Hyperbola Calculator
Center, vertices, foci, eccentricity and asymptotes from a, b and the center.
Enter your values
Choose one of: Horizontal — opens left/right, Vertical — opens up/down.
Enter the a (distance center → vertex) you are working with.
Enter the b (conjugate axis) you are working with.
Enter the center h (x) you are working with.
Enter the center k (y) you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- Standard equation
- (x − 0)²/9 − (y − 0)²/16 = 1
- Center
- (0, 0)
- Vertices
- (-3, 0) and (3, 0)
- Foci
- (-5, 0) and (5, 0)
- c
- 5
- Eccentricity e = c/a
- 1.666667
- Latus rectum 2b²/a
- 10.666667
- Asymptotes
- y − 0 = ±1.333333(x − 0)
Working
- 1c² = a² + b² = 9 + 16 = 25
- 2c = 5
- 3e = c / a = 1.666667 (always greater than 1 for a hyperbola)
Quick answer
A hyperbola is the set of points whose distances to two foci differ by a constant. For a horizontal hyperbola (x−h)²/a² − (y−k)²/b² = 1, c² = a² + b², the foci sit c units from the center and the asymptotes have slope ±b/a.
(x−h)²/a² − (y−k)²/b² = 1, c² = a² + b²
At a glance
| What it does | Center, vertices, foci, eccentricity and asymptotes from a, b and the center. |
|---|---|
| Category | Math |
| Inputs needed | Orientation, a (distance center → vertex), b (conjugate axis), Center h (x), Center k (y) |
| Main output | Standard equation |
| Formula | (x−h)²/a² − (y−k)²/b² = 1, c² = a² + b² |
| Cost | Free — no sign-up, no download |
How to use the Hyperbola Calculator
- 1Pick the orientation of the transverse axis.
- 2Enter a, b and the center coordinates.
- 3Read the vertices, foci, eccentricity and asymptote equations.
Inputs explained
- Orientation
- Choose one of: Horizontal — opens left/right, Vertical — opens up/down.
- a (distance center → vertex)
- Enter the a (distance center → vertex) you are working with.
- b (conjugate axis)
- Enter the b (conjugate axis) you are working with.
- Center h (x)
- Enter the center h (x) you are working with.
- Center k (y)
- Enter the center k (y) you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- OrientationHorizontal — opens left/right
- a (distance center → vertex)3
- b (conjugate axis)4
- Center h (x)0
- Center k (y)0
Results
- Standard equation(x − 0)²/9 − (y − 0)²/16 = 1
- Center(0, 0)
- Vertices(-3, 0) and (3, 0)
- Foci(-5, 0) and (5, 0)
- c5
- Eccentricity e = c/a1.666667
- Latus rectum 2b²/a10.666667
- Asymptotesy − 0 = ±1.333333(x − 0)
- c² = a² + b² = 9 + 16 = 25
- c = 5
- e = c / a = 1.666667 (always greater than 1 for a hyperbola)
Frequently asked questions
How do you find the foci of a hyperbola?
Compute c = √(a² + b²), then move c units from the center along the transverse axis in both directions.
What is the eccentricity of a hyperbola?
e = c / a, which is always greater than 1. Larger eccentricity means the branches open more sharply.
What do I need to enter into the Hyperbola Calculator?
Just 5 values: orientation, a (distance center → vertex), b (conjugate axis), center h (x), center k (y). Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Hyperbola Calculator work out the answer?
A hyperbola is the set of points whose distances to two foci differ by a constant. For a horizontal hyperbola (x−h)²/a² − (y−k)²/b² = 1, c² = a² + b², the foci sit c units from the center and the asymptotes have slope ±b/a. It applies the formula (x−h)²/a² − (y−k)²/b² = 1, c² = a² + b² and shows the working so you can check each step by hand.
Is the Hyperbola Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.