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CBSE Class 11-science Answered

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Asked by sagar285.rmgm | 11 Oct, 2020, 04:11: PM
answered-by-expert Expert Answer
begin mathsize 14px style L e t space fraction numerator left parenthesis space 1 space plus space i space right parenthesis space left parenthesis space 2 space plus space 3 i space right parenthesis space left parenthesis space 3 space minus space 4 space i space right parenthesis over denominator left parenthesis space 2 space minus space 3 space i space right parenthesis space left parenthesis space 1 space minus space i space right parenthesis space left parenthesis space 3 space plus space 4 space i space right parenthesis end fraction space equals space a space plus space i space b end style  .....................(1)
Let us calculate the numerator
 
begin mathsize 14px style left parenthesis space 1 space plus space i space right parenthesis space left parenthesis space 2 space plus space 3 space i space right parenthesis space equals space 2 space plus space 5 i space plus space 3 space i squared space equals space minus 1 space plus space 5 space i end style
begin mathsize 14px style left parenthesis space minus 1 space plus space 5 space i space right parenthesis space left parenthesis space 3 space minus space 4 space i space right parenthesis space equals space 15 space i space minus space 3 space minus space 20 space i to the power of 2 to the power of space end exponent plus space 4 space i space equals space 17 space plus space 19 space i end style
Let us calculate the denominator
 
begin mathsize 14px style left parenthesis space 2 space minus space 3 space i space right parenthesis left parenthesis space 1 space minus space i space right parenthesis space equals space 2 space minus space 3 i space minus 2 i space plus space 3 i to the power of 2 to the power of space end exponent space equals space minus 1 space minus space 5 space i end style
begin mathsize 14px style negative left parenthesis space 1 space plus space 5 space i space right parenthesis left parenthesis space 3 space plus space 4 i space right parenthesis space equals space minus space left square bracket space 3 space plus space 15 i space plus space 4 i space plus space 20 space i to the power of 2 to the power of space end exponent right square bracket space equals space 17 space minus space 19 space i end style
 
hence eqn.(1) becomes
 
begin mathsize 14px style fraction numerator 17 space plus space 19 space i over denominator 17 space minus space 19 space i end fraction space equals fraction numerator 17 space plus space 19 space i over denominator 17 space minus space 19 space i end fraction cross times fraction numerator 17 space plus space 19 space i over denominator 17 space plus space 19 space i end fraction space equals space fraction numerator left parenthesis space 17 space plus space 19 space i space right parenthesis squared over denominator 17 squared plus 19 squared end fraction space equals space fraction numerator negative 72 space plus space 646 space i over denominator 650 end fraction space equals space a space plus space b space i end style
 
hence a = -72/650   and  b = ( 646 / 650 )
 
we get from above values of a and b,  a2 + b2 = 1
Answered by Thiyagarajan K | 14 Oct, 2020, 10:53: PM

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