12. Let the circle $C: x^{2}+y^{2}-\left(2 a^{2}-4\right) x-4 a^{2} y+5 a^{4}-4=0$. Then the equation of the locus of the center of circle $C$ is
12. $2 x-y+4=0(-2 \leqslant x0$, we get the range of real number $a$ as $(-\sqrt{2}, \sqrt{2})$. Let the center of the circle be $C(x, y)$, then $x=a^{2}-2, y=2 a^{2}$. Eliminating $a$, we can obtain the equation of the trajectory of the center $C$.