If A.M. and G.M. between two numbers is in the ratio `m : n` then prove that the numbers are in the ratio `(m+sqrt(m^2-n^2)):sqrt((m-m^2-n^2))dot`

Explanation

Let the two numbers be a and b `(agtb)`.
Also, let A and G be, respectively, the arthimatic and geometric means between a and b.
So, `a=A+sqrt(A^(2)-G^(2))`
and `b=A-sqrt(A^(2)-G^(2))`
Given that `A/G=m/n`
Let `A=mlamdaand G=nlamda`.
`thereforea/b=(A+sqrtA^(2)-G^(2))/(A-sqrt(A^(2)-G^(2)))`
`=(mlamda+sqrt(m^(2)lamda^(2)-n^(2)lamda^(2)))/(mlamda-sqrt(m^(2)lamda^(2)-n^(2)lamda^(2)))`
`=(m+sqrt(m^(2)-n^(2)))/(m-sqrt(m^(2)-n^(2)))`