General equation of hyperbola:
(x-h)²/a²-(y-k)²/b²=1 where (h,k) is the centre (0,-4):
x²/a²-(y+4)²/b²=1.
The asymptotes intersect at the centre.
For a left-right hyperbola, the foci, vertices and centre all have the same y-coordinate k=-4, so from the given info we know that we do in fact have such a hyperbola. (An up-down hyperbola would have a vertical transverse axis so the foci, vertices and centre would be aligned vertically and have the same x-coordinate.)
When x=±a we have a vertex, that is, at (-a,k) and (a,k). So, since the given vertex is (3,-4), a=3.
The focal distance f=√(a²+b²)=√(9+b²) and the foci lie on y=-4. Therefore,
One focus is at (4,-4) which is 4 units from the centre (0,-4) at x=0. Therefore the focal distance=4=√(9+b²), 16-9=b², b=√7.
The equation of the hyperbola is:
x²/9-(y+4)²/7=1.
