SATHEE: Complex Numbers 2 Question 6

Complex Numbers 2 Question 6

6. A complex number $z$ is said to be unimodular, if $| z | \neq 1$ . If $z_{1}$ and $z_{2}$ are complex numbers such that $\frac{z_{1} - 2 z_{2}}{2 - z_{1} {\bar{z}}_{2}}$ is unimodular and $z_{2}$ is not unimodular.

Then, the point $z_{1}$ lies on a

(2015 Main)

(a) straight line parallel to $X$ -axis

(b) straight line parallel to $Y$ -axis

(d) circle of radius $\sqrt{2}$

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Answer:

Correct Answer: 6. (c)

Solution:

PLAN If $z$ is unimodular, then $| z | = 1$ . Also, use property of modulus i.e. $z \bar{z} = {z |}^{2}$

Given, $z_{2}$ is not unimodular i.e. $| z_{2} | \neq 1$

and $\frac{z_{1} - 2 z_{2}}{2 - z_{1} {\bar{z}}_{2}}$ is unimodular.

$\begin{aligned} \Rightarrow | \frac{z_{1} - 2 z_{2}}{2 - z_{1} {\bar{z}}_{2}} | = 1 \Rightarrow {| z_{1} - 2 z_{2} |}^{2} = 2 - {z_{1} {\bar{z}}_{2} |}^{2} \\ \Rightarrow (z_{1} - 2 z_{2}) ({\bar{z}}_{1} - 2 {\bar{z}}_{2}) = (2 - z_{1} {\bar{z}}_{2}) (2 - {\bar{z}}_{1} z_{2}) [z \bar{z} = | z |^{2}] \\ \Rightarrow {| z_{1} |}^{2} + 4 {| z_{2} |}^{2} - 2 {\bar{z}}_{1} z_{2} - 2 z_{1} {\bar{z}}_{2} \\ = 4 + {| z_{1} |}^{2} {| z_{2} |}^{2} - 2 {\bar{z}}_{1} z_{2} - 2 z_{1} {\bar{z}}_{2} \Rightarrow ({| z_{2} |}^{2} - 1) ({| z_{1} |}^{2} - 4) = 0 \\ ∵ | z_{2} | \neq 1 \\ ∴ z_{1} ∣= 2 \\ Let z_{1} = x + i y \Rightarrow x^{2} + y^{2} = (2)^{2} \end{aligned}$

$∴$ Point $z_{1}$ lies on a circle of radius 2 .

Complex Numbers 2 Question 6

6. A complex number $z$ is said to be unimodular, if $| z | \neq 1$ . If $z_{1}$ and $z_{2}$ are complex numbers such that $\frac{z_{1} - 2 z_{2}}{2 - z_{1} {\bar{z}}_{2}}$ is unimodular and $z_{2}$ is not unimodular.

Answer:

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Complex Numbers 2 Question 6

6. A complex number z is said to be unimodular, if |z|≠1. If z1 and z2 are complex numbers such that z1−2z22−z1z¯2 is unimodular and z2 is not unimodular.

Answer:

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6. A complex number $z$ is said to be unimodular, if $| z | \neq 1$ . If $z_{1}$ and $z_{2}$ are complex numbers such that $\frac{z_{1} - 2 z_{2}}{2 - z_{1} {\bar{z}}_{2}}$ is unimodular and $z_{2}$ is not unimodular.