if the sum of two no.s is equal to b times there geometric means show that the show that the no. are in the ratio 3+2root2 :3_2root2

Let the two numbers be a and b.

Then ,

a+b = 6 x root ab

a+b = 3 x 2 root ab

a+b / 2 rootab = 3

By componendo and dividendo,

a+b+2 rootab/ a+b - 2root ab = 3+1/3-1

(root a+root b)2 / (root a - root b)2 = 2

root a+ root b / root a - root b = root 2

root a+ root b = root 2 x root a - root2 x root b

root b (1+root2) = root a (root2 - 1)

(1+ root2) / (root2 -1) = root a / root b

(root 2+1)2 / (root2 -1)2 = (root a)2 / (root b)2

3+ 2root 2 / 3- root2 = a/b