Example 8 Given the parabolas $C_{1}: y=x^{2}+2 x$ and $C_{2}: y=-x^{2}+a$, if a line $l$ is simultaneously tangent to $C_{1}$ and $C_{2}$, we call $l$ a common tangent of $C_{1}$ and $C_{2}$. The line segment between the two points of tangency on the common tangent is called the common tangent segment.
(1) For what value of $a$ does $C_{1}$ and $C_{2}$ have exactly one common tangent? Write the equation of this common tangent;
(2) If $C_{1}$ and $C_{2}$ have two common tangents, prove that the corresponding two common tangent segments bisect each other.
Solve As shown in Figure 10-1, let the line $l: y=k x+b$ be a common tangent line of the parabolas $C_{1}$ and $C_{2}$, and let points $A\left(x_{1}, y_{1}\right)$ and $C\left(x_{2}, y_{2}\right)$ be the points of tangency of line $l$ with $C_{1}$ and $C_{2}$, respectively.
The derivative of $y=x^{2}+2 x$ is $y^{\prime}=2 x+2$; the derivative of $y=-x^{2}+a$ is $y^{\prime}=-2 x$.
According to the geometric meaning of the derivative, we have $2 x_{1}+2=k, -2 x_{2}=k$,
Thus, $2 x_{1}+2=-2 x_{2}$, which means $x_{1}+x_{2}=-1$.
(1) Since $C_{1}$ and $C_{2}$ have exactly one common tangent line, points $A$ and $C$ must coincide,

i.e., $x_{1}=x_{2}$, and since $x_{1}+x_{2}=-1$,
we have $A\left(-\frac{1}{2},-\frac{3}{4}\right), C\left(-\frac{1}{2},-\frac{3}{4}\right)$.
Since $C$ lies on $C_{2}$, we have $a=-\frac{1}{2}$.
At this point, $k=-2 \times\left(-\frac{1}{2}\right)=1$.
Therefore, when $a=-\frac{1}{2}$, there is exactly one common tangent line, and the equation of this common tangent line is $y=x-\frac{1}{4}$.
(2) From $x_{1}+x_{2}=-1$ we get $x_{2}=-1-x_{1}$,

Thus,
$$
\begin{aligned}
y_{1}+y_{2} & =x_{1}^{2}+2 x_{1}+\left(-x_{2}^{2}+a\right)=x_{1}^{2}+2 x_{1}-\left(x_{1}+1\right)^{2}+a \\
& =a-1,
\end{aligned}
$$

Hence, the midpoint of the common tangent segment $A C$ is $\left(-\frac{1}{2}, \frac{a-1}{2}\right)$.
Similarly, the midpoint of the other common tangent segment $B D$ is also $\left(-\frac{1}{2}, \frac{a-1}{2}\right)$.
Therefore, the two common tangent segments bisect each other.