2. Inside the circle $x^{2}+y^{2}-5 x=0$, there are three chords passing through the point $\left(\frac{5}{2}, \frac{3}{2}\right)$ whose lengths form a geometric sequence. Then the range of the common ratio is ( ).
(A) $\left[\frac{\sqrt{2}}{\sqrt[4]{5}}, \frac{2}{\sqrt{5}}\right]$
(B) $\left[\sqrt[3]{\frac{4}{5}}, \frac{2}{\sqrt{5}}\right]$
(C) $\left[\frac{2}{\sqrt{5}}, \frac{\sqrt{5}}{2}\right]$
(D) $\left(\frac{2}{\sqrt[3]{5}}, \frac{\sqrt{5}}{2}\right)$
2. (C).

The equation $x^{2}+y^{2}-5 x:=0$ has a diameter of 5, so the maximum length of the chord passing through the point $\left(\frac{5}{2}, \frac{3}{2}\right)$ is 5. The chord passing through the point $\left(\frac{5}{2}, \frac{3}{2}\right)$ and perpendicular to the diameter through that point is the shortest, with a length of 4. From $a_{3}=a_{1} q^{2}$ we get $q=\sqrt{\frac{a_{3}}{a_{1}}}$, and since $4 \leqslant a_{3} \leqslant 5,4 \leqslant a_{1} \leqslant 5$, we have:
$$
q=\sqrt{\frac{a_{3}}{a_{1}}} \leqslant \sqrt{\frac{5}{4}}=\frac{\sqrt{5}}{2} \text {, and } q \geqslant \sqrt{\frac{4}{5}}=\frac{2}{\sqrt{5}} \text {. }
$$