Geometric mean of a and b=√ab ⇒√ab=16‌‌ (given) ‌ ⇒ab=256‌...‌ (i) ‌ And harmonic mean of a and b=‌
2ab
a+b
∴‌‌‌
2ab
a+b
=‌
64
5
(given) ⇒‌
2×256
a+b
‌=‌
64
5
‌‌‌‌‌ [from Eq. (i)] ‌ ⇒‌a+b‌=40 ‌‌ Now, ‌‌...‌..(ii) ‌ ‌=√(40)2−4×256 ‌=√1600−1024 ‌=√(a+b)2−4ab ⇒‌a−b‌=24 On solving Eqs. (ii) and (iii), we get ‌a=32‌ and ‌b=8 ‌∴‌‌a:b=32:8 ‌=4:1 ‌