11. Let $P$ be any point on the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ other than the endpoints of the major axis, $F_{1}$ and $F_{2}$ be its left and right foci respectively, and $O$ be the center. Then $\left|P F_{1}\right| \cdot\left|P F_{2}\right|+|O P|^{2}=$ $\qquad$ .
11. Fill in 25. In $\triangle P F_{1} F_{2}$, by the median length formula, we get
$$
\left|P F_{1}\right|^{2}+\left|P F_{2}\right|^{2}=2|O P|^{2}+2\left|O F_{2}\right|^{2} \text {, }
$$

i.e., $\left(\left|P F_{1}\right|+\left|P F_{2}\right|\right)^{2}=2|O P|^{2}+2\left|O F_{2}\right|^{2}+2\left|P F_{1}\right| \cdot\left|P F_{2}\right|$, by the definition of an ellipse, we get
$$
\left|P F_{1}\right| \cdot\left|P F_{2}\right|+|O P|^{2}=2 a^{2}-\left|O F_{2}\right|^{2}=2 a^{2}-c^{2}=a^{2}+b^{2}=16+9=25 .
$$