The value is=?

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Answer 1 · 2018-06-26T10:09:27

The Right Option is $(4) : \frac{5 \sqrt{6}}{12}$ $(4) : (5sqrt6)/12$ .

Let, $a,b in RR^+" such that, "a > b$ .

Their $"AM"=(a+b)/2," and the GM"=sqrt(ab)$ .

It is given that, $(a+b)/2=5sqrt(ab)$ .

$:. (a+b)=10sqrt(ab)$ .

$:. (a+b)^2=100ab........................(ast^1)$ .

Further, $(a-b)^2=(a+b)^2-4ab$ .

$=100ab-4ab..........[because, (ast^1)]$ .

$:. (a-b)^2=96ab..............................(ast^2)$ .

From $(ast^1) and (ast^2)$ , we have,

$(a+b)^2/(a-b)^2=(100ab)/(96ab)=25/24$ .

$:. (a+b)/(a-b)=+sqrt(5^2/(2^2*6))............[because, (a-b) gt 0]$ ,

$=5/(2sqrt6)*sqrt6/sqrt6$ .

$rArr (a+b)/(a-b)=(5sqrt6)/12$ , showing that,

The Right Option is $(4) : (5sqrt6)/12$ .