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If ( 1 + I ) Z = ( 1 − I ) ¯ Z ,Then Show that Z = − I ¯ Z . - Mathematics

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Question

If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].

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Solution

\[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\]

\[ \Rightarrow \frac{z}{\bar{z}} = \frac{1 - i}{1 + i}\]

\[ \Rightarrow \frac{z}{\bar{z}} = \frac{1 - i}{1 + i} \times \frac{1 - i}{1 - i}\]

\[ \Rightarrow \frac{z}{\bar{z}} = \frac{1 + i^2 - 2i}{1 - i^2}\]

\[ \Rightarrow \frac{z}{\bar{z}} = \frac{1 - 1 - 2i}{1 + 1} [ \because i^2 = - 1]\]

\[ \Rightarrow \frac{z}{\bar{z}} = \frac{- 2i}{2}\]

\[ \Rightarrow \frac{z}{\bar{z}} = - i\]

\[ \Rightarrow z = - i \bar{z}\]

Hence,  

\[z = - i \bar{z}\].

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Chapter 13: Complex Numbers - Exercise 13.2 [Page 33]

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RD Sharma Mathematics [English] Class 11
Chapter 13 Complex Numbers
Exercise 13.2 | Q 18 | Page 33

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