Find an equation for the hyperbola whose graph has vertices (+/-5,0) and foci (+/-7,0)

To find the equation of a hyperbola with vertices and foci, we can use the standard form equation:

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

or

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

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

Given:
Vertices: (+/-5,0)
Foci: (+/-7,0)

1. Step 1: Determine the center
The center of the hyperbola is the midpoint between the vertices. In this case, both vertices lie on the x-axis, so the center is at (0,0).

2. Step 2: Determine the distance from the center to the vertices.
The distance from the center to one of the vertices is equal to the value of 'a' in the standard form equation.

The distance from the center to one of the vertices is 5 units.

3. Step 3: Determine the distance from the center to the foci.
The distance from the center to one of the foci is equal to 'c' in the standard form equation.

The distance from the center to one of the foci is 7 units.

4. Step 4: Determine 'b'
We can use the formula c^2 = a^2 + b^2 to find 'b'.

Since a = 5 and c = 7, we can solve for b:

7^2 = 5^2 + b^2
49 = 25 + b^2
b^2 = 49 - 25
b^2 = 24
b = sqrt(24) = 2sqrt(6)

5. Step 5: Write the equation
Since the vertices lie along the x-axis, this is a horizontal hyperbola. Thus, we will use the horizontal form of the equation:

(x-0)^2/5^2 - (y-0)^2/(2sqrt(6))^2 = 1

Simplifying, the equation is:

x^2/25 - y^2/24 = 1

So, the equation of the hyperbola is x^2/25 - y^2/24 = 1.