11. 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
$$
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)
$$

Let $P$ and $M$ be two moving points on the hyperbola and the ellipse, respectively, different from $A$ and $B$, and satisfy
$$
\overrightarrow{A P}+\overrightarrow{B P}=\lambda(\overrightarrow{A M}+\overrightarrow{B M}),
$$

where $\lambda \in \mathbf{R},|\lambda|>1$. Let the slopes of the lines $A P, B P, A M, B M$ be $k_{1}, k_{2}, k_{3}, k_{4}$, respectively, and $k_{1}+k_{2}=5$. Then $k_{3}+k_{4}=$ $\qquad$
11. -5 .

Let $A(-a, 0), B(a, 0), P\left(x_{1}, y_{1}\right), M\left(x_{2}, y_{2}\right)$.
From $\overrightarrow{A P}+\overrightarrow{B P}=\lambda(\overrightarrow{A M}+\overrightarrow{B M})$, we know that points $O, P, M$ are collinear, and we can find that
$$
\begin{array}{l}
k_{1}+k_{2}=\frac{y_{1}}{x_{1}+a}+\frac{y_{1}}{x_{1}-a} \\
=\frac{2 x_{1} y_{1}}{x_{1}^{2}-a^{2}}=\frac{2 b^{2}}{a^{2}} \cdot \frac{x_{1}}{y_{1}} .
\end{array}
$$

Similarly, $k_{3}+k_{4}=-\frac{2 b^{2}}{a^{2}} \cdot \frac{x_{2}}{y_{2}}$.
Therefore, $k_{3}+k_{4}=-\left(k_{1}+k_{2}\right)=-5$.