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If the arithmetic means of two positive ...

If the arithmetic means of two positive number a and b `(a gt b )` is twice their geometric mean, then find the ratio a: b

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The correct Answer is:
`(2+ sqrt(3)):(2-sqrt(3))`
Let A be the A.M. and G be the G.M. of a and b. Then the numbers a and b are roots of the equation `x^(2)-4Gx+G^(2)=0`b `[because a=2G]`
`rArrx=(4Gpmsqrt(16G^(2)-4G^(2)))/2`
`=(4Gpm2sqrt3G)/2=(2pmsqrt3)G`
Hence, a:b=`(2+sqrt3):(2-sqrt3)`
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  • If H_(1) , H_(2) are two harmonic means between two positive numbers a and b , (a != b) , A and G are the arithmetic and geometric means between a and b , then (H_(2) + H_(1))/(H_(2) H_(1)) is

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