If the harmonic mean between two positive numbers is to their geometric mean as 12:13

If the harmonic mean between two positive numbers is to their geometric mean as 12 : 13; then the numbers could be in the ratio

  1. 12 : 13
  2. 4 : 9
  3. 2 : 3
  4. 1/12 : 1/13

Answer

Let a and b be two numbers.

Harmonic Mean, HM = 2ab(a+b)

geometric Mean, GM = √(ab)

2ab(a+b) : √(ab) = 12:13

(a+b)/√(ab) = 13/6

√(a/b) +√(b/a) = 13/6

Let √(a/b) = x

x + 1/x = 13/6

6x2 - 13x + 6 = 0

x = 2/3 or 3/2

a/b = x2

a/b = 4/9 

The correct option is B.