Understanding the Equation of the Hyperbola
To find the equation of a hyperbola with the specified foci and vertices, we start by identifying the characteristics of hyperbolas. A hyperbola is defined by two sets of points that are equidistant from two points called foci. In this case, the foci are located at (5, 0) and (-5, 0), while the vertices are at (4, 0) and (-4, 0).
Step 1: Determine the Center of the Hyperbola
The center of the hyperbola is the midpoint between the vertices or the foci. Since we have vertices at (4, 0) and (-4, 0), the center (h, k) will be at:
Center (h, k) = ((4 + (-4))/2, (0 + 0)/2) = (0, 0)
Step 2: Identify the Values of a and c
The distance from the center to each vertex is represented as ‘a’. The distance from the center to each focus is represented as ‘c’. We can calculate these distances as follows:
- a: The distance from the center (0, 0) to the vertex (4, 0) is 4. Therefore, a = 4.
- c: The distance from the center (0, 0) to the focus (5, 0) is 5. Therefore, c = 5.
Step 3: Calculate b
For hyperbolas, we have the relationship: c2 = a2 + b2. Thus:
52 = 42 + b2
25 = 16 + b2
Solving for b2, we get:
b2 = 25 – 16 = 9
Hence, b = 3.
Step 4: Write the Equation
Since the hyperbola opens horizontally (the foci are aligned along the x-axis), the standard form of the equation for a hyperbola is:
(y – k)2 / b2 – (x – h)2 / a2 = 1
Substituting the Values:
Given h = 0, k = 0, a = 4, and b = 3, the final equation becomes:
(y – 0)2 / 32 – (x – 0)2 / 42 = 1
Or simply:
y2 / 9 – x2 / 16 = 1
Conclusion
Thus, the equation of the hyperbola with foci (5, 0) and vertices (4, 0) is:
y2 / 9 – x2 / 16 = 1