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The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio (3+2√2) : (3− 2√2 ) .

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The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio (3+2√2) : (3− 2√2 ) .
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Solution:

Let the two numbers be a and b.

G.M. = square root of a b end root

According to the given condition,

a space plus space b space equals space 6 square root of a b end root space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesisrightwards double arrow open parentheses a space plus space b close parentheses squared space equals space 36 open parentheses a b close parentheses

Also,

open parentheses a minus b close parentheses squared space equals space open parentheses a plus b close parentheses squared space minus space 4 a b space equals space 36 a b space minus 4 a b space equals space 32 a brightwards double arrow a minus b space equals space square root of 32 space square root of a b end root spaceequals space 4 space square root of 2 space square root of a b end root space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis

Adding (1) and (2), we obtain

2 a space equals space open parentheses 6 space plus space 4 square root of 2 close parentheses square root of a b end rootrightwards double arrow a space equals space open parentheses 3 space plus space 2 square root of 2 close parentheses square root of a b end root

Substituting the value of a in (1), we obtain

b space equals space 6 square root of a b end root space minus space open parentheses 3 plus 2 square root of 2 close parentheses square root of a b end rootrightwards double arrow b space equals space open parentheses 3 space minus space 2 square root of 2 close parentheses space square root of a b end roota over b space equals space fraction numerator open parentheses 3 space plus 2 square root of 2 close parentheses square root of a b end root over denominator open parentheses 3 minus 2 square root of 2 close parentheses square root of a b end root end fraction space equals fraction numerator 3 space plus space 2 square root of 2 over denominator 3 minus 2 square root of 2 end fraction space

Thus, the required ratio is (3+2√2) : (3− 2√2 ) .

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