Find equation of hyperbola with asymptotes slopes of +/-4 and foci of (4,0) and (-2,0)

Question

How do you find the equation of a hyperbola with the slopes of asymptotes of +/-4 and foci of (4,0) and (-2,0)?  I don't know how to find a and b but I was able to find c.

Answer 1

Hyperbola :

A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from two fixed points, called the foci, is constant.

Given :

A hyperbola at the right has foci at (4, 0) and (- 2, 0).

Asymptotes slopes = ± 4.

The standard form of the equation of a hyperbola with center (where a and b are not equals to 0) is (Transverse axis is horizontal) or (Transverse axis is vertical).

The vertices and foci are, respectively a and c units from the center and the relation between a, b and c is .

Since the y - coordinate is constant in the foci, this is a horizontal hyperbola.

The center of the hyperbola lies at the midpoint of its vertices or foci.

So, the center = == (1, 0).

To find the value of c, find the distance between the center (1, 0) and a focus (- 2, 0).

This distance is 3, so .

Answer 2

From the given data, Asymptotes slopes = = ± 4.(Because, this is a horizontal hyperbola).

So, .

Substitute the values of   and in .

.

Substitute the value in .

.

Substitute the values of ,  , = (1, 0) and in .

.

An equation of this hyperbola is  .