If the A.M is twice the G.M. of the numbers a and b, then a:b will be

Given that a and b are in GP. 

Let a = ar and b=a/r

Also, given that AM = 2 (GM)

(a+b)/2 = 2√(ab)

a+b = 4√ab …(1)

Now, substitute a and b values in (1),

ar + (a/r) = 4√[(ar)(a/r)]

a[r+(1/r)] = 4√(a)2

a [r+(1/r)] = 4a

r + (1/r) = 4

r2 – 4r +1 =0

On factorising the equation, we get

r = 2+√3 and 2-√3

Hence, a = 2+√3 and b=2-√3.

Therefore, a: b = (2+√3)/(2-√3).

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