To find the equation of a hyperbola in standard form, we start by identifying key values like the center, vertices, and foci based on the given information.
1. **Identify the Center**: The vertices of the hyperbola are at (0, 2) and since hyperbolas are symmetric, the center is the midpoint between the vertices. The distance between the vertices gives us the length of the transverse axis.
– The distance from (0, 2) to (0, 2) along the y-axis is 2, meaning the center is also at (0, 2) and thus is the midpoint:
Center = (0, 2)
2. **Distance to Vertices (a)**: The distance from the center (0, 2) to each vertex is calculated as follows:
– Distance from (0, 2) to (0, 2) is 0 (obviously) and from (0, 2) to (0, 11) is 9.
– Therefore, a = 9. Since the vertices are on the y-axis, we will use a2 = 92=81.
3. **Distance to Foci (c)**: The foci are given to be at (0, 11), and since we already established the center is (0, 2):
– The distance from (0, 2) to (0, 11) is 9, yielding:
– Therefore, c = 9 as well, so c2 = 92 = 81.
4. **Finding b**: The relationship between a, b, and c in hyperbolas is given by the equation:
c2 = a2 + b2
From this, we can substitute the known values:
81 = 81 + b2
This simplifies to b2 = 0, which tells us that b = 0 in terms of our standard form model.
5. **Standard Form Equation**: The standard form of a vertical hyperbola is given by:
(y – k)2 / a2 – (x – h)2 / b2 = 1
Substituting h = 0, k = 2, a2 = 81, and b2 = 0 into the equation, we arrive at:
(y – 2)2 / 81 – (x – 0)2 / 0 = 1
However, since we cannot have division by zero, it indicates a degenerate case, essentially reducing to a point. Typically, hyperbolas require b to not be zero.
Therefore, while it’s a bit unusual, the standard form of the hyperbola based on the provided vertices and foci relates to:
(y – 2)2 / (1) – (x – 0)2 / (any small positive number) = 1
This means the hyperbola will appear as two vertical lines.