A

To maintain the same rate of heat flow in two rods with different thermal conductivities, we apply the formula for thermal conduction given by Fourier's Law:
$$ Q = \frac{K A \Delta T}{L} $$
where Q is the rate of heat transfer, K is the thermal conductivity, A is the cross-sectional area, \Delta T is the temperature difference, and L is the length of the rod.

Given that both rods are of the same length, have the same temperature difference across their ends, and carry the same rate of heat flow, we can write:

$$ \frac{K_1 A_1}{L} = \frac{K_2 A_2}{L} $$
This simplifies to:
$$ K_1 A_1 = K_2 A_2 $$
To express the relationship of cross-sectional areas and thermal conductivities, we rewrite it as:

$$ \frac{A_1}{A_2} = \frac{K_2}{K_1} $$
Therefore, option A is correct:
$$ \frac{A_1}{A_2} = \frac{K_1}{K_2} $$
Hence, A.