To find the equation of the hyperbola, we start by identifying the key components from the information given.
The vertices of the hyperbola are at points (0, 4) and since they are vertically aligned, we can determine their coordinates as:
- Vertex 1: (0, 4)
- Vertex 2: (0, -4)
The distance between the center of the hyperbola and each vertex is denoted by ‘a’. Since the vertices are given as (0, 4) and through symmetry, we can conclude:
- Center: (0, 0)
- a = 4
The foci are located at (0, 5). The distance from the center to each focus is denoted by ‘c’. Here, we observe that:
- c = 5
To find ‘b’, we use the relationship between ‘a’, ‘b’, and ‘c’ in a hyperbola, given by the equation:
c2 = a2 + b2
Substituting our values:
- 52 = 42 + b2
- 25 = 16 + b2
- b2 = 25 – 16
- b2 = 9
- b = 3
Now that we have a and b, we can write the equation of the hyperbola in standard form. The standard form for a hyperbola that opens vertically is:
\( \frac{x^2}{b^2} – \frac{y^2}{a^2} = -1 \)
Substituting ‘a’ and ‘b’ into the equation gives us:
\( \frac{x^2}{9} – \frac{y^2}{16} = -1 \)
Thus, the equation in standard form for the hyperbola with the specified vertices and foci is:
\( \frac{x^2}{9} – \frac{y^2}{16} = -1 \)