2. Consider a tangent line to the ellipse $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{3^{2}}=1$, which intersects the two symmetry axes of the ellipse at points $A$ and $B$. Then the minimum length of segment $AB$ is $\qquad$ .
2. 8 .

Let the point of tangency be \( P(5 \cos \theta, 3 \sin \theta) \). Then the equation of the tangent line to the ellipse at point \( P \) is
$$
\frac{\cos \theta}{5} x + \frac{\sin \theta}{3} y = 1,
$$

which intersects the \( x \)-axis and \( y \)-axis at
$$
A\left(\frac{5}{\cos \theta}, 0\right), B\left(0, \frac{3}{\sin \theta}\right) \text{. }
$$

Therefore, \( A B^{2} = \frac{5^{2}}{\cos ^{2} \theta} + \frac{3^{2}}{\sin ^{2} \theta} \).
By the Cauchy-Schwarz inequality, we have
$$
\begin{aligned}
& \frac{5^{2}}{\cos ^{2} \theta} + \frac{3^{2}}{\sin ^{2} \theta} \\
& = \left( \frac{5^{2}}{\cos ^{2} \theta} + \frac{3^{2}}{\sin ^{2} \theta} \right) \left( \cos ^{2} \theta + \sin ^{2} \theta \right) \\
\geqslant & (5 + 3)^{2} = 8^{2} .
\end{aligned}
$$

Thus, \( |A B| \geqslant 8 \).
The equality holds when \( \tan ^{2} \theta = \frac{3}{5} \).