Q:

Proving non CFL with pumping lemma

I can't seem to figure out how to prove this as not a CFL:
$$\left\{x^{a}y^{b}\mid a=kb\space\text{for some positive integer k}\right\}$$
I've tried a bunch of "s"'s to pump such as $a^{2p}b^{p}$ but everything seems to be able to be pumped. Could anyone give me a hint on how to continue this problem?

A:

HINT: If $p$ is the pumping length, start with the word $w=a^{p(p+1)}b^{p+1}$. It decomposes as $w=uvxyz$, where $|vxy|\le p$, $|vy|\ge 1$, and $uv^kxy^kz\in L$ for all $k\ge 0$. Let $n=|vxy|\le p$; then $vxy=a^rb^{n-r}$ for some $r$ such that $0\le r\le n$. Note that this is enough to guarantee that $r\ne p(n-r)$, since $n\le p$. Use this to show that pumping $w$ down to $uxz$ must then give you a word not in $L$.

