How do you write the equation of the hyperbola given Foci: (0,-8),(0,8) and vertices (0,-5), (0,5)?

Mar 14, 2017

Equation of hyperbola is ${y}^{2} / 25 - {x}^{2} / 39 = 1$

Explanation:

As the focii and vertices are symmetrically placed on $y$-axis,

its center is $\left(0 , 0\right)$ and the equation of hyperbola is of the type

${y}^{2} / {a}^{2} - {x}^{2} / {b}^{2} = 1$

As the distance between center and either vertex is $5$, we have $a = 5$

and as distance between center and either focus is $8$, we have $c = 8$

As ${c}^{2} = {a}^{2} + {b}^{2}$, ${b}^{2} = {8}^{2} - {5}^{2} = 39$

and equation of hyperbola is ${y}^{2} / 25 - {x}^{2} / 39 = 1$
graph{y^2/25-x^2/39=1 [-40, 40, -20, 20]}