z1 and z2 are two complex numbers such that (z1-2 z2/2-z1 barz2) is unimodular whereas z2 is not unimodular. Then |z1|=

Question

z1 and z2 are two complex numbers such that (z1-2 z2/2-z1 barz2) is unimodular whereas z2 is not unimodular. Then |z1|= (A) 1 (B) 2 (C) 3 (D) 4

Tardigrade Admin · Accepted Answer

Clearly |(z1-2 z2/2-z1 barz2)|=1 ⇒ ((z1-2 z2/2-z1 barz2))(( barz1-2 barz2/2- barz1 z2))=1 ⇒ z1 barz1-2 z1 barz2-2 barz1 z2+4 z2 barz2 =4-2 barz1 z2-2 z1 barz2+z1 z2 barz1 barz2 ⇒ |z1|2+4|z2|2=4+|z1|2|z2|2 ⇒ |z1|2[1-|z2|2]=4[1-|z2|2] ⇒ |z1|2=4 (∵|z2| ≠ 1) ⇒ |z1|=2

Q. z1​ and z2​ are two complex numbers such that 2−z1​zˉ2​z1​−2z2​​ is unimodular whereas z2​ is not unimodular. Then ∣z1​∣=

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Q. $z_{1}$ and $z_{2}$ are two complex numbers such that $\frac{z _{1} - 2 z _{2}}{2 - z _{1} z ˉ _{2}}$ is unimodular whereas $z_{2}$ is not unimodular. Then $∣ z_{1} ∣ =$