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 $k$.
19. Let $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$, and the equation of the line passing through point $P$ with slope $k$ be $y=k(x-a)$.
$$
\begin{array}{l}
\text { From }\left\{\begin{array}{l}
y=k(x-a), \\
\frac{x^{2}}{25}+\frac{y^{2}}{16}=1,
\end{array}\right. \text { we get } \\
x_{1}+x_{2}=\frac{50 a k^{2}}{16+25 k^{2}}, \\
x_{1} x_{2}=\frac{25 a^{2} k^{2}-400}{16+25 k^{2}} . \\
\text { Therefore, }|P A|^{2}+|P B|^{2} \\
=\left(1+k^{2}\right)\left[\left(x_{1}-a\right)^{2}+\left(x_{2}-a\right)^{2}\right] \\
=\left(k^{2}+1\right) \frac{\left(512-800 k^{2}\right) a^{2}+800\left(16+25 k^{2}\right)}{\left(16+25 k^{2}\right)^{2}} .
\end{array}
$$

Let $512-800 k^{2}=0$. Solving for $k$ gives $k= \pm \frac{4}{5}$.