17. Let $P, Q, R, S$ be four integer points on the circle $\odot O: x^{2}+y^{2}=25$, and $|P Q|, |R S|$ are both irrational numbers. Then the maximum possible value of $\frac{|P Q|}{|R S|}$ is ( ).
(A) 3
(B) 5
(C) $3 \sqrt{5}$
(D) 7
(E) $5 \sqrt{2}$
17. D.

From the problem, we know
$$
\begin{array}{c}
P, Q, R, S \in \{( \pm 3, \pm 4), (\pm 4, \pm 3), \\
(0, \pm 5), (\pm 5,0)\} .
\end{array}
$$

Since $|P Q|$ and $|R S|$ are both irrational numbers, points $P$ and $Q$ are selected from different quadrants, and points $R$ and $S$ are selected from the same quadrant.
Thus, $|P Q| \leqslant \sqrt{(4+3)^{2}+(3+4)^{2}}=7 \sqrt{2}$,
$$
\begin{array}{l}
|R S| \geqslant \sqrt{(4-3)^{2}+(3-4)^{2}}=\sqrt{2} . \\
\text { Therefore, } \frac{|P Q|}{|R S|} \leqslant \frac{7 \sqrt{2}}{\sqrt{2}}=7 .
\end{array}
$$