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.

Options

A. 6, 3
B. 9, 5
C. 12, 7
D. 3, 1

Answer

A

Explanation

Let the two numbers be a, b.
Given, `H.M.=4implies(2ab)/(a+b)=4impliesab=2(a+b)" "...(i)`
Also, given `2A+G^(2)=27`
`implies2((a+b)/(2))+ab=27`
`impliesa+b+ab=27`
`impliesa+b+2(a+b)=27" "("from (i)")`
`implies3a+3b=27`
`impliesa+b=9" "...(ii)`
From (i), `ab=2(9)=18" "...(iii)`
Solving (ii), (iii) we get
`a=3,b=6" or "a=6,b=3`