To find the equation of a hyperbola in standard form, we need to analyze the given information about the vertices and asymptotes. In this case, we have:
- Vertices at (0, 8)
- Asymptotes at y = 1/2 x
Step 1: Identify the Center of the Hyperbola
The vertices are located at (0, 8). Since a hyperbola has two vertices, we can determine the center by finding the midpoint of the vertices. Assuming the other vertex is at (0, 8 + 2a), where ‘a’ is the distance from the center to each vertex:
Let’s find the coordinates of the second vertex. Since both vertices are aligned vertically, we can write:
- First Vertex: (0, 8)
- Second Vertex: (0, 8 – 2a) = (0, 8 – 2) = (0, 6)
The center is the average of the y-values of the vertices:
Center: (0, (8 + 6)/2) = (0, 7)
Step 2: Determining ‘a’ and ‘b’
From the vertices, we can see:
- Distance ‘a’ from the center to each vertex is 1 (because the vertices are 2 units apart, which gives us a = 1).
Next, we use the slopes of the asymptotes, which are given by the equation:
- y = 1/2 x
This format indicates that the slopes of the asymptotes are ±b/a. Setting this equal to 1/2 gives:
- 1/2 = b/a
Since we found that ‘a’ is 1, we can solve for ‘b’:
- b = 1/2 * 1 = 1/2
Step 3: Write the Standard Form Equation
The standard form of a vertical hyperbola is:
(y - k)²/a² - (x - h)²/b² = 1
Where (h, k) is the center. Substituting the values we have:
- h = 0
- k = 7
- a = 1
- b = 1/2
Plugging these into the standard form gives:
(y - 7)²/1² - (x - 0)²/(1/2)² = 1,
This simplifies to:
(y - 7)² - 4x² = 1.
Final Equation
Thus, the equation of the hyperbola in standard form is:
(y - 7)² - 4x² = 1.
By following these steps, we derived the standard form equation for the hyperbola based on the given vertices and asymptotes.