To find an equation in standard form for the hyperbola with vertices at (0, 8) and asymptotes defined by the equations y = (1/2)x, we first need to analyze the given information.
The vertices at (0, 8) suggest that the center of the hyperbola is at (0, 8), and since the vertices indicate a vertical hyperbola, we can begin to form the standard equation. The general form for the equation of a vertical hyperbola centered at (h, k) is:
(y – k)²/a² – (x – h)²/b² = 1
In our case, h = 0 and k = 8, so we have:
(y – 8)²/a² – x²/b² = 1
Next, we need to find the value of a, which is the distance from the center to each vertex. Since the vertices are at (0, 8) and we assume other vertex at (0, 8 + 2a), we can find the value of ‘a’ knowing the vertices. The distance from the center to the vertices (moving vertically) is determined from the hyperbola’s configuration.
Now, we also have the slopes of the asymptotes. For a vertical hyperbola, the slopes of the asymptotes are given by ±a/b. The asymptotes y = (1/2)x indicate that the positive slope is 1/2. Therefore:
a/b = 1/2
Next, we know that the distance from a center to a vertex (in this case is a) is 2 units due to the positions of the vertices (assuming the vertices to be at (0, 9) and (0, 7) making it a = 1). We can then substitute a = 1 into the equation a/b = 1/2:
1/b = 1/2, thus b = 2.
Now, we have found values for a and b. Therefore:
- a = 1
- b = 2
Substituting ‘a’ and ‘b’ into the standard form equation gives us:
(y – 8)²/1² – x²/2² = 1
Which simplifies to:
(y – 8)² – x²/4 = 1
This is the equation in standard form for the hyperbola with the specified vertices and asymptotes. Therefore, the final equation of the hyperbola is:
(y – 8)² – (x²/4) = 1