Let us consider L(M) first :
So going by the automata from start state to final state , we get : a*b*a*
Now we consider L(N) :
Collapsing to single state , we get two incoming loops on q0 ,
i.e. a and bb* So we can write the regular expression as : (a + bb*)*
Now setting b* to ϵ , we get : (a + b)*
Now obviously L(M) is the subset of L(N).
Hence L(M) ∩ L(N) = a*b*a*
Hence A) option should be correct .