Equilibrium constants K1 and K2 for the following equilibria :

are related as :

To determine the relationship between the equilibrium constants K1 and K2, we need to analyze the equilibria and the reactions involved. So, let's consider the equilibria: $NO(g)+\frac{1}{2}{O}_2(g)\rightleftharpoons N{O}_2(g)$ and $2NO(g)+O_2(g)\rightleftharpoons 2N{O}_2(g)$. Notice that the second equilibrium is the reverse of the first equilibrium multiplied by 2. Think of it this way: the equilibrium constant for the reverse of a reaction is the reciprocal of the equilibrium constant for the forward reaction. In other words, if we multiply the first equilibrium by 2 and reverse it, we get the second equilibrium. Therefore, the correct answer is the option that involves the relationship $K_2=\frac{1}{K_1^2}$, which is option 2.