9. (15 points) Find all values of $a$ for which the equation

$$
\sin \left(\sqrt{a^{2}-x^{2}-2 x-1}\right)=0.5
$$

has exactly seven distinct solutions.
Answer: $: \frac{17 \pi}{6}$

Solution. The number of solutions to the original equation coincides with the number of solutions to the system

$$
\left\{\begin{array}{l}
y=\sqrt{a^{2}-x^{2}-2 x-1} \\
\sin y=0.5
\end{array}\right.
$$

Which is equivalent to

$$
\left\{\begin{array}{l}
y \geq 0 \\
y^{2}+(x+1)^{2}=a^{2} \\
{\left[\begin{array}{l}
y=\frac{\pi}{6}+2 \pi n, n \in \mathbb{Z} \\
y=\frac{5 \pi}{6}+2 \pi n, n \in \mathbb{Z}
\end{array}\right.}
\end{array}\right.
$$

The solution to the equation $\sin y=0.5$ is represented by a family of horizontal lines (red lines), while the equation $y=\sqrt{a^{2}-x^{2}-2 x-1}$ represents a semicircle with center at the point $(-1 ; 0)$ and radius $|a|$. The radius of the circle is chosen such that the semicircle touches the fourth line above zero. Hence, $|a|=\frac{5 \pi}{6}+2 \pi$.

![](https://cdn.mathpix.com/cropped/2024_05_06_c5ac26fa1d2ba726211fg-4.jpg?height=708&width=1077&top_left_y=1342&top_left_x=501)