Relation Between Am, Gm And Hm

For two positive numbers the arithmetic, geometric, and harmonic means satisfy the identity GM^2 = AM × HM, a relation tested heavily at JEE Advanced. Rearranging gives HM = GM^2 / AM = 8^2 / 10 = 64/10 = 6.4. This also respects the ordering AM ≥ GM ≥ HM, since 10 ≥ 8 ≥ 6.4 holds for the distinct numbers in question. Option 8 wrongly equates HM with GM, which only happens when the numbers are equal. Option 9 averages AM and GM rather than using the multiplicative relation. Option 5 mistakenly divides AM by 2. Hence the harmonic mean is 6.4. The identity GM^2 = AM × HM means the geometric mean is itself the geometric mean of the other two means, so knowing any two of the three means immediately determines the third without ever recovering the original numbers, a shortcut JEE Advanced expects students to apply fluently. Plausibility check: solving for the numbers from AM = 10 and GM = 8 gives roots of t^2 - 20t + 64 = 0, namely 4 and 16, whose harmonic mean 2·4·16/(4+16) = 128/20 = 6.4, confirming the identity directly.