Prove that $L = \{ xy \in \{a , b \}\textbf{*} \mid |x|_a = 2|y|_b \}$ is not regular
Here are several routes to proving this:


Intersect with $a^*b^*$ and apply an appropriate inverse homomorphism to reduce to the language $\{a^nb^n : n \ge 0 \}$, which you know isn't regular.
Use Myhill&ndash;Nerode theory: the words $\{a^n : n \geq 0\}$ are pairwise inequivalent.
Use the pumping lemma on the word $a^{2p} b^p$, where $p$ is the pumping constant.