Pre Calculus

Question

Given the center, a vertex, and one focus, find an equation for the hyperbola: center: (4, −2); vertex (4, 2); one focus (4, -2+4√5)

Answer 1

Leslie,

 

The best way to do this is to draw the three points you are given and go from there.

 

Notice that the three points all have the same x-coordinate, which means they are on the vertical line x = 4. This also means that the transverse axis is this same vertical line.

 

The value of a = 4, since this is the distance from the center to a vertex.

The value of c = 4√5, since this is the distance from the center to a focus.

 

You can find b using the formula c² = a² + b². This formula only applies to hyperbolas. You will get the value b = 8.

 

The center is (4,-2), so k = 4, k = -2.

 

The standard form of a hyperbola with center (h,k) and vertical transverse axis is

 

(y - k)²/a² - (x - h)²/b² = 1.

 

Plugging in what you know at this point, you will get the answer.

 

If you need more help with this, please post a reply.