hyperbola conic sections

Question

need help walking through the steps

this hyperbola is centered at the origin find its equation

foci: (-4,0) and (4,0)

vertices: (-3,0) and (3,0)

Answer 1

It helps to graph the vertices and foci, because it becomes evident that the hyperbola is running horizontally. The general equation of a horizontal hyperbola centered at (h, k) is:

(x-h)^2 (y-k)^2

--------- -- --------- = 1 (the y term is the negative term for a horizontal hyperbola)

a^2 b^2

Since the hyperbola is centered about the origin (h,k = 0), the equation becomes:

x^2 y^2

--------- -- --------- = 1

a^2 b^2

Since a is the distance from the center of the hyperbola to each vertex, a = 3

To find b, we need to use the relationship that c^2 = a^2 + b^2, where c is the distance from the center of the hyperbola to each focus.

The equation then becomes 4^2 = 3^2 + b^2

16 = 9 + b^2

b^2 = 7

Substituting a^2 and b^2 into the general equation, this gives:

x^2 y^2

--------- -- --------- = 1

9 7