEE Main 2015 · 4 Apr · Shift 1, Mathematics question 62

A complex number zz is said to be unimodular if ∣z∣=1|z|=1. Suppose z1z_1 and z2z_2 are complex numbers such that z1−2z22−z1zˉ2\dfrac{z_1-2z_2}{2-z_1\bar z_2} is unimodular and z2z_2 is not unimodular. Then the point z1z_1 lies on a :

Choose one option
Answer and worked solution

Official answer: C

∣z1−2z2∣2=∣2−z1zˉ2∣2|z_1-2z_2|^2=|2-z_1\bar z_2|^2 expands to
∣z1∣2+4∣z2∣2=4+∣z1∣2∣z2∣2  ⟺  (∣z1∣2−4)(1−∣z2∣2)=0.|z_1|^2+4|z_2|^2=4+|z_1|^2|z_2|^2\iff(|z_1|^2-4)(1-|z_2|^2)=0.
Since ∣z2∣≠1|z_2|\ne1, ∣z1∣=2|z_1|=2: a circle of radius 2.

Topics:Modulus and argument
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