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CBSE Class 11-science Answered

If the Geometric Mean is 1/n times the Harmonic Mean between two positive numbers then show that these numbers are in the ratio 1+√1-n^2 : 1-√1-n^2 (ISC ML AGARWAL)
Asked by Vidushi412 | 18 Jan, 2019, 02:09: PM
answered-by-expert Expert Answer
Let a and b are two numbers.
 
Geometric mean of a and b = begin mathsize 12px style square root of a space b end root end style
Harmonic mean of a nd b = begin mathsize 12px style fraction numerator 2 over denominator begin display style 1 over a plus 1 over b end style end fraction space equals space fraction numerator 2 space a space b over denominator a plus b end fraction end style

we are given : begin mathsize 12px style square root of a space b end root space equals space 1 over n open parentheses fraction numerator 2 space a space b over denominator a space plus space b end fraction close parentheses end style ...................(1)
eqn.(1) is rewritten as begin mathsize 12px style open parentheses a plus b close parentheses squared space equals space 1 over n squared space left parenthesis space 4 space a space b space right parenthesis end style  ............................(2)
begin mathsize 12px style open parentheses a space plus space b close parentheses squared space minus space 4 space a space b space equals space left parenthesis space a space minus space b space right parenthesis squared space equals space 4 space a space b space open parentheses 1 over n squared space minus space 1 close parentheses end style.........................(3)
by dividing eqn.(2) by eqn.(3) and taking square root on both sides, we get   begin mathsize 12px style fraction numerator open parentheses a plus b close parentheses over denominator open parentheses a minus b close parentheses end fraction space equals space fraction numerator 1 over denominator square root of 1 minus n squared end root end fraction end style  ...................(4)
By componendo divideno rule applying to eqn.(4), we get begin mathsize 12px style a over b space equals space fraction numerator 1 plus square root of 1 minus n squared end root over denominator 1 minus square root of 1 minus n squared end root end fraction end style
Answered by Thiyagarajan K | 18 Jan, 2019, 08:20: PM

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