To solve this problem, we need to consider the condition that the difference in lengths $(l_2 - l_1)$ is maintained the same at all temperatures.$\\$The change in length of a rod due to temperature change is given by:$\\ \Delta l = \alpha \cdot l \cdot \Delta T \\$For the brass rod, the change in length is:$\\ \Delta l_1 = \alpha_1 \cdot l_1 \cdot \Delta T \\$For the steel rod, the change in length is:$\\ \Delta l_2 = \alpha_2 \cdot l_2 \cdot \Delta T \\$Since the difference in lengths $(l_2 - l_1)$ remains the same at all temperatures, we have:$\\ \Delta l_2 - \Delta l_1 = 0 \\$Substitute the expressions for $\Delta l_1$ and $\Delta l_2: \\ \alpha_2 \cdot l_2 \cdot \Delta T - \alpha_1 \cdot l_1 \cdot \Delta T = 0 \\$Factor out $\Delta T: \\ (\alpha_2 \cdot l_2 - \alpha_1 \cdot l_1) \cdot \Delta T = 0 \\$Since $\Delta T \neq 0$ (temperature change is non-zero), we must have:$\\ \alpha_2 \cdot l_2 = \alpha_1 \cdot l_1 \\$Therefore, the correct relation is:$\\ \alpha_1 \cdot l_1 = \alpha_2 \cdot l_2 \\$This corresponds to Option 2.