10.2. Find all values of the parameter $b$, for each of 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. $\quad 0 \leq b \leq 2$.

Solution. On the coordinate plane, the first equation represents a circle $S$ : $(x-1)^{2}+y^{2}=1$ with a radius of 1 centered at the point ( $1 ; 0$ ). The second equation is the equation of a line $l$ passing through the point $M(b ; 0)$ with a slope of $a$. If the point $M$ lies inside $S$ (i.e., when $0 \leq b \leq 2$), then the line $l$ will intersect the circle $S$ for any slope $a$. If, however, $M$ lies outside $S$, then there always exists a sufficiently large absolute value of the slope $a$ such that $l$ does not intersect $S$.