We know :
a) Harmonic mean of two positive numbers a and b = 2ab/(a + b)
b) Geometric mean of two positive numbers a and b = √ab
So given , H.M. / G.M. = 12 / 13
==> [2ab / (a+b)] / √ab = 12 / 13
==> √ab / (a+b) = 6 / 13
==> ab / (a+b)2 = 36 / 169
==> a = 4k and b = 9k
Hence the ratio a : b = 4 : 9
Hence C) should be the correct option.