write and equation of a hyperbola, given foci is (3,5), (3,-3), vertex is (3,4)

write and equation of a hyperbola, given foci is (3,5), (3,-3), vertex is (3,4)

Since one vertex is given as (3,4), the other is at (3,-2).

There are 6 units between the vertices. That means the center is at (3,1).

c is the distance from the center to the foci. A and b are the lengths of the semi-major and semi-minor axes.

b=\sqrt{c^{2}-a^{2}}

c=4, \;\ a=3

b=\sqrt{4^{2}-3^{2}}=\sqrt{16-9}=\sqrt{7}

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

\frac{(y-1)^{2}}{9}-\frac{(x-3)^{2}}{7}=1

Unless you have a calcualtor that graphs conics, we must solve for y to graph.