# $\{<M>: M \text{ is a finite automata and L(M) contains a word of form } a^ib^j\}$ is decidable?

How can we show that the language $$K = \{: M \text{ is a finite automata and L(M) contains a word of form } a^ib^j\}$$ is decidable?

Let $$\hat M$$ be the DFA for the language $$L(a^*b^*)$$ (aka, words of the form $$a^ib^j$$).
Construct the intersection DFA between $$M$$ and $$\hat M$$, and call it $$M'$$. Now, check if $$L(M')=\emptyset$$. If it does, reject. Otherwise, accept.