AMC 10 2017a Problem 17

Distinct points $P$, $Q$, $R$, $S$ lie on the circle $x^{2}+y^{2}=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\frac{PQ}{RS}$? $\textbf{(A) } 3 \textbf{(B) } 5 \textbf{(C) } 3\sqrt{5} \textbf{(D) } 7 \textbf{(E) } 5\sqrt{2}$