Proving that the language Lq = {x.y : x and y are binary strings and [x.y]2 ≤ q} is not regular if q is irrational

Question

I have this assignment on finite automatas and tried to solve it but I have a question on the string s to use for the pumping lemma. I have the intuition that for q = square root of 2, I might want to use a troncature of q and then pump the y part (from the format x.y). In my case can I use for example 1.011 as s and then from this pump the part 011 to show the iregularity or am I on the wrong path? I appreciate any help I could get here!