12 Given the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ with eccentricity $e=\sqrt{3}$, and the distance from its left focus $F_{1}$ to the asymptote is $\sqrt{2}$.
(1) Find the equation of the hyperbola $C$;
(2) If a line $l$ passing through the point $D(2,0)$ intersects the hyperbola $C$ at points $A$ and $B$, and the circle with diameter $AB$ passes through the origin $O$. Find the equation of the line $AB$.