1. The function $f(x)=\frac{\sqrt{a^{2}-x^{2}}}{|x+a|-a}$ is an odd function if and only if ( ).
(A) $a<0$
(B) $a>0$
(C) $-1<a<0$ or $0<a<1$
(D) $-1 \leqslant a<0$ or $0<a \leqslant 1$
,$- \mathrm{C}$.
Sufficiency. If $a>0$, then the domain of $f(x)$ is $[-a, 0) \cup (0, a]$. In this case, $f(x)=\frac{\sqrt{a^{2}-x^{2}}}{x}$, which is clearly an odd function.

Necessity. If $f(x)=\frac{\sqrt{a^{2}-x^{2}}}{|x+a|-a}$ is an odd function, then $a \neq 0$ (otherwise, the domain of $f(x)$ is an empty set).
From $f(-x)=-f(x)$, we get
$$
\begin{array}{l}
\frac{\sqrt{a^{2}-(-x)^{2}}}{|1-x+a|-a}=-\frac{\sqrt{a^{2}-x^{2}}}{|x+a|-a} \\
\Rightarrow|x-a|-a=a-|x+a| \\
\Rightarrow|x-a|+|x+a|=2 a \Rightarrow a>0 .
\end{array}
$$