# How do you write the equation of the hyperbola given Foci: (0,-9),(0,9) and vertices (0,-3sqrt5), (0,3sqrt5)?

Jan 14, 2017

Because it is the y coordinate that is changing for the given points, use the vertical transverse axis form:
${\left(y - k\right)}^{2} / {a}^{2} - {\left(x - h\right)}^{2} / {b}^{2} = 1 \text{ }$
vertices: $\left(h , k \pm a\right)$
foci: $\left(h , k \pm \sqrt{{a}^{2} + {b}^{2}}\right)$

#### Explanation:

Using the given points, write the following equations:

$h = 0 \text{ }$
$k - a = - 3 \sqrt{5} \text{ }$
$k + a = 3 \sqrt{5} \text{ }$
$k - \sqrt{{a}^{2} + {b}^{2}} = - 9 \text{ }$
$k + \sqrt{{a}^{2} + {b}^{2}} = 9 \text{ }$

To obtain the value of k, add equations  and :

$2 k = 0$

$k = 0$

To obtain the value of a, substitute 0 for k into equation :

$a = 3 \sqrt{5}$

Substitute the known values of "k" and "a" into equation 6 and solve for b:

$0 + \sqrt{{\left(3 \sqrt{5}\right)}^{2} + {b}^{2}} = 9$

${b}^{2} + 45 = 81$

${b}^{2} = 36$

$b = 6$

Substitute the known values into equation :

${\left(y - 0\right)}^{2} / {\left(3 \sqrt{5}\right)}^{2} - {\left(x - 0\right)}^{2} / {6}^{2} = 1 \text{ }$

Here is a graph of equation  with vertices and foci: 