Formula Used
The calculator first checks whether the foci and vertices have the same midpoint. That midpoint is the center, written as (h, k).
The semi-transverse axis is a = distance between vertices / 2. The focal distance is c = distance between foci / 2.
For every valid hyperbola, c must be greater than a. The semi-conjugate value comes from b² = c² - a².
The standard local form is x′² / a² - y′² / b² = 1. If the axis is vertical, the local axes rotate accordingly.
The eccentricity is e = c / a. The directrix distance from the center is a² / c. The latus rectum length is 2b² / a.
How To Use This Calculator
- Enter the coordinates of both foci.
- Enter the coordinates of both vertices.
- Choose decimal precision and tolerance if needed.
- Enter a test point only when you want point checking.
- Press Calculate to show the result above the form.
- Use the CSV or PDF buttons to save the current calculation.
Example Data Table
| Case |
Foci |
Vertices |
Center |
a |
c |
b² |
Equation |
| Horizontal |
(-5, 0), (5, 0) |
(-3, 0), (3, 0) |
(0, 0) |
3 |
5 |
16 |
x² / 9 - y² / 16 = 1 |
| Vertical |
(0, -10), (0, 10) |
(0, -6), (0, 6) |
(0, 0) |
6 |
10 |
64 |
y² / 36 - x² / 64 = 1 |
| Shifted |
(2, -6), (2, 10) |
(2, -2), (2, 6) |
(2, 2) |
4 |
8 |
48 |
(y - 2)² / 16 - (x - 2)² / 48 = 1 |
Understanding The Hyperbola From Given Points
A hyperbola becomes easier to read when its foci and vertices are known. The vertices mark the closest points on both branches. The foci control the shape and the constant distance rule. This calculator first finds the shared center. It then measures the transverse semi axis. That value is called a. It also measures the focal distance. That value is called c. A valid hyperbola needs c greater than a. The calculator then solves b squared from c squared minus a squared.
Why Foci And Vertices Matter
These four points define the main geometry. The midpoint of the vertices gives the center. The midpoint of the foci should match it. The vertices and foci should lie on one straight line. When they do, the calculator can build a local coordinate system. It works for horizontal, vertical, and rotated hyperbolas. This makes the tool useful for graphing, analytic geometry, and classroom checking.
What The Result Shows
The result includes the center, a, b, c, eccentricity, and axis length. It also gives the standard equation. For rotated inputs, it gives a rotated equation using local x prime and y prime. The expanded conic form is also shown. You can review asymptote lines, directrices, conjugate endpoints, and latus rectum length. These values help you sketch the branches with better accuracy.
Checking A Point
An optional point test is included. Enter any point, and the calculator evaluates the left side of the hyperbola equation. A value near one means the point is on the curve. A larger value sits outside the opening region. This is useful when checking plotted points or homework answers.
Practical Uses
Use this calculator when a problem gives coordinates instead of an equation. It reduces repeated work and shows each value. The export buttons help save the calculation for worksheets, notes, or reports. You can compare your values with the example table before entering new data. Teachers can show why invalid point sets fail. Students can change one coordinate and compare new geometry. Watch eccentricity, directrix distance, and asymptote slope respond. This supports fast exploration without losing the geometric idea. Try examples before saving work. Use the exports when you need a clean record.
FAQs
What does this calculator find?
It finds the hyperbola center, equation, axis values, eccentricity, asymptotes, directrices, and related geometry from two foci and two vertices.
What points are required?
You need two foci and two vertices. Their midpoints must match, and all four points must lie on the same transverse axis.
Why must c be greater than a?
For a hyperbola, the foci sit farther from the center than the vertices. That means c must be greater than a.
Can it handle vertical hyperbolas?
Yes. It detects vertical input and writes the equation with the y term as the positive transverse term.
Can it handle rotated hyperbolas?
Yes. It builds local x prime and y prime axes, then gives the rotated equation and expanded conic form.
What does the point test mean?
The point test evaluates the equation. A value near one means the point lies on the hyperbola within your selected tolerance.
What is the tolerance field for?
Tolerance controls how strictly the calculator checks midpoint matching, collinearity, and optional point testing.
What do the export buttons save?
The CSV and PDF buttons save the computed equation, center, axes, eccentricity, asymptotes, directrices, and point test result.