(19) Let $P$ be a moving point on the major axis of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$. A line passing through $P$ with slope $k$ intersects the ellipse at points $A$ and $B$. If $|P A|^{2}+|P B|^{2}$ depends only on $k$ and not on $P$, find the value of $k$.

Let $P$ be a moving point on the major axis of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$. A line passing through $P$ with slope $k$ intersects the ellipse at points $A$ and $B$. If $|P A|^{2}+|P B|^{2}$ depends only on $k$ and not on $P$, find the value of $k$.
19 Let $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$, and the equation of the line passing through point $P$ with slope $k$ is $y = k(x-a)$. From
$$
\left\{\begin{array}{l}
y=k(x-a), \\
\frac{x^{2}}{25}+\frac{y^{2}}{16}=1,
\end{array}\right.
$$

we get $x_{1}+x_{2}=\frac{50 a k^{2}}{16+25 k^{2}}, x_{1} x_{2}=-\frac{25 a^{2} k^{2}-400}{16+25 k^{2}}$, so
$$
\begin{array}{c}
y_{1}+y_{2}=-\frac{32 a k}{16+25 k^{2}}, \\
y_{1} y_{2}=k^{2}\left(x_{1}-a\right)\left(x_{2}-a\right)=\frac{\left(16 a^{2}-400\right) k^{2}}{16+25 k^{2}} .
\end{array}
$$

Therefore,
$$
\begin{aligned}
& |P A|^{2}+|P B|^{2} \\
= & \left(x_{1}-a\right)^{2}+y_{1}^{2}+\left(x_{2}-a\right)^{2}+y_{2}^{2} \\
= & \left(x_{1}+x_{2}\right)^{2}-2 x_{1} x_{2}-2 a\left(x_{1}+x_{2}\right)+\left(y_{1}+y_{2}\right)^{2}-2 y_{1} y_{2}+2 a^{2} \\
= & \left(k^{2}+1\right) \cdot \frac{\left(512-800 k^{2}\right) a^{2}+800\left(16+25 k^{2}\right)}{\left(16+25 k^{2}\right)^{2}},
\end{aligned}
$$
$$
\text { Let } 512-800 k^{2}=0 \Rightarrow k= \pm \frac{4}{5} \text {. }
$$