A

Step 1: The rate of heat flow through a rod can be expressed using Fourier's law of heat conduction: \[ Q = \frac{KA(\Delta T)}{L} \] where \( Q \) is the heat transfer per unit time, \( K \) is the thermal conductivity, \( A \) is the cross-sectional area, \( \Delta T \) is the temperature difference, and \( L \) is the length of the rod.
Step 2: Given that both rods have the same length and temperature difference at their ends, we can equate the heat conduction rates for both rods: \[ Q_1 = Q_2 \]. Thus, we have:
\[ \frac{K_1 A_1 (T_1 - T_2)}{L} = \frac{K_2 A_2 (T_1 - T_2)}{L} \]
Step 3: Simplifying, we cancel out the common factors and rearranging gives us the relationship:
\[ \frac{A_1}{A_2} = \frac{K_2}{K_1} \] (this is the first option given).
Therefore, A.