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)):(m-sqrt(m^2-n^2))`

Explanation

Given, `AM=m, GM=n`


We know, `AM=(a+b)/2`, `GM=sqrt(ab)`


Dividing AM and GM


`(AM)/(GM)=m/n`


`((a+b)/2)/(sqrt(ab))=m/n`


Applying componendo and dividendo


`(a+b+2sqrt(ab))/(a+b-2sqrt(ab))=(m+n)/(m-n)`


`((sqrt(a)+sqrt(b))^2)/((sqrt(a)-sqrt(b))^2)=(m+n)/(m-n)`


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