11.2. Find all parameters $b$, for which the system of equations $\left\{\begin{array}{l}x^{2}-2 x+y^{2}=0 \\ a x+y=a b\end{array}\right.$ has a solution for any $a$.
Answer: $b \in[0 ; 2]$. Solution. The first equation of the system represents a circle $(x-1)^{2}+y^{2}=1$, its center is at the point $(1 ; 0)$, and the radius is 1. The second equation is the equation of a line with a slope of (-a). Note that this line passes through the point $M$ with coordinates $(b ; 0)$. Therefore, if $M$ lies on the specified circle or inside it (i.e., when $b \in[0 ; 2]$), then any line passing through $M$ will intersect the circle. If, however, $M$ lies outside the circle, then there will be a line passing through $M$ that does not intersect the circle. Hence, the result follows.