1. A parabolic pond is shown in Figure 1, where $A$ and $B$ are two corresponding points on the edge of the pond, and $O$ is the center of the pond bottom (i.e., the vertex of the parabola). Then for point $C$ in the pond,
$$
\tan \angle C A B + \tan \angle C B A
$$

should satisfy ( ).
(A) When point $C$ is closer to point $O$, its value increases
(B) When point $C$ is closer to point $O$, its value decreases
(C) Regardless of where point $C$ is, its value remains constant
(D) Cannot be determined
- 1.C.

As shown in Figure 4, based on the given conditions, establish a rectangular coordinate system $x O^{\prime} y$ with the midpoint of $A B$ as the origin and $A B$ as the $x$-axis. Suppose the equation of the parabola is
$$
y=a x^{2}+b(a>0, b<0).
$$
We have $A\left(-\sqrt{\frac{-b}{a}}, 0\right)$,
$B\left(\sqrt{\frac{-b}{a}}, 0\right)$. Let $C\left(x_{0}, a x_{0}^{2}+b\right)$. Then
$$
\begin{array}{l}
\tan \angle C A B=-k_{u:}=-\left(\frac{a x_{0}^{2}+b}{x_{0}+\sqrt{\frac{-b}{a}}}\right), \\
\tan \angle C B A=k_{H:}=\frac{a x_{0}^{2}+b}{x_{0}-\sqrt{\frac{-b}{a}}} . \\
\text { Therefore, } \tan \angle C A B+\tan \angle C B A \\
=\left(a x_{0}^{2}+b\right)\left(\frac{-1}{x_{0}+\sqrt{\frac{-b}{a}}}+\frac{1}{x_{0}-\sqrt{\frac{-b}{a}}}\right) \\
=\left(a x_{0}^{2}+b\right) \frac{2 \sqrt{\frac{-b}{a}}}{x_{0}^{2}+\frac{b}{a}}=2 \sqrt{-a b}
\end{array}
$$

which is a constant.