Focus of a Hyperbola
A hyperbola has two focus points. These points guide its shape. They sit on the transverse axis. The calculator finds them from the center, orientation, and semi-axis lengths. It also gives c, eccentricity, vertices, conjugate endpoints, and asymptote equations.
Why the Focus Matters
The focus points show how the curve opens. They help students graph conic sections with less guesswork. They also support optics, navigation, orbital models, and analytic geometry checks. When the hyperbola is horizontal, the foci move left and right from the center. When it is vertical, they move up and down.
Input Details
Use a positive value for a. This is the semi-transverse axis. Use a positive value for b. This is the semi-conjugate axis. Enter h and k for the center. Choose the orientation that matches your equation. The tool accepts decimals, so measured data can be tested.
Result Meaning
The value c is the distance from the center to each focus. For every standard hyperbola, c squared equals a squared plus b squared. Eccentricity is c divided by a. It is always greater than one for a real hyperbola. Larger eccentricity means a wider opening shape.
Practical Use
After calculation, review the ordered pairs. Copy them into a graphing tool, worksheet, or report. Use the CSV button when you need spreadsheet data. Use the PDF button when you need a printable summary. The example table gives sample values and expected foci for quick checking.
Accuracy Tips
Check units before you start. Keep both axis values in the same unit. Do not enter squared axis values unless your source already lists a and b. If an equation shows a squared term under the denominator, take the square root first. Round final values only after the calculation finishes.
Common Mistakes
Many errors come from switching a and b. Remember that a belongs to the transverse direction. It follows the opening direction. Another error is placing the foci on the wrong axis. Read the plus and minus term carefully. The positive squared term usually shows the opening direction. Finally, avoid early rounding. Small decimal changes can move the focus point and affect graph labels. This helps keep reports consistent and easy to verify later.