Solution

Given paths traced by the wheel of two trains are
$$x + 2y - 4 = 0$$ ...(i)
$$2x + 4y - 12 = 0$$
$$\Rightarrow x +2 y - 6 = 0$$ ...(ii)
As we can see that $$a_1 = 1, b_1 = 2$$ and $$c_1 = -4$$
and $$a_2 = 1, b_2 = 2$$ and $$c_2 = -6$$
$$\dfrac{a_1}{a_2} = \dfrac{1}{1} = 1, \dfrac{b_1}{b_2} = \dfrac{2}{2} = 1$$ and $$\dfrac{c_1}{c_2} = \dfrac{-4}{-6} = \dfrac{2}{3}$$
$$\Rightarrow \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$$
$$\therefore$$ Given pair of equations has no solution.
Hence, two paths will not cross each other.