2. The tangent line of $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ intersects the $x$-axis at $A$ and the $y$-axis at $B$, then the minimum value of $|A B|$ is ( ).
(A) $2 \sqrt{a^{2}+b^{2}}$
(B) $a+b$
(C) $\sqrt{2 a b}$
(D) $4 \sqrt{a b}$
2. (B).

A circle is a special case of an ellipse. Let's first examine the case when $a=b=1$, i.e., the unit circle. Suppose $AB$ is tangent to $\odot O$ at $C$, and let $\angle AOC = \alpha (\alpha \in (0, \frac{\pi}{2}))$. Then, $|AC| = \tan \alpha$ and $|BC| = \cot \alpha$, so $|AB| = |AC| + |BC| = \tan \alpha + \cot \alpha \geqslant 2$. This means that when $a=b=1$, the minimum value of $|AB|$ is 2.