Find the foci of the hyperbola 9y^{2}-x^{2}-36y-6x-18=0

9y^{2}-x^{2}-36y-6x-18=0

9y^{2}-36y-x^{2}-6x=18

9(y^{2}-4y)-(x^{2}+6x)=18

9(y^{2}-4y+4)-(x^{2}+6x+9)= 18+36-9

9(y-2)^{2}-(x+3)^{2}=45

Since the y^{2} term is positive, we know the
transverse axis is vertical and a^{2}=5 and b^{2}=45.

From c^{2}=a^{2}+b^{2}=45+5=50

The center of this hyperbola is at (-3, 2) and the foci are therefore at (-3, 2+√50) and (-3, 2-√50)=

(-3, 2+5√2) and (-3, 2-5√2)