Write an equation of the hyperbola with vertices at (–7, 0) and (7, 0) and foci at (–9, 0) and (9, 0)
To find the equation of a hyperbola given the vertices and foci, we can start by determining the center of the hyperbola. The center is the midpoint between the vertices, which we can find using the following formula:
Center = ((x₁ + x₂)/2, (y₁ + y₂)/2)
In this case, the coordinates of the vertices are (-7, 0) and (7, 0). So, applying the formula:
Center = ((-7 + 7)/2, (0 + 0)/2)
Center = (0, 0)
The center of the hyperbola is at the origin (0, 0).
Next, we can determine the distance between the center and one of the foci. We know that the distance between the center and each focus is equal to the distance between the center and each vertex. This distance is known as the distance "c".
In this case, using the coordinates of the center (0, 0) and one focus (-9, 0):
c = distance between (0, 0) and (-9, 0)
c = |-9 - 0|
c = 9
Now, since we have the center (0, 0) and the value of "c" as 9, we can determine another important value known as "a". "a" represents the distance between the center and each vertex, which we already know is 7. Thus, we have:
a = 7
The last step is determining the value of "b". The relationship between "a", "b", and "c" is defined by the equation of a hyperbola:
c² = a² + b²
Substituting the known values:
9² = 7² + b²
81 = 49 + b²
b² = 81 - 49
b² = 32
b = √32
Now, we have all the necessary values to write the equation of the hyperbola in standard form. The standard form for a hyperbola with a horizontal transverse axis is:
(x - h)²/a² - (y - k)²/b² = 1
In this case, the center (h, k) is (0, 0), "a" is 7, and "b" is √32.
Therefore, the equation of the hyperbola is:
(x - 0)²/7² - (y - 0)²/ (√32)² = 1
Simplifying:
x²/49 - y²/32 = 1
Thus, the equation of the hyperbola is x²/49 - y²/32 = 1.