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 fractions (b) atleast one root which is negative fraction (c) exactly one positive root (d) atleast one root which is an integer

Options

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

Answer

B, C

Explanation

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.