Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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If ( a 2 + 1 ) 2 2 a − I = X + I Y Find the Value of X 2 + Y 2 . - Mathematics

If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of  \[x^2 + y^2\].

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Solution

\[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy . . . . (1)\]

\[ \Rightarrow \left[ \bar{\frac{\left( a^2 + 1 \right)^2}{2a - i}} \right] = \bar{{x + iy}}\]

\[ \Rightarrow \frac{\left( a^2 + 1 \right)^2}{2a + i} = x - iy . . . . (2)\]

\[\text { On multiplying (1) and (2), we get }\]

\[\frac{\left( a^2 + 1 \right)^2}{2a - i} \times \frac{\left( a^2 + 1 \right)^2}{2a + i} = \left( x + iy \right)\left( x - iy \right)\]

\[ \Rightarrow \frac{\left( a^2 + 1 \right)^4}{\left( 2a \right)^2 - i^2} = x^2 - i^2 y^2 \]

\[ \Rightarrow \frac{\left( a^2 + 1 \right)^4}{\left( 2a \right)^2 + 1} = x^2 + y^2\]

Hence, 

\[x^2 + y^2 = \frac{\left( a^2 + 1 \right)^4}{4 a^2 + 1}\].

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Q 14 | Page 63
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