The harmonic mean of two numbers is 4. Their arithmetic mean `A` and the geometric mean `G` satisfy the relation `2A+G^2=27.` Find two numbers.

Explanation

Let the numbers be a and b.
Given, `H=4`
`therefore " " G^(2)=AH=4A" " "…..(i)"`
and given `2A+G^(2)=27`
`implies 2A+4A=27" " [" From Eq. (i) " ]`
`therefore A=(9)/(2)`
From Eq.(i), `G^(2)=4xx(9)/(2)=18`
Now, from important theorem of GM lt brgt `a,b=Apm sqrt(A^(2)-G^(2))=(9)/(2) pm sqrt(((81)/(4)-18))`
`=(9)/(2)pm(3)/(2)=6,3 " or " 3,6`