MathNet 2013 Question 24

Arpon chooses a positive real number $k$. For each positive integer $n$, he places a marker at the point $(n, n k)$ in the $(x, y)$ plane. Suppose that two markers whose $x$ coordinates differ by $4$ have distance $31$. What is the distance between the markers at $(7,7 k)$ and $(19,19 k)$?

Supplemental illustration (AI-generated):