Understanding a Hyperbola: A Guide to Finding Its Equation

When working with conic sections, one of the key elements to understanding a hyperbola is determining its equation based on given foci and vertices. This article delves into the step-by-step process of finding the equation of a hyperbola using a specific example.

Step-by-Step Guide to Finding the Hyperbola Equation

The process of determining the equation of a hyperbola involves several key steps: identifying the center, determining the orientation, calculating the parameters a and c, and then finding b. Once all these parameters are known, you can plug them into the standard form of the hyperbola equation. Here is a detailed walkthrough using a sample problem.

Step 1: Identify the Center

The first step in finding the equation of a hyperbola is determining its center. The center is the midpoint of the segment connecting the foci. In this example, the foci are at (-4, 2) and (2, 2).

Center (frac{-4 2}{2}, frac{2 2}{2}) (-1, 2)

Step 2: Determine the Orientation

The orientation of a hyperbola can be vertical or horizontal. In this case, since both the foci and the vertices have the same y-coordinate (2), the hyperbola opens horizontally.

Step 3: Calculate a and c

The next step involves calculating b and c. The vertices are at (0, 2) and (-2, 2). The distance between the center (-1, 2) and a vertex (0, 2) gives us a:

a |0 - (-1)| 1

The distance c is the distance from the center (-1, 2) to a focus (-4, 2):

c |-4 - (-1)| 3

Step 4: Calculate b

The relationship between a, b, and c in hyperbolas is given by the equation: c2 a2 b2. Substituting the known values:

32 12 b2
9 1 b2
b2 8
b sqrt{8} 2sqrt{2}

Step 5: Write the Equation

The standard form of the equation of a horizontally oriented hyperbola is:

frac{(y - k2)}{b2} - frac{(x - h2)}{a2} 1

Where h and k are the center. Substituting h -1, k 2, a2 1, and b2 8:

frac{(y - 22)}{8} - frac{(x 2)}{1} 1

Final Equation: Therefore, the equation of the hyperbola is:

frac{(y - 22)}{8} - x 2 1

Conclusion

By following these steps, you can determine the equation of a hyperbola when given its foci and vertices. This method is crucial for understanding and working with conic sections in various mathematical contexts.

References

MathIsFun: Conic Sections - Hyperbola Khan Academy: Hyperbola Equation Cuemath: Hyperbola