SATHEE: Complex Numbers 2 Question 43

Complex Numbers 2 Question 43

44.

A relation $R$ on the set of complex numbers is defined by $z_{1} R z_{2}$ , if and only if $\frac{z_{1} - z_{2}}{z_{1} + z_{2}}$ is real.

Show that $R$ is an equivalence relation.

(1982, 2M)

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Solution:

Here, $z_{1} R z_{2} \Leftrightarrow \frac{z_{1} - z_{2}}{z_{1} + z_{2}}$ is real

(i) Reflexive $z_{1} R z_{1} \Leftrightarrow \frac{z_{1} - z_{1}}{z_{1} + z_{2}} = 0$ [purely real]

$∴ z_{1} R z_{1} is reflexive.$

(ii) Symmetric $z_{1} R z_{2} \Leftrightarrow \frac{z_{1} - z_{2}}{z_{1} + z_{2}}$ is real

$\begin{aligned} \Rightarrow \frac{- (z_{2} - z_{1})}{z_{1} + z_{2}} is real \Rightarrow z_{2} R z_{1} \\ ∴ z_{1} R z_{2} \Rightarrow z_{2} R z_{1} \end{aligned}$

Therefore, it is symmetric.

(iii) Transitive $z_{1} R z_{2}$

$\begin{array}{ll} \Rightarrow & \frac{z_{1} - z_{2}}{z_{1} + z_{2}} is real \\ and & z_{2} R z_{3} \\ \Rightarrow & \frac{z_{2} - z_{3}}{z_{2} + z_{3}} is real \end{array}$

Here, let $z_{1} = x_{1} + i y_{1}, z_{2} = x_{2} + i y_{2}$ and $z_{3} = x_{3} + i y_{3}$

$∴ \frac{z_{1} - z_{2}}{z_{1} + z_{2}}$ is real $\Rightarrow \frac{(x_{1} - x_{2}) + i (y_{1} - y_{2})}{(x_{1} + x_{2}) + i (y_{1} + y_{2})}$ is real

$\Rightarrow \frac{(x_{1} - x_{2}) + i (y_{1} - y_{2}) (x_{1} + x_{2}) - i (y_{1} + y_{2})}{{(x_{1} + x_{2})}^{2} + {(y_{1} + y_{2})}^{2}}$

$\Rightarrow (y_{1} - y_{2}) (x_{1} + x_{2}) - (x_{1} - x_{2}) (y_{1} + y_{2}) = 0$

$\Rightarrow 2 x_{2} y_{1} - 2 y_{2} x_{1} = 0$

$\Rightarrow \frac{x_{1}}{y_{1}} = \frac{x_{2}}{y_{2}}$

Similarly,

$\begin{matrix} z_{2} R z_{3} \\ \frac{x_{2}}{y_{2}} = \frac{x_{3}}{y_{3}} \end{matrix}$

From Eqs. (i) and (ii), we have $\frac{x_{1}}{y_{1}} = \frac{x_{3}}{y_{3}} \Rightarrow z_{1} R z_{3}$

Thus, $z_{1} R z_{2}$ and $z_{2} R z_{3} \Rightarrow z_{1} R z_{3}$ .

Hence, $R$ is an equivalence relation.

Complex Numbers 2 Question 43

44.

Solution:

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Complex Numbers 2 Question 43

44.

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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