If AM of two numbers is twice of their GM, then the ratio of greatest number to smallest number is

Options

A. `7-4sqrt(3)`
B. `7+4sqrt(3)`
C. 21
D. 5

Answer

b

Explanation

Let the numbers be x and y , such that `x gt y`
Since , `(x+y)/2 = 2sqrt(xy)`
` rArr x+y = 4 sqrt(xy)`
` rArr (x+y)^(2) = 16xy" "` …(i)
Also, `(x-y)^(2) = (x+y)^(2) - 4xy`
` :. (x-y)^(2) = 16xy - 4xy = 12 xy`
` rArr x - = 2 sqrt(3xy)" "` ... (iii)
On solving Eqs. ( i ) and (ii) , we get
` x = (2+sqrt(3)) sqrt(xy)`
and ` y = (2- sqrt(3)) sqrt(xy)`
` :. "Required ratio " = x/y = ((2+sqrt(3))sqrt(xy))/((2-sqrt(3))sqrt(xy))`
` = (2+sqrt(3))^(2) = 7 + 4sqrt(3)`