(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 $\overrightarrow{A P}+\overrightarrow{B P}=\lambda(\overrightarrow{A M}+\overrightarrow{B M})(\lambda \in \mathbf{R}$, $|\lambda|>1)$. Let the slopes of the lines $A P$, $B P$, $A M$, and $B M$ be $k_{1}$, $k_{2}$, $k_{3}$, and $k_{4}$, respectively, and $k_{1}+k_{2}=5$, then $k_{3}+k_{4}=$
$11-5$ Hint: Let $A(-a, 0), B(a, 0), P\left(x_{1}, y_{1}\right), Q\left(x_{2}\right.$, $\left.y_{2}\right)$, from $\overrightarrow{A P}+\overrightarrow{B P}=\lambda(\overrightarrow{A M}+\overrightarrow{B M})$ we know, points $O, P, M$ are collinear, thus we can obtain
$$
k_{1}+k_{2}=\frac{y_{1}}{x_{1}+a}+\frac{y_{1}}{x_{1}-a}=\frac{2 x_{1} y_{1}}{x_{1}^{2}-a^{2}}=\frac{2 b^{2}}{a^{2}} \cdot \frac{x_{1}}{y_{1}},
$$

Similarly, we obtain
$$
k_{3}+k_{4}=-\frac{2 b^{2}}{a^{2}} \cdot \frac{x_{2}}{y_{2}}
$$

Therefore,
$$
k_{3}+k_{4}=-\left(k_{1}+k_{2}\right)=-5 .
$$