Equation of Hyperbola Calculator
Formula Used
Horizontal hyperbola:
((x - h)² / a²) - ((y - k)² / b²) = 1
Vertical hyperbola:
((y - k)² / a²) - ((x - h)² / b²) = 1
Core values:
c = √(a² + b²)
e = c / a
Latus rectum length = 2b² / a
Transverse axis length = 2a
Conjugate axis length = 2b
How to Use This Calculator
Enter the center values h and k. These values move the hyperbola left, right, up, or down.
Enter a for the semi-transverse axis. Enter b for the semi-conjugate axis.
Choose horizontal when the branches open left and right. Choose vertical when the branches open upward and downward.
Press Calculate to view the standard equation, general equation, foci, vertices, asymptotes, directrices, and related measurements.
Use the CSV or PDF buttons to save the calculation result.
Example Data Table
| Center h | Center k | a | b | Orientation | Standard Equation |
|---|---|---|---|---|---|
| 0 | 0 | 5 | 3 | Horizontal | x² / 25 - y² / 9 = 1 |
| 2 | -1 | 4 | 6 | Vertical | ((y + 1)² / 16) - ((x - 2)² / 36) = 1 |
| -3 | 4 | 7 | 2 | Horizontal | ((x + 3)² / 49) - ((y - 4)² / 4) = 1 |
Understanding the Equation of a Hyperbola
What This Calculator Does
A hyperbola is a conic curve with two separate branches. It appears when a plane cuts both halves of a double cone. This calculator builds the standard equation from the center, axis size, and opening direction. It also gives key curve properties. These include vertices, foci, asymptotes, directrices, eccentricity, and axis lengths.
Why the Center Matters
The center is written as h, k. It is the midpoint between the two vertices. It also sits halfway between the two foci. When h changes, the curve moves sideways. When k changes, the curve moves upward or downward. The shape stays the same unless a or b changes.
Horizontal and Vertical Forms
A horizontal hyperbola opens left and right. Its x term is positive in standard form. A vertical hyperbola opens upward and downward. Its y term is positive in standard form. This sign pattern is important. It tells you the direction of the transverse axis.
Important Measurements
The value a measures from the center to each vertex. The value b controls the conjugate axis and asymptote slope. The value c measures from the center to each focus. For every hyperbola, c squared equals a squared plus b squared. Because of this, c is always larger than a.
Using Results in Graphing
Start graphing with the center. Then mark the vertices. Next, draw a guide rectangle using a and b. The asymptotes pass through the rectangle corners and the center. The curve gets close to these lines but does not touch them. This makes the equation easier to check and draw.
Advanced Output
The general equation is useful in algebra work. The standard equation is better for graphing. The eccentricity describes how open the hyperbola is. The latus rectum helps describe focal width. Together, these outputs give a fuller picture of the curve.
FAQs
1. What is a hyperbola?
A hyperbola is a conic section with two open branches. It has a center, vertices, foci, asymptotes, and two axes.
2. What is the standard equation of a hyperbola?
The standard equation depends on direction. Horizontal form has the x term first. Vertical form has the y term first.
3. What does a mean in a hyperbola?
The value a is the distance from the center to each vertex. It also defines half of the transverse axis.
4. What does b mean in a hyperbola?
The value b defines the semi-conjugate axis. It also helps set the slope of each asymptote.
5. How are foci calculated?
First calculate c using c = √(a² + b²). Then place each focus c units from the center along the transverse axis.
6. What are asymptotes?
Asymptotes are guide lines. The branches approach them but do not cross them in the standard graphing model.
7. What is eccentricity?
Eccentricity equals c divided by a. For a hyperbola, eccentricity is always greater than one.
8. Can I download the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple printable report.