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.....(ii) Now, ‌‌(a−b)=√(a+b)2−4ab =√(40)2−4×256 =√1600−1024 =√576 ⇒‌‌a−b=24.......(jii) On solving Eqs. (ii) and (iii), we get a=32 and b=8 a:b=32:8 =4:1