Question

How do you find the equation of the hyperbola with center at the origin and vertices `(+-1,0)` and asymptotes `y=+-3x`?

Answer 1

Please see the explanation.

Explanation:

Please observe that the vertices are horizontally oriented, `(-1, 0) and (1,0)`, therefore, the hyperbola is the horizontal transverse axis type . From the reference, the standard Cartesian form for the equation of a hyperbola with a horizontal transverse axis is:

`(x - h)^2/a^2 - (y - k)^2/b^2 = 1" [1]"`

where h and k are the center point `(h,k)`, "a" is the distance from the center to a vertex, and `+-b/a` is the slope of asymptotes.

We are given that the center is the origin so substitute 0 for h and k into equation [1]:

`(x - 0)^2/a^2 - (y - 0)^2/b^2 = 1" [2]"`

We can see that the distance from the center to either vertex is 1, therefore substitute 1 for a into equation [2]:

`(x - 0)^2/1^2 - (y - 0)^2/b^2 = 1" [3]"`

We are given that the slope of the vertices is `+-3`; this allows us to solve for b:

`+-b/a = +-3`

Substitute 1 for a:

`+-b/1 = +-3`

`b = 3`

To substitute 3 for b into equation [3]:

`(x - 0)^2/1^2 - (y - 0)^2/3^2 = 1" [4]"` Answer

You can simplify equation [4] if you like, but I recommend that you leave it in this form.