Write an equation for the hyperbola. It's center is (0,0).
Foci:(-4,0), (4,0)
vertices:(-3,0), (3,0)
To write the equation for a hyperbola with a center at (0,0), foci at (-4,0) and (4,0), and vertices at (-3,0) and (3,0), we need to determine the values of a, b, and c.
First, let's find the value of c, which is the distance from the center to each focus. In this case, c = 4.
Next, we can find the value of a, which is the distance from the center to each vertex. In this case, a = 3.
The equation for a hyperbola with horizontal transverse axis (center at the origin) is given by:
(x^2 / a^2) - (y^2 / b^2) = 1
Since the center is at (0,0) and the transverse axis is horizontal, we can substitute the known values of a^2 = 3^2 = 9 and c^2 = 4^2 = 16 into the equation:
(x^2 / 9) - (y^2 / b^2) = 1
Now, we need to find the value of b^2. The relationship between a, b, and c for a hyperbola is given by:
c^2 = a^2 + b^2
Substituting the known values:
16 = 9 + b^2
Rearranging the equation:
b^2 = 16 - 9
b^2 = 7
Therefore, the equation for the hyperbola is:
(x^2 / 9) - (y^2 / 7) = 1