The sum of two numbers is 6 times their geometric mean. Show that the numbers are in the ratio 3 + 2?2 : 3 - 2?2 .

 

let the two numbers be a,b

then a+b=6[(ab)^1/2].....

next    a+b/2[(ab)^1/2]=3

now apply componendo and dividendo......

a+b+2[(ab)^1/2] / a+b-2[(ab)^1/2]=3+1/3-1

(roota+rootb)  /  (roota-rootb)=root2

now again applying componendo and dividendo...

roota / rootb=root2+1/ root2-1

now squaring both sides....

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