Question

A hyperbola has vertices (4,0) and one focus (5,0). What is the standard-form equation of the hyperbola?

Answer 1

Since the vertices are on the x-axis, we know that this is a horizontal hyperbola. The distance from the center to each vertex is 4 units, so the center of the hyperbola is at (4, 0). The distance from the center to one focus is 5 units. We can use this information to find c, the distance from the center to the focus:

c² = a² + b²
5² = 4² + b²
25 = 16 + b²
b² = 9
b = 3

The standard form equation for a horizontal hyperbola with center at (h, k) and vertices a units to the left and right of the center is:

(x - h)² / a² - (y - k)² / b² = 1

Plugging in our values, we get:

(x - 4)² / 16 - y² / 9 = 1

So the standard-form equation of the hyperbola is:

(x - 4)² / 16 - y² / 9 = 1