3. For which $a$ does the system $\left\{\begin{array}{l}x \cos a+y \sin a=5 \cos a+2 \sin a \\ -3 \leq x+2 y \leq 7,-9 \leq 3 x-4 y \leq 1\end{array}\right.$ have a unique solution?
Solution. The inequalities in the system limit a parallelogram:

$$
y=\frac{-x-3}{2}, y=\frac{-x+7}{2}, y=\frac{3 x+9}{4}, y=\frac{3 x-1}{4} \text {. }
$$

Let's find its vertices:

$$
\begin{aligned}
& A:\left\{\begin{array}{l}
x+2 y+3=0 \\
3 x-4 y-1=0
\end{array} \rightarrow A(-1 ;-1) \quad B:\left\{\begin{array}{c}
x+2 y+3=0 \\
3 x-4 y+9=0
\end{array} \rightarrow B(-3 ; 0)\right.\right. \\
& C:\left\{\begin{array}{c}
x+2 y-7=0 \\
3 x-4 y+9=0
\end{array} \rightarrow C(1 ; 3) \quad D:\left\{\begin{array}{l}
x+2 y-7=0 \\
3 x-4 y-1=0
\end{array} \rightarrow D(3 ; 2)\right.\right.
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_dcb518c8ba3f5cf57563g-2.jpg?height=540&width=828&top_left_y=1729&top_left_x=474)

The equation $x \cos a+y \sin a=5 \cos a+2 \sin a$ defines a line.

Case 1. $\sin a=0 \rightarrow a=\pi k \rightarrow x=5-$ no solutions.

![](https://cdn.mathpix.com/cropped/2024_05_06_dcb518c8ba3f5cf57563g-3.jpg?height=800&width=1359&top_left_y=277&top_left_x=311)

Case 2. $\sin a \neq 0 \rightarrow y=\operatorname{ctg} a(5-x)+2$.- a family of lines passing through the point $E(5 ; 2)$ with the slope $\operatorname{ctg} a$. The unique solution: the line passes through $\mathrm{C}(1 ; 3)$ or $\mathrm{A}(-1 ;-1)$.

![](https://cdn.mathpix.com/cropped/2024_05_06_dcb518c8ba3f5cf57563g-3.jpg?height=808&width=1336&top_left_y=1338&top_left_x=363)

$$
\left[\begin{array}{c}
3=\operatorname{ctg} a(5-1)+2 \rightarrow \operatorname{ctg} a=\frac{1}{4} \rightarrow a=\operatorname{arctg} 4+\pi k, k \in Z \\
-1=\operatorname{ctg} a(5+1)+2 \rightarrow \operatorname{ctg} a=-\frac{1}{2} \rightarrow a=-\operatorname{arctg} 2+\pi k, k \in Z
\end{array}\right.
$$

Answer: $a_{1}=\operatorname{arctg} 4+\pi k, a_{2}=-\operatorname{arctg} 2+\pi k, k \in Z$.