6. Let $A$ and $B$ be the common vertices of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ and the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. $P$ and $M$ are two moving points on the hyperbola and the ellipse, respectively, different from $A$ and $B$, and satisfy $\boldsymbol{A P}+\boldsymbol{B P}=\lambda(\boldsymbol{A M}+\boldsymbol{B} \boldsymbol{M})(\lambda \in \mathbf{R}$ and $|\lambda|>1)$. Let the slopes of $A P$, $B P$, $A M$, and $B M$ be $k_{1}$, $k_{2}$, $k_{3}$, and $k_{4}$, respectively. Then the value of $k_{1}+k_{2}+k_{3}+k_{4}$ is ( ).
(A) 0
(B) $-\frac{b}{2 a}$
(C) $-\frac{2 b}{a}$
(D) $-\frac{b^{2}}{a^{2}}$
6.A.

Let's take $A(-a, 0)$ and $B(a, 0)$. Suppose $P\left(x_{1}, y_{1}\right)$ and $M\left(x_{2}, y_{2}\right)$. From the given equation, we know that $O$, $P$, and $M$ are collinear. Therefore,
$$
\begin{array}{l}
k_{1}+k_{2}=\frac{y_{1}}{x_{1}+a}+\frac{y_{1}}{x_{1}-a}=\frac{2 x_{1} y_{1}}{x_{1}^{2}-a^{2}} \\
=\frac{2 x_{1} y_{1}}{a^{2}+\frac{a^{2}}{b^{2}} y_{1}^{2}-a^{2}}=\frac{2 b^{2}}{a^{2}} \cdot \frac{x_{1}}{y_{1}} .
\end{array}
$$

Similarly, $k_{3}+k_{4}=-\frac{2 b^{2}}{a^{2}} \cdot \frac{x_{2}}{y_{2}}$.
Thus, $k_{1}+k_{2}+k_{3}+k_{4}=0$.