10. If the curve symmetric to the hyperbola $x^{2}-y^{2}=a^{2}(a>0)$ with respect to the line $y=x-2$ is tangent to the line $2 x+3 y-6=0$, then the value of $a$ is ( ).
(A) $\frac{4 \sqrt{5}}{5}$
(B) $\frac{8 \sqrt{5}}{5}$
(C) $\frac{12 \sqrt{5}}{5}$
(D) $\frac{16 \sqrt{5}}{5}$
10.B.

It is easy to find the equation of the symmetric curve as
$$
(y+2)^{2}-(x-2)^{2}=a^{2} \text { . }
$$

Solving the system of equations $\left\{\begin{array}{l}(y+2)^{2}-(x-2)^{2}=a^{2}, \\ 2 x+3 y-6=0 .\end{array}\right.$
Eliminating $x$ yields $-\frac{5}{4} y^{2}+7 y+3-a^{2}=0$.
From $\Delta=64-5 a^{2}=0$, solving for $a$ gives $a=\frac{8 \sqrt{5}}{5}$ (the negative value is discarded)