Fun!
The center of the hyperbola is the midpoint of its foci. The distance between the foci is 43-9=34. So they're 17 from the center.
We have coordinates
F1(0,17)
F2(0,-17)
Let's call the vertex of the hyperbola V. We have
V(0,13)
This is a vertical hyperbola so the equation is of the form
[tex] \dfrac{y^2}{a^2}- \dfrac{x^2}{b^2}=1[/tex]
The vertices for a vertical hyperbola like this are (0, a) and (0,-a)
So we have a=13
The distance c from the foci to the center is given by
[tex]c^2 = a^2+b^2[/tex]
[tex]b^2 = c^2-a^2 = 17^2 - 13^2 = 120[/tex]
That says our hyperbola's equation is
[tex] \dfrac{y^2}{169}- \dfrac{x^2}{120}=1[/tex]
That's our answer.
