To find the equation of a hyperbola when you have the center, focus, and vertex, you can follow these steps:
- Identify the Center: Let’s denote the center of the hyperbola as (h, k).
- Determine the Orientation: Hyperbolas can open either horizontally or vertically. If the vertices and foci are aligned horizontally, the standard form of the equation is:
(y – k)²/a² – (x – h)²/b² = -1
If they are aligned vertically, the form is:
(x – h)²/a² – (y – k)²/b² = -1 - Find ‘a’ and ‘c’: The distance from the center to a vertex gives you ‘a’, and the distance from the center to a focus gives you ‘c’.
- Calculate ‘b’: Use the relationship c² = a² + b² to solve for ‘b’.
- Write the Equation: Substitute the values of h, k, a, and b back into the equation based on the orientation you determined in step 2.
Example: Suppose the center is (2, 3), the vertex is (2, 5) and the focus is (2, 7). Here ‘a’ (the distance from center to vertex) is 2, and ‘c’ (the distance from center to focus) is 4. Thus, we have:
- Center (h, k): (2, 3)
- Vertex: (2, 5) gives ‘a’ = 2
- Focus: (2, 7) gives ‘c’ = 4
- Calculate ‘b’: c² = a² + b² -> 16 = 4 + b² -> b² = 12
Since the hyperbola is vertical (same x-coordinates), the equation is:
(x – 2)²/12 – (y – 3)²/4 = -1