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Explanation:

The harmonic mean of two numbers x and y is $\frac{2xy}{(x+y)}$and the geometric mean is $\sqrt{xy}$

∴ $\frac{{\displaystyle \frac{2xy}{(x+y)}}}{\sqrt{xy}}$ = $\frac{12}{13}$

⇒ $\frac{4xy}{{(x+y)}^2}$ = $\frac{144}{169}$

Although this can be simplified to get the answer, the best way to proceed from here would be to look out for the answer choices and figure out which pair of x & y satisfies the above equation.

Option (c) satisfies the above expression

Alternately,
Please note that this sum is a classic example of how you could have gone for intelligent guess work. Since we know that the denominator of the ratio is the geometric mean, which is $\sqrt{xy}$, the two numbers should be in such a ratio that their product should be a perfect square.

The only pair from the answer choices that supports this is 4 & 9, as $\sqrt{4\times 9}$ = $\sqrt{36}$ = 6

Hence, option (c).