The ratio of the A.M. and G.M. of two positive numbers a and b, is m : n. Show that a : b = `(m+sqrt(m^2-n^2)):(m-sqrt(m^2-n^2))`.

Answer

N/a

Explanation

A.M. of a and b = `(a + b)/(2)` and G.M. `= sqrt(ab)`
Given that, `((a + b)//2)/(sqrt(ab)) = (m)/(n)`
`rArr` `(a + b)/(2sqrt(ab)) = (m)/(n)`
From componendo/dividendo rule
`(a + b + 2sqrt(ab))/(a + b - 2sqrt(ab)) = (m + n)/(m - n)`
`rArr` `((sqrta + sqrtb)/(sqrta - sqrtb))^(2) = (m + n)/(m - n)`
`rArr` `(sqrta + sqrtb)/(sqrta - sqrtb) = (sqrt(m + n))/(sqrt(m - n))`
Again, from componendo/dividendo rule
`((sqrta + sqrtb) +(sqrta - sqrtb))/((sqrta + sqrtb) - (sqrta - sqrtb)) = (sqrt(m + n) + sqrt(m - n))/(sqrt(m + n) - sqrt(m - n))`
`rArr` `(2sqrt(a))/(2sqrt(b)) = (sqrt(m + n) + sqrt(m - n))/(sqrt(m + n) - sqrt(m - n))`
`rArr` `(sqrt(a))/(sqrt(b)) = (sqrt(m + n) + sqrt(m - n))/(sqrt(m + n) - sqrt(m - n))`
Squaring both sides
`(a)/(b) = (m + n + m - n + 2sqrt(m^(2) - n^(2)))/(m + n + m - n - 2sqrt(m^(2) - n^(2)))`
`= (m + sqrt(m^(2) - n^(2)))/(m - sqrt(m^(2) - n^(2)))`
`rArr` `a : b= (m + sqrt(m^(2) - n^(2))):(m - sqrt(m^(2) - n^(2)))`
Hence Proved.