11. Given the ellipse $\frac{x^{2}}{2}+y^{2}=1$, two mutually perpendicular moving lines passing through the fixed point $C(1,0)$ intersect the ellipse at points $P, Q$ respectively. $F_{1}, F_{2}$ are the left and right foci, and $O$ is the origin.
(1) Find the minimum value of the vector $\left|\overrightarrow{P F_{1}}+\overrightarrow{P F_{2}}\right|$;
(2) When the vectors $\overrightarrow{P F_{1}}+\overrightarrow{P F_{2}}$ and $\overrightarrow{Q F_{1}}+\overrightarrow{Q F_{2}}$ are perpendicular to each other, find the slope of the line passing through points $P, Q$.
11. Solution: (1) $\overrightarrow{P F_{1}}+\overrightarrow{P F_{2}}=2 \overrightarrow{P O}$, so $\left|\overrightarrow{P F_{1}}+\overrightarrow{P F_{2}}\right|=2|\overrightarrow{P O}|$. The minimum value is $2 b=2$. When point $P$ is at the top of the minor axis, the equality holds.
(2) $\overrightarrow{P F_{1}}+\overrightarrow{P F_{2}}=2 \overrightarrow{P O}, \overrightarrow{Q F}_{1}+\overrightarrow{Q F}_{2}=2 \overrightarrow{Q O}$, so $\overrightarrow{P O}$ and $\overrightarrow{Q O}$ are perpendicular to each other, which means segment $P Q$ is the common hypotenuse of right $\triangle P O Q$ and right $\triangle P C Q$. Let the midpoint of segment $P Q$ be $M$, then $M C=M O$, i.e., $x_{P}+x_{Q}=2 x_{M}=x_{C}=1$. (1) Suppose the equation of line $P Q$ is $y=k x+b$, combined with $\frac{x^{2}}{2}+y^{2}=1$ we get:
$\left(1+2 k^{2}\right) x^{2}+4 k b x+2 b^{2}-2=0$, from (1) we have: $1+2 k^{2}+4 k b=0$.
Also, from the fact that $\overrightarrow{P O}$ and $\overrightarrow{Q O}$ are perpendicular, we know $\frac{y_{P}}{x_{P}} \cdot \frac{y_{Q}}{x_{Q}}=-1$.
Combining line $P Q$ with $\frac{x^{2}}{2}+y^{2}=1$ we get: $\frac{x^{2}}{2}+y^{2}=\left(\frac{y-k x}{b}\right)^{2}$, i.e., $\left(2-\frac{2}{b^{2}}\right)\left(\frac{y}{x}\right)^{2}+\frac{4 k}{b} \cdot \frac{y}{x}+1-\frac{2 k^{2}}{b^{2}}=0$,

From (3) we get $\left(2-\frac{2}{b^{2}}\right)+\left(1-\frac{2 k^{2}}{b^{2}}\right)=0$ (4), solving (2) and (4) we get $k= \pm \sqrt{\frac{2 \sqrt{10}-5}{10}}$.