11. Given the circle $P: x^{2}+y^{2}=2 x$ and the parabola $S: y^{2}=4 x$, a line $l$ is drawn through the center of circle $P$. This line intersects the two curves at four points, which are sequentially labeled as $A, B, C, D$ from top to bottom. If the lengths of segments $A B, B C, C D$ form an arithmetic sequence in this order, find the equation of line $l$.
Solve:
$|A B|+|C D|=2|B C|=4 \Rightarrow|A D|=6$, Let the line $l: x=n y+1$, combining $\left\{\begin{array}{l}x=n y+1, \\ y^{2}=4 x\end{array} \Rightarrow y^{2}-4 n y-4=0\right.$. Then $|A D|=\sqrt{n^{2}+1} \cdot \sqrt{16 n^{2}+16}$ $=4\left(n^{2}+1\right)=6 \Rightarrow n^{2}=\frac{1}{2} \Rightarrow n= \pm \frac{\sqrt{2}}{2}$. Therefore, the equation of the line $l$ is $x= \pm \frac{\sqrt{2}}{2} y+1$.