Answer
$\frac{y^{2}}{16} - \frac{(x-1)^{2}}{9} = 1$
Work Step by Step
General Equation of a Hyperbola is: $\frac{(y-k)^{2}}{a^{2}} - \frac{(x-h)^{2}}{b^{2}} = 1$
If vertices are on the vertical axis
Foci are (1, -5) and (1,5), so the midpoint of the foci is the center, which is at (1,0)
Thus c = 5 (distance from center to focus)
Thus h = 1 and k = 0
So we know this: $\frac{y^{2}}{a^{2}} - \frac{(x-1)^{2}}{b^{2}} = 1$
(1,4) is a point on this graph, so substitute x for 1 and y for 4
$\frac{4^{2}}{a^{2}} - \frac{(1-1)^{2}}{b^{2}} = 1$
$\frac{16}{a^{2}} = 1$
Thus a = 4
$c^{2} = a^{2} + b^{2}$
$25 = 16 + b^{2}$
$b^{2} = 9$
So the final equation is $\frac{y^{2}}{16} - \frac{(x-1)^{2}}{9} = 1$
