The sum of two numbers is 6 times their geometric means, show that numbers are in the ratio `(3+2sqrt(2)):(3-2sqrt(2))`.

Explanation

Let the two numbers be a and b.
`G.M.=sqrt ab`
According to question ,
`a+b=6sqrtab `----------(1)
`(a+b)^2=36ab `
Also,
`(a-b)^2=(a+b)^2-4ab
implies (a-b)^2=36ab-4ab`
`(a-b)^2=32ab `
`(a-b)=sqrt32 sqrt ab `
`(a-b)=4sqrt 2 sqrtab `---------(2)
Adding (1) and (2)
`2a=(6+4sqrt2) sqrtab `
`implies a=(3+2sqrt2)sqrtab`
Substituting the value of a in (1)
`b=6sqrt ab -(3+2sqrt 2)sqrtab `
`b=6sqrtab -(3+2sqrt2)ab`
`b=(3-2sqrt2)ab `
`a/b=((3+2sqrt2)sqrtab)/((3-2sqrt2)sqrtab)`
`a/b=(3+2sqrt2)/(3-2sqrt2)`