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

$$
\left\{\begin{array}{l}
x^{2}+y^{2}+2 b(b+x+y)=81 \\
y=4 \cos (x+3 a)-3 \sin (x+3 a)
\end{array}\right.
$$

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

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 $(-b ;-b)$.

By introducing an auxiliary angle, the second equation of the system can be reduced to the form $y=5 \cos (x+3 a-\theta)$. For all possible $a \in \mathrm{R}$, the graphs of these functions sweep out the strip $-5 \leq y \leq 5$.

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 the given strip, from which $b \in[-14 ; 14]$.