Question

If the A.M. of two positive numbers `aa n db(a > b)` is twice their geometric mean. Prove that : `a : b=(2+sqrt(3)):(2-sqrt(3))dot`

Answer 1

`AM=2(GM) hArr 1/2 (a+b)=2 sqrt(ab) hArr (a+b)/(2sqrt(ab))=2/1`
`hArr (a+b+2sqrt(ab))/(a+b-2sqrt(ab))=(2+1)/(2-1) hArr ((sqrt(a)+sqrt(b))^(2))/((sqrt(a)-sqrt(b))^(2))=((sqrt(3))^(2))/((1)^(2))`
`hArr (sqrt(a)+sqrt(b))/(sqrt(a)-sqrt(b))=sqrt(3)/1 hArr ((sqrt(a)+sqrt(b))+(sqrt(a)-sqrt(b)))/((sqrt(a)+sqrt(b))-(sqrt(a)-sqrt(b)))=(sqrt(3)+1)/(sqrt(3)-1)`
`hArr sqrt(a)/sqrt(b)=(sqrt(3)+1)/(sqrt(3)-1) hArr a/b =((sqrt(3)+1)^(2))/((sqrt(3)-1)^(2))=(2+sqrt(3))/(2-sqrt(3))`.