How do you find the coordinates of the vertices, foci, and the equation of the asymptotes for the hyperbola x^2-y^2=4?

1 Answer
Jun 23, 2018

Vertices: (+-2, 0)
Foci: (+-2sqrt2, 0)
Asymptotes: y=+-x

Explanation:

First, we need to have our equation in standard form:

x^2/4-y^2/4=1

That's better! To get to our answer, we need to know that our hyperbola is a horizontal hyperbola, which means that the vertices and foci will be have the same y-coordinate as the center.

Next, we need to know our the values of a, b, and c. These help us find out the dimensions of our graph. The formula for a horizontal hyperbola is

(x-h)^2/a^2-(y-k)^2/b^2=1

In our case, the h and k values don't exist, so our center is at the origin. a^2 and b^2 =4, so both a and b = 2. To find c, we will use an equation you've probably seen before:

a^2+b^2=c^2
4+4=c^2
8=c^2
2sqrt2=c

To find our vertices and foci, we move out a certain amount of units in both directions: a for the vertices and c for the foci.

Vertices: (+-2, 0)
Foci: (+-2sqrt2, 0)

Finally, let's get the equations of the hyperbola's asymptotes. The equations are y=+-b/a x for a horizontal hyperbola at (0,0). Therefore, our equations are y=+-x.