The midpoint of the line joining the vertices $(5,7),(-3,-1)$ is the centre of the hyperbola.
In the standard form (i.e asymptotes parallel to the coordinate axis),its equation is $(x-1)(y-3)=c^2$
Since the vertices lie on the hyperbola,$(5,7)$ is a point on the rectangular hyperbola.
Therefore the equation of the rectangular hyperbola is $(x-1)(y-3)=16$
The combined equation of the asymptotes differs only in the constant.
But the asymptotes pass through the centre $(1,3)$.
Therefore the combined equation of the asymptotes is $(x-1)(y-3)=0$ and the asymptotes are $x=1,y=3$