2. There are the following four propositions:
(1) Through a point on a line, there is one and only one line perpendicular to this line;
(2) The equation $x \sqrt{(x+2)^{2}}+1=0$ has three different real solutions;
(3) If $\odot O_{1}$ and $\odot O_{2}$ intersect at points $A$ and $B$, $\odot O_{1}$ has a radius of $5$, $O_{1} O_{2}=4+\sqrt{7}$, and $A B=8$, then the radius of $\odot O_{2}$ is $\sqrt{24+2 \sqrt{7}}$;
(4) If the quadratic function $y=x^{2}+a x+a$ intersects the coordinate axes at only one point, then $a=0$ or 4.
Among the above four propositions, the number of true propositions is ( . ) .
(A) 0
(B) 1
(C) 2
(D) 3
2. A.

For a line in space, an infinite number of perpendicular lines can be drawn through a point on the line, so statement (1) is a false statement.
For statement (2), in the original equation, it is clear that $x<0$.
When $x<-2$, the original equation can be transformed into
$$
-x(x+2)+1=0 \text {. }
$$

Solving gives $x_{1}=-1+\sqrt{2}$ (discard), $x_{2}=-1-\sqrt{2}$;
When $-2<x<0$, the original equation can be transformed into
$$
x(x+2)+1=0 \Rightarrow x=-1 \text {. }
$$

Therefore, the original equation has two real solutions.
Statement (2) is a false statement.
For statement (3), as shown in Figure 1, when points $O_{1}$ and $O_{2}$ are on the same side of the common chord $AB$, extend $\mathrm{O}_{2} \mathrm{O}_{1}$ to intersect $AB$ at point $C$, and connect $\mathrm{O}_{1} A$ and $\mathrm{O}_{2} A$. Then $O_{2} C \perp AB$, and $AC=\frac{1}{2} AB=4$.

In the right triangle $\triangle A O_{1} C$, calculate to get $O_{1} C=3$.
Therefore, $O_{2} C=O_{1} O_{2}+O_{1} C=7+\sqrt{7}$.
In the right triangle $\triangle A O_{2} C$,
$O_{2} A=\sqrt{O_{2} C^{2}+A C^{2}}=\sqrt{72+14 \sqrt{7}}$.
Thus, the radius of $\odot \mathrm{O}_{2}$ could also be $\sqrt{72+14 \sqrt{7}}$.
Therefore, statement (3) is a false statement.
For statement (4), the graph of the quadratic function $y=x^{2}+a x+a$ must intersect the $y$-axis at one point.

When this intersection point is the origin, we have $a=0$, at this time, $y=x^{2}$ intersects the coordinate axes at only one point;

When this intersection point is not the origin, and there is no intersection with the $x$-axis, then
$$
\Delta=a^{2}-4 a<0 \Rightarrow 0<a<4 .
$$

In summary, $0 \leqslant a<4$.
Therefore, statement (4) is also a false statement.