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If A, G and H are the arithmetic mean, g...

If A, G and H are the arithmetic mean, geometric mean and harmonic mean between unequal positive integers. Then, the equation `Ax^2 -|G|x- H=0` has

A

both roots are fractioins

B

atleast one root which is negative fraction

C

exactly one positive root

D

atleast one root which is an integer

Text Solution

Verified by Experts

The correct Answer is:
B, C
Given equation is
`Ax^(2)-|G|x-H=0`………i
`:.` Discriminant `=(-|G|)^(20)-4A(-H)`
`=G^(@)+4AH`
`=G^(2)+4G^(2)[:'G^(2)=AH]`
`=5G^(2)gt0`
`:.` Roots of Eq. (i) are real and distinct.
`:'A=(a+b)/2gt0, G=sqrt(ab)gt0,H=(2ab)/(a+b)gt0`
[ `:'` a and b are two unequal positive integers]
Let `alpha` and `beta` be the roots of Eq. (i). Then
`alpha+ beta=(|G|)/Agt0`
and `alpha beta =-H/Alt0`
and `alpha-beta=(sqrt(D))/A=(Gsqrt(5))/Agt0`
`:.alpha=(|G|+Gsqrt(5))/(2A)gt0`
and `beta=(|G|-Gsqrt(5))/(2A)lt0`
Exactly one positive root and atleat one root which is negative fraction.
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