Question

A hyperbola has vertices (±5, 0) and one focus (6, 0). What is the standard-form equation of the hyperbola?..

Answer 1

To find the standard form equation of the hyperbola, we can use the formula for the distance between the center of the hyperbola and the vertices:

c^2 = a^2 + b^2

where a is the distance between the center and a vertex, b is the distance between the center and the foci, and c is the distance between the center and a focus.

In this case, the distance between the center and the vertices is a = 5, the distance between the center and the focus is c = 6, and the distance between the center and the co-vertex is b.

Since the distance between the center and the focus is c = 6, we have:

c^2 = a^2 + b^2
6^2 = 5^2 + b^2
36 = 25 + b^2
b^2 = 11

Now that we have a, b, and c, we can write the standard form equation of the hyperbola:

(x-h)^2/a^2 - (y-k)^2/b^2 = 1

where (h, k) is the center of the hyperbola.

Since the hyperbola is centered at (0, 0), the equation becomes:

x^2/25 - y^2/11 = 1

Therefore, the standard form equation of the hyperbola is:

x^2/25 - y^2/11 = 1.