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ICSE Class 10 Answered

The arithmetic mean between a and b is twice the geometric mean between a and b. Prove that:a/b=7+4√3 or 7-4√3.  
Asked by gsitabhanu24 | 06 Feb, 2019, 08:43: PM
answered-by-expert Expert Answer
arithmetic mean between a and b  = (a+b)/2 .
 
geometric mean between a and b = begin mathsize 12px style square root of a space b end root end style
 
Given :   begin mathsize 12px style fraction numerator a plus b over denominator 2 end fraction space equals space 2 square root of a space b end root end style
begin mathsize 12px style a space plus space b space equals space 4 square root of a space b end root space.................. left parenthesis 1 right parenthesis open parentheses a space plus space b close parentheses squared space equals space 16 space a space b space left parenthesis a space plus space b right parenthesis squared space minus space 4 space a space b space equals space 12 space a space b open parentheses a space minus space b close parentheses squared space equals plus-or-minus space 2 square root of 3 space a space b l e t space u s space c o n s i d e r space comma space space open parentheses a space minus space b close parentheses space equals space 2 square root of 3 space square root of a space b end root space................ left parenthesis 2 right parenthesis B y space s o l v i n g space e q n s. left parenthesis 2 right parenthesis space a n d space left parenthesis 1 right parenthesis comma space space w e space g e t space comma a space equals space open parentheses 2 space plus space square root of 3 close parentheses space square root of a space b end root space........... left parenthesis 3 right parenthesis space b space equals space open parentheses 2 space minus space square root of 3 close parentheses square root of a space b end root space............ left parenthesis 4 right parenthesis h e n c e space a over b space equals space fraction numerator 2 space plus space square root of 3 over denominator 2 space minus space square root of 3 end fraction space equals space fraction numerator 2 space plus space square root of 3 over denominator 2 space minus space square root of 3 end fraction cross times fraction numerator 2 space plus space square root of 3 over denominator 2 space plus space square root of 3 end fraction space equals space open parentheses 2 space plus space square root of 3 close parentheses squared space equals space 7 plus 4 square root of 3 l e t space u s space c o n s i d e r space comma space space open parentheses a space minus space b close parentheses space equals space minus space 2 square root of 3 space square root of a space b end root space................ left parenthesis 5 right parenthesis B y space s o l v i n g space e q n s. left parenthesis 5 right parenthesis space a n d space left parenthesis 1 right parenthesis comma space space w e space g e t space comma a space equals space open parentheses 2 space minus space square root of 3 close parentheses space square root of a space b end root space........... left parenthesis 6 right parenthesis space b space equals space open parentheses 2 space plus space square root of 3 close parentheses square root of a space b end root space............ left parenthesis 7 right parenthesis h e n c e space a over b space equals space fraction numerator 2 space minus space square root of 3 over denominator 2 space plus space square root of 3 end fraction space equals space fraction numerator 2 space minus space square root of 3 over denominator 2 space plus space square root of 3 end fraction cross times fraction numerator 2 space minus space square root of 3 over denominator 2 space minus space square root of 3 end fraction space equals space open parentheses 2 space minus space square root of 3 close parentheses squared space equals space 7 minus 4 square root of 3 end style
Answered by Thiyagarajan K | 06 Feb, 2019, 11:30: PM

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