6. Find all values of the parameter $b$, for each of which there exists a number $a$ such that the system

$$
\left\{\begin{array}{l}
x^{2}+y^{2}+2 b(b+x+y)=81 \\
y=\frac{5}{(x-a)^{2}+1}
\end{array}\right.
$$

has at least one solution $(x ; y)$.
Answer. $b \in[-14 ; 9)$.

Solution. The first equation of the system can be transformed into the form $(x+b)^{2}+(y+b)^{2}=9^{2}$, hence it represents a circle of radius 9 with center $(-b ;-b)$.

Consider the function defined by the second equation when $a=0$. At the point $x=0$, it takes the maximum value of 5. As $x$ increases in magnitude, the function decreases and approaches zero as $x \rightarrow \infty$. If we vary $a$, the graph shifts by $|a|$ units to the left or right. For all possible $a \in \mathrm{R}$, the graphs of these functions sweep out the strip $0<y \leq 5$.

For the condition of the problem to be satisfied, it is necessary and sufficient that the circle defined by the first equation has at least one common point with this strip, from which $b \in[-14 ; 9)$.