Standard form equation

Notice that the vertices and foci have common x-values, x=1, which tells us that the graph of this hyperbola has a vertical transverse axis. The standard form of the equation of a hyperbola with a vertical transverse axis is as follows:

(y - k)^{2}/a^{2} - (x - h)^{2}/b^{2} = 1

where (h, k) is the center of the hyperbola, the vertices are at (h, k+a) and (h, k-a), and the foci are at (h, k+c) and (h, k-c).

Vertices: (1, 5) = (h, k+a) ==> k + a = 5 ==> k = 5 - a

(1, 11) = (h, k-a) ==> k - a = 11 ==> k = 11 + a

5 - a = 11 + a ==> 2a = -6 ==> a = -3 ==> a^{2} = 9

Foci: (1, 4) = (h, k+c) ==> k + c = 4 ==> k = 4 - c

(1, 12) = (h, k-c) ==> k - c = 12 ==> k = 12 + c

4 - c = 12 + c ==> 2c = -8 ==> c = -4 ==> c^{2} = 16

Find b using the following formula: b^{2} = c^{2} - a^{2}

b^{2} = 16 - 9 ==> b^{2} = 7 ==> b = √7

Solve for k by plugging in appropriate variable into one of the equations determined for k:

k = 5 - a ==> k = 5 - (-3) = 5 + 3 = 8 ==> k = 8

Thus, given that h = 1 , k = 8 , a^{2} = 9 , and b^{2} = 7 , the equation of the hyperbola is as follows:

(y - 8)^{2}/9 - (x - 1)^{2}/7 = 1