12. (1) Let the hyperbola $C: (y+a)^{2}-(x-a)^{2}=2a$, with its asymptotes passing through the point $(3,1)$. Find the equations of the asymptotes of $C$.
(2) A certain hyperbola has the lines $y= \pm(x+1)$ as its asymptotes, and the focal distance is 4. Find its equation.
12. (1) Regarding the required asymptotes as a degenerate hyperbola \((y+a)^{2}-(x-a)^{2}=0\), substituting the point \((3,1)\) yields \(a=1\), thus the asymptotes of \(C\) are \(y=x-2\) and \(y=-x\).
(2) Similarly to (1), regarding the given line as a degenerate hyperbola, the equation of the hyperbola can be set as \(y^{2}-(x+1)^{2}=k\). Since the focal distance is 4, we have \(c^{2}=a^{2}+b^{2}=2k=4\), yielding \(k=2\). The required hyperbola equation is \(y^{2}-(x+1)^{2}=2\).