Find the equation of the hyperbola with the following information:
vertices at (0,8) and (0,-8), foci at (0,9) and (0,-9).
To find the equation of the hyperbola, we first need to determine the center of the hyperbola.
The center of a hyperbola is the midpoint between its two vertices. In this case, the vertices are located at (0, 8) and (0, -8).
To find the midpoint, we use the formula:
Midpoint = [(x₁ + x₂) / 2 , (y₁ + y₂) / 2]
For the x-coordinates, we have:
x₁ = 0 and x₂ = 0
So, (x₁ + x₂) / 2 = (0 + 0) / 2 = 0
For the y-coordinates, we have:
y₁ = 8 and y₂ = -8
So, (y₁ + y₂) / 2 = (8 + (-8)) / 2 = 0
Therefore, the midpoint (center) of the hyperbola is (0, 0).
Next, we need to find the distance between the center and one of the foci to determine the value of 'c', which is the distance between the center and the foci.
For this hyperbola, the foci are located at (0, 9) and (0, -9).
Using the distance formula:
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
For the x-coordinates, we have:
x₁ (center) = 0 and x₂ = 0
So, (x₂ - x₁)² = (0 - 0)² = 0
For the y-coordinates, we have:
y₁ (center) = 0 and y₂ = 9
So, (y₂ - y₁)² = (9 - 0)² = 81
Applying the distance formula:
Distance = √[0 + 81] = √81 = 9
Therefore, the value of 'c' is 9, which represents the distance between the center and the foci.
Now, we can determine the value of 'a', which is the distance between the center and each vertex. In this case, the distance between the center (0, 0) and each vertex (0, 8) or (0, -8) is 8.
Now, we can use the information obtained to write the equation of the hyperbola in standard form:
(x - h)² / a² - (y - k)² / b² = 1
The center is at (h, k) = (0, 0).
The value of 'a' is 8.
The value of 'c' is 9.
Since 'a' is the distance from the center to the vertex, we can write a² = 8² = 64.
Also, we know that c² = a² + b².
Substituting the values, we get:
9² = 8² + b²
81 = 64 + b²
81 - 64 = b²
17 = b²
Now, we have all the values needed to write the equation of the hyperbola:
(x - 0)² / 8² - (y - 0)² / √17² = 1
Simplifying further, we have:
x² / 64 - y² / 17 = 1
Therefore, the equation of the hyperbola is:
x² / 64 - y² / 17 = 1