Q. If ,G, and H be the A.M;G.M; and H.M. respectively of two distinct positive integers,then the equation `Ax^2-|G|x - H=0` has:-
A) both roots as fractions
B) atleast one root as a negative fraction
C) exactly one positive root
D) atleast one root as integer
Given `Ax^2-|G|x-H=0` where A,G,H are the arithmetic mean, geometric mean, and the harmonic mean of two positive integers:
I. Note that `H=G^2/A` ; we can use the quadratic formula to look at the roots:
Now A,G>0. `1+sqrt(5)>0,1-sqrt(5)=0` so there is exactly 1 positive root, and 1 negative root.
The answer is C. (Typically fractions are defined as a ratio of integers; the roots here will never be rational so A,B, and D cannot be correct.)
II. An alternative: let a,b be positive integers. Then `A=(a+b)/2,G=sqrt(ab),H=(2ab)/(a+b)` so we have:
`(a+b)/2 x^2-sqrt(ab)x-(2ab)/(a+b)=0` Use the quadratic formula:
Again, `sqrt(ab)>0,a+b>0,1+sqrt(5)>0,1-sqrt(5)<0` so we have a positive and negative root, neither of which is rational.
An example: let a=6, b=8 then A=7, `G=4sqrt(3),H=48/7`
The roots are `x=(4sqrt(3)+-4sqrt(15))/14=(4sqrt(3)(1+-sqrt(5)))/14`
Sorry,it will be `A,G and H` are the A.M. ; G.M.; and H.M.