Let Z1 and Z2 be two complex numbers such that Z1-2Z2-2-Z1Z2 is unimodular and Z2 is not unimodular- find the modulus of Z1-

Modulus of Z_1.A complex number Z is unimodular, if |Z|=1.Given that |Z_1 2Z_2/2 Z_1Z_2|=1 and |Z_2| 1|Z_1 2Z_2|^2=|2 Z_1Z_2|^2Use modulus property |Z|^2=ZZ to ...

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Byju's Answer

Standard XII

Mathematics

Properties of Modulus

Let Z1 and Z2...

Question

Let $Z_{1}$ and $Z_{2}$ be two complex numbers such that $\frac{Z_{1} - 2 Z_{2}}{2 - Z_{1} {\bar{Z}}_{2}}$ is unimodular and $Z_{2}$ is not unimodular, find the modulus of $Z_{1}$ .

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Solution

Modulus of $Z_{1}$ .
A complex number $Z$ is unimodular, if $|Z| = 1$ .
Given that $|\frac{Z_{1} - 2 Z_{2}}{2 - Z_{1} {\bar{Z}}_{2}}| = 1$ and $|Z_{2}| \neq 1$
${|Z_{1} - 2 Z_{2}|}^{2} = {|2 - Z_{1} \bar{Z_{2}}|}^{2}$
Use modulus property ${|Z|}^{2} = Z \bar{Z}$ to determine modulus of $Z_{1}$ :

$\begin{array}{rcl} (Z_{1} - 2 Z_{2}) (\bar{Z_{1}} - 2 \bar{Z_{2}}) & = & (2 - Z_{1} \bar{Z_{2}}) (2 - \bar{Z_{1}} Z_{2}) \\ \Rightarrow & Z_{1} \bar{Z_{1}} - 2 Z_{1} \bar{Z_{2}} - 2 \bar{Z_{1}} Z_{2} + 4 Z_{2} \bar{Z_{2}} = 4 - 2 Z_{1} \bar{Z_{2}} - 2 \bar{Z_{1}} Z_{2} + Z_{1} \bar{Z_{1}} Z_{2} \bar{Z_{2}} \\ \Rightarrow & {|Z_{1}|}^{2} + 4 {|Z_{2}|}^{2} = 4 + {|Z_{1}|}^{2} {|Z_{2}|}^{2} \\ \Rightarrow & {|Z_{1}|}^{2} [1 - {|Z_{2}|}^{2}] = 4 [1 - {|Z_{2}|}^{2}] \end{array} \Rightarrow {\begin{array}{rcl} |Z_{1}| \end{array}}^{2} \begin{array}{rcl} = \end{array} \begin{array}{rcl} 4 \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} [∵ |Z_{2}| \neq 1] \end{array} \Rightarrow \begin{array}{rcl} |Z_{1}| \end{array} \begin{array}{rcl} = \end{array} 2 \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \end{array}$
Therefore, $| Z_{1} | = 2$
Hence, modulus of $Z_{1}$ is $2$ .

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Let $Z_{1}$ and $Z_{2}$ be two complex numbers such that $\frac{Z_{1} - 2 Z_{2}}{2 - Z_{1} {\bar{Z}}_{2}}$ is unimodular and $Z_{2}$ is not unimodular, find the modulus of $Z_{1}$ .