A complex number z is said to be unimodular if |z| = 1. Suppose z1 and z2 are complex numbers such that \frac{z_{1} - 2 z_{2}}{2 - z_{1} \left(\bar{z}\right)_{2}} is unimodular and z2 is not unimodular. Then the point z1 lies on a

Engineering

Mathematics

Conjugate Modulus and Argument of a Complex Number

Question

A complex number z is said to be unimodular if |z| = 1. Suppose z₁ and z₂ are complex numbers such that $\frac{z_{1} - 2 z_{2}}{2 - z_{1} {\bar{z}}_{2}}$ is unimodular and z₂ is not unimodular. Then the point z₁ lies on a

straight line parallel to y-axis

straight line parallel to x-axis

circle of radius 2

circle of radius

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⇒ |z₁ – 2z₂| = |2 – z_{1 ${\bar{z}}_{2}$}|

${| z_{1} |}^{2} - 2 z_{1} {\bar{z}}_{2} - 2 z_{2} {\bar{z}}_{1} + 4 | z_{1} |^{2} = 4 - 2 {\bar{z}}_{1} {\bar{z}}_{2} - 2 z_{1} {\bar{z}}_{2} + | z_{1} |^{2} | z_{2} |^{2}$

(|z₂|² – 1) (|z₁|² – 4) = 0 ⇒ |z₁| = 2.