2. Given the lengths of line segments $A B$ and $C D$ are $a$ and $b$ $(a, b > 0)$. If line segments $A B$ and $C D$ slide on the $x$-axis and $y$-axis respectively, and such that points $A, B, C, D$ are concyclic, then the equation of the locus of the centers of these circles is $\qquad$ .
2. $4 x^{2}-4 y^{2}-a^{2}+b^{2}=0$.

Let the center of the required circle be $O^{\prime}(x, y)$. Then
$$
\begin{array}{l}
A\left(x-\frac{a}{2}, 0\right), B\left(x+\frac{a}{2}, 0\right), \\
C\left(0, y-\frac{b}{2}\right), D\left(0, y+\frac{b}{2}\right) .
\end{array}
$$

Notice that,
$A, B, C, D$ are concyclic
$$
\begin{array}{l}
\Leftrightarrow\left|O^{\prime} B\right|^{2}=\left|O^{\prime} D\right|^{2} \\
\Leftrightarrow\left(\frac{a}{2}\right)^{2}+y^{2}=x^{2}+\left(\frac{b}{2}\right)^{2} \\
\Leftrightarrow 4 x^{2}-4 y^{2}-a^{2}+b^{2}=0 .
\end{array}
$$