10. (20 points) Find all values of the parameter $a$, for each of which the system

$$
\left\{\begin{array}{l}
|2 x+y-b|+\sqrt{5(x-2)^{2}+5(y-a)^{2}}=a+4-b \\
(x-2)^{2}+(y-c)^{2}=r^{2}
\end{array}\right.
$$

has a unique solution.
# Solution:

Transform the first equation:

$$
\frac{|2 x+y-b|}{\sqrt{5}}+\sqrt{(x-2)^{2}+(y-a)^{2}}=\frac{2 \cdot 2+a-b}{\sqrt{5}}
$$

$\frac{|2 x+y-b|}{\sqrt{5}}$ defines the distance from point $M(x ; y)$ to the line $n: 2 x+y-b=0$, $\sqrt{(x-2)^{2}+(y-a)^{2}}$ the distance from $M$ to point $C(2 ; a)$, and $\frac{2 \cdot 2+a-b}{\sqrt{5}}$ the distance from $C$ to the line $n$ (the case when point $C$ is above the line $n$). On the diagram, this looks like

$$
M G+M C=C D
$$

This equality holds if and only if point $M$ belongs to the segment $C D$, i.e., the solution to the first equation is the segment $C D$. The second equation defines a circle with center at point $A(2 ; c)$ and radius $r$. In all cases, the circle does not intersect the line. As the parameter $a$ changes, point $C$ moves along the line $x=2$, and for $c-r<a<c+r$ it lies inside the circle, and the segment $C D$ intersects the circle. We check the endpoints directly: $a=c-r$ fits, $a=c+r$ does not. When $a=c+\frac{\sqrt{5}}{2} r$, the segment $C D$ is tangent to the circle from above. From this, we get the answer.

Answers:

![](https://cdn.mathpix.com/cropped/2024_05_06_e38b8a325740b6549434g-10.jpg?height=837&width=834&top_left_y=1146&top_left_x=954)

| Option | 1 | 2 | 3 | 4 |
| :---: | :---: | :---: | :---: | :---: |
| Answer | $[3 ; 5) \cup\left\{4+\frac{\sqrt{5}}{2}\right\}$ | $[2 ; 6) \cup\{4+\sqrt{5}\}$ | $[4 ; 6) \cup\left\{5+\frac{\sqrt{5}}{2}\right\}$ | $[3 ; 7) \cup\{5+\sqrt{5}\}$ |


| Option | 5 | 6 | 7 | 8 |
| :---: | :---: | :---: | :---: | :---: |
| Answer | $[3 ; 7) \cup\{5+\sqrt{5}\}$ | $[2 ; 8) \cup\left\{5+\frac{3 \sqrt{5}}{2}\right\}$ | $[3 ; 5) \cup\left\{4+\frac{\sqrt{5}}{2}\right\}$ | $[2 ; 4) \cup\left\{3+\frac{\sqrt{5}}{2}\right\}$ |


| Option | 9 | 10 |
| :---: | :---: | :---: |
| Answer | $[1 ; 5) \cup\{3+\sqrt{5}\}$ | $[2 ; 6) \cup\{4+\sqrt{5}\}$ |