To find the equation of a hyperbola when you have been given the center, focus, and vertex, you’ll follow a series of steps that relate these elements to the general form of a hyperbola’s equation. Here’s how:
Step 1: Identify the Components
A hyperbola can be characterized by its center, foci (plural of focus), vertices, and the distances associated with them. The general forms of a hyperbola are:
- For a horizontal hyperbola: (x – h)²/a² – (y – k)²/b² = 1
- For a vertical hyperbola: (y – k)²/a² – (x – h)²/b² = 1
Here, (h, k) is the center of the hyperbola.
Step 2: Determine the Center
The center (h, k) is the midpoint between the vertices and also between the foci. From the information provided, identify the coordinates of the center. For example, if your focus is at (c, k) and your vertex is at (a, k), then it implies that the center will be at (h, k), where h = (x-coordinate of focus + x-coordinate of vertex) / 2.
Step 3: Calculate Values for a and c
– The distance from the center to a vertex is represented by ‘a’. This is the distance from the center to the vertex. If the vertex is at (h + a, k), then a equals this distance.
– The distance from the center to a focus is represented by ‘c’. If the focus is at (h + c, k), then c is likewise computed. Generally, the relationship between ‘a’, ‘b’, and ‘c’ in a hyperbola is given by:
c² = a² + b²
Thus, if you have ‘a’ and ‘c’, you can find ‘b’ using the above relationship.
Step 4: Write the Equation
Now that you have identified ‘h’, ‘k’, ‘a’, and ‘b’, plug them into the general form based on whether the hyperbola is horizontal or vertical:
– For a horizontal hyperbola, the equation would be:
(x – h)²/a² – (y – k)²/b² = 1
– For a vertical hyperbola, the equation would be:
(y – k)²/a² – (x – h)²/b² = 1
Example
Suppose the center of the hyperbola is at (2, 3), a vertex is at (5, 3), and a focus is at (6, 3). In this case:
- Center (h, k) = (2, 3)
- Distance a = 5 – 2 = 3
- Distance c = 6 – 2 = 4
Now, we can find ‘b’:
c² = a² + b² => 4² = 3² + b² => 16 = 9 + b² => b² = 7
Since the hyperbola opens horizontally:
The equation will be:
(x – 2)²/9 – (y – 3)²/7 = 1
In conclusion, with the information about the center, focus, and vertex, you can successfully derive the equation of a hyperbola. Just remember to identify whether it opens horizontally or vertically, determine your distances accurately, and plug them into the correct equation format.