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

Answer:

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#.