Example Data Table
| Foci |
Asymptote Input |
Center |
b/a |
a |
b |
Equation Form |
| (-5, 0), (5, 0) |
-0.75x + y = 0 |
(0, 0) |
0.75 |
4 |
3 |
x²/16 - y²/9 = 1 |
| (0, -10), (0, 10) |
Local slope 0.5 |
(0, 0) |
0.5 |
8.9443 |
4.4721 |
Rotated by 90 degrees |
| (1, 2), (7, 6) |
Angle 35 degrees |
(4, 4) |
0.7002 |
3.7596 |
2.6326 |
Centered rotated form |
Formula Used
The calculator treats the conic as a hyperbola. The center is the midpoint of the foci:
(h, k) = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
The focal distance from center is:
c = distance between foci / 2
The focal axis angle is:
θ = atan2(y₂ - y₁, x₂ - x₁)
If the local asymptote ratio is r = b/a, then:
a = c / √(1 + r²), b = r × a, and e = c / a
Local coordinates are built as:
u = cosθ(x - h) + sinθ(y - k)
v = -sinθ(x - h) + cosθ(y - k)
The final centered equation is:
u²/a² - v²/b² = 1
How to Use This Calculator
- Enter the coordinates of the two foci.
- Select how you want to describe the asymptote.
- Use line coefficients for a full line equation.
- Use local slope when you already know b/a.
- Use angle when the asymptote angle is measured from the focal axis.
- Choose decimal precision for the displayed answer.
- Press calculate to show the result above the form.
- Download the CSV or PDF result for your records.
Understanding the Hyperbola Behind the Calculator
A hyperbola is built from distances. Every point on one branch keeps a constant difference between its distances to two foci. The midpoint of the foci is the center. The line through the foci gives the transverse axis. Once an asymptote direction is known, the calculator can find the opening shape.
Why Foci and an Asymptote Are Enough
The focal distance is called c. It is measured from the center to either focus. A centered hyperbola also has two scale values. The transverse scale is a. The conjugate scale is b. Its asymptote slope in local coordinates is b divided by a. Since c squared equals a squared plus b squared, one valid asymptote direction gives the missing ratio. Then both scales follow directly.
Rotated Equation Support
Many textbook examples use axes aligned with the page. Real problems may be rotated. This tool handles that case. It creates local coordinates u and v around the center. The u axis points from the center toward a focus. The v axis is perpendicular. The equation is then written as u squared over a squared minus v squared over b squared equals one.
Input Checks Improve Accuracy
The calculator checks whether the two foci are distinct. It also checks whether the asymptote data can define a real angle. When a global line is entered, its direction is used. If the line misses the center, the matching parallel asymptote through the center is reported. This prevents a misplaced constant from breaking the geometry.
Using the Results
The result shows the center, focal distance, axis angle, a, b, eccentricity, asymptote equations, local equation, and expanded quadratic form. Students can compare the local equation with the expanded form. Teachers can use the CSV and PDF options for records, assignments, or solution notes.
Practical Study Notes
Always enter foci in the same unit. Keep the asymptote line consistent with the drawing. A small rounding setting is useful for classroom work. More decimal places are better for engineering sketches. If the asymptote angle is very small, the hyperbola becomes narrow. If the angle is large, the branches spread wider. Review each warning before using final values. Save exports for later checking too.
FAQs
1. What does this calculator find?
It finds the equation of a hyperbola from two foci and one asymptote description. It also gives center, axis angle, a, b, eccentricity, asymptote lines, and expanded form.
2. Why are two foci needed?
The two foci define the center and the focal axis. Without them, the calculator cannot know where the hyperbola is centered or how it is rotated.
3. What is the local slope b/a?
It is the asymptote slope measured in axes centered at the hyperbola. The u axis follows the foci, and the v axis is perpendicular to it.
4. Can the hyperbola be rotated?
Yes. The calculator builds a rotated coordinate system from the foci. It then writes both the local equation and the expanded equation in x and y.
5. What happens if the line misses the center?
A true asymptote must pass through the center. If the entered line misses it, the calculator uses the same direction through the center and shows a warning.
6. Why must the asymptote angle be nonzero?
A zero angle would make b equal zero. That is not a valid hyperbola. The angle must also stay below ninety degrees.
7. Is the expanded form always shown?
Yes. After calculation, the tool shows the standard rotated form and the expanded quadratic form. This helps with graphing and algebra checks.
8. Can I save the result?
Yes. Use the CSV download for spreadsheet work. Use the PDF download for a printable solution summary with the main equations and values.