Calculator Form
Formula Used
Horizontal hyperbola: ((x - h)² / a²) - ((y - k)² / b²) = 1
Vertical hyperbola: ((y - k)² / a²) - ((x - h)² / b²) = 1
Focal relation: c² = a² + b²
Horizontal vertices: (h - a, k), (h + a, k)
Horizontal foci: (h - c, k), (h + c, k)
Vertical vertices: (h, k - a), (h, k + a)
Vertical foci: (h, k - c), (h, k + c)
Eccentricity: e = c / a
Directrix offset: a² / c
Latus rectum length: 2b² / a
How to Use This Calculator
- Enter the center coordinates as h and k.
- Select whether the hyperbola is horizontal or vertical.
- Choose whether your inputs are axis lengths or squared denominators.
- Enter the a and b values, or enter a² and b².
- Choose the number of decimal places.
- Press the calculate button.
- Review the result above the form.
- Use CSV or PDF download for saving the result.
Example Data Table
| Center | Orientation | a | b | c | Vertices | Foci |
|---|---|---|---|---|---|---|
| (0, 0) | Horizontal | 5 | 3 | 5.8310 | (-5, 0), (5, 0) | (-5.8310, 0), (5.8310, 0) |
| (2, -1) | Vertical | 4 | 6 | 7.2111 | (2, -5), (2, 3) | (2, -8.2111), (2, 6.2111) |
| (-3, 2) | Horizontal | 7 | 2 | 7.2801 | (-10, 2), (4, 2) | (-10.2801, 2), (4.2801, 2) |
About the Vertices and Foci of a Hyperbola
A hyperbola is a conic section with two open branches. It appears in analytic geometry, optics, navigation, and many engineering models. The most important points are the center, vertices, and foci. The center is the midpoint of the main axis. The vertices sit on the transverse axis. The foci sit farther from the center and control the curve shape.
Why These Points Matter
Vertices show the nearest points of both branches. Foci help define the distance relationship of the hyperbola. For any point on the curve, the absolute difference of distances from the two foci is constant. That constant equals two times a. This makes the foci useful when drawing, checking, or converting equations.
Horizontal and Vertical Forms
A horizontal hyperbola opens left and right. Its x term is positive in standard form. A vertical hyperbola opens up and down. Its y term is positive. This calculator lets you choose the orientation directly. It then places vertices, foci, directrices, and asymptotes in the correct direction.
Advanced Output
The tool also returns c, eccentricity, semi-conjugate length, focal distance, latus rectum length, and asymptote equations. These values help students verify homework steps. They also help teachers create examples. CSV export is useful for records. PDF export is useful for printing or sharing.
Practical Use
Use exact positive values for a and b. If your equation shows squared denominators, choose the squared input mode. If your equation already gives semi-axis lengths, choose direct input mode. Enter the center as h and k. Pick horizontal or vertical orientation. Then press calculate. The result appears above the form for quick review.
Checking Your Equation
Before using the tool, rewrite the equation in standard form. Make sure the right side equals one. Move constants carefully. Divide every term when needed. The positive squared term decides the opening direction. The denominator below that positive term gives a squared. The other denominator gives b squared. Small algebra mistakes can move every point, so review signs.
Interpreting Results
After calculation, compare the vertices with the foci. Foci must be farther from the center than vertices. The eccentricity must be greater than one. These checks reveal most input errors quickly and reliably.
FAQs
What are vertices of a hyperbola?
Vertices are the two closest branch points to the center. They lie on the transverse axis. Their position depends on a and the chosen orientation.
What are foci of a hyperbola?
Foci are two fixed points inside the hyperbola layout. They sit farther from the center than the vertices. They help define the curve.
How is c calculated?
For every standard hyperbola, c² equals a² plus b². So c is the square root of a² + b².
How do I know if the hyperbola is horizontal?
A horizontal hyperbola has the x expression first and positive. It opens left and right. Its vertices change along the x direction.
How do I know if the hyperbola is vertical?
A vertical hyperbola has the y expression first and positive. It opens upward and downward. Its vertices change along the y direction.
Can I enter squared denominators?
Yes. Choose the squared input mode. Then enter a² and b² directly from the standard equation denominators.
Why is eccentricity greater than one?
For a hyperbola, c is always greater than a. Since eccentricity equals c divided by a, the value is always greater than one.
What export options are included?
The calculator includes CSV and PDF download buttons. Use CSV for spreadsheet records. Use PDF for printing, sharing, or saving results.