# How do you write the equation of the hyperbola given Foci: (-6,0),(6,0) and vertices (-4,0), (4,0)?

##### 1 Answer

There are two standard Cartesian forms for the equation of a hyperbola. I will explain how one knows which one to use and how to use it in the explanation.

#### Explanation:

The standard Cartesian form for the equation of a hyperbola with a vertical transverse axis is:

Its vertices are located at the points,

Its foci are located at the points,

The standard Cartesian form for the equation of a hyperbola with a horizontal transverse axis is:

Its vertices are located at the points,

Its foci are located at the points,

Given:

Foci:

Vertices:

Please observe that it is the x coordinates of the vertices and foci that changes, while the y coordinate remains fixed; this matches the the horizontal transverse type, equation [2]. This is how one knows which equation to use. If it were the y coordinates that were changing and the x coordinates were fixed, then it would be equation [1].

We can use

To find the value of h, add equation [4] to equation [5]:

To find the value of a, subtract equation [4] from equation [5]:

Using, h = 0, a = 4, and the point

and then solve for b:

Substitute the values for h, k, a, and b into equation [2]:

Equation [6] is the desired equation.