Calculator Inputs
Formula Used
The parabola is the set of points equally distant from a fixed focus and a fixed directrix.
Distance from (x, y) to focus = Distance from (x, y) to directrix
For focus F(a, b) and directrix Ax + By + C = 0:
√((x - a)² + (y - b)²) = |Ax + By + C| / √(A² + B²)
After squaring and expanding:
(A² + B²)((x - a)² + (y - b)²) - (Ax + By + C)² = 0
This calculator expands the expression into Dx² + Exy + Fy² + Gx + Hy + I = 0. It also computes the vertex, axis, focal parameter, focal length, latus rectum, and graph points.
How to Use This Calculator
- Enter the x and y coordinates of the focus.
- Write the directrix in the form
Ax + By + C = 0. - Enter A, B, and C into the matching fields.
- Select decimal precision for rounded output.
- Adjust graph span and steps if you need a wider plot.
- Press the calculate button.
- Review the equation, vertex, axis, and graph.
- Use CSV or PDF export for records and assignments.
Example Data Table
| Focus | Directrix | Expected Equation | Vertex | Opening |
|---|---|---|---|---|
| (2, 1) | y + 3 = 0 | x² - 4x - 8y - 4 = 0 | (2, -1) | Upward |
| (4, 0) | x + 2 = 0 | y² - 12x + 12 = 0 | (1, 0) | Right |
| (0, 5) | y - 1 = 0 | x² - 8y + 24 = 0 | (0, 3) | Upward |
| (1, 2) | x + y - 4 = 0 | Expanded rotated parabola | Computed by midpoint rule | Along directrix normal |
Understanding Focus and Directrix Equations
What the Calculator Solves
A parabola can be defined without first knowing its standard equation. You only need one focus point and one directrix line. Every point on the curve has the same distance to both. This calculator uses that distance rule directly. It handles horizontal, vertical, and slanted directrices.
Why the Directrix Form Matters
The directrix is entered as Ax + By + C = 0. This form is flexible. It can describe many line positions. For example, y + 3 = 0 creates a horizontal line. The equation x + 2 = 0 creates a vertical line. A line such as x + y - 4 = 0 creates a rotated parabola.
How the Vertex Is Found
The vertex sits halfway between the focus and the directrix. More exactly, it lies on the perpendicular path from the focus to the directrix. The calculator finds the perpendicular foot first. It then takes the midpoint between that foot and the focus. This gives the exact vertex.
General and Rotated Equations
Some parabolas have simple forms, such as x² = 4py. Others include an xy term. That term appears when the axis is rotated. The calculator keeps the full expanded form, so rotated results are not lost. The discriminant also confirms the curve is a parabola.
Practical Uses
This tool is useful for analytic geometry, conic sections, engineering sketches, optics, and classroom checking. It shows intermediate values, not just the final equation. That helps students understand each transformation. The graph also makes the direction and shape easier to inspect.
Export and Review
The CSV export stores coefficients and main results. The PDF export creates a readable report. These options help with homework, notes, and project documentation. Always check that the focus does not lie on the directrix. In that case, no valid parabola exists.
FAQs
1. What does this calculator find?
It finds the equation of a parabola from a given focus point and directrix line. It also shows the vertex, axis, focal parameter, focal length, latus rectum, and graph.
2. Which directrix format should I use?
Use the line format Ax + By + C = 0. For y = -3, enter A = 0, B = 1, and C = 3. For x = -2, enter A = 1, B = 0, and C = 2.
3. Can this handle a slanted directrix?
Yes. A slanted directrix creates a rotated parabola. The result may include an xy term because the curve is not aligned with the coordinate axes.
4. Why is my result not in standard form?
The calculator gives the expanded general form. This form works for horizontal, vertical, and rotated parabolas. Standard form is simpler only when the axis matches the x or y direction.
5. What is the focal parameter?
The focal parameter p is the directed distance from the vertex to the focus. Its sign helps describe the opening direction of the parabola.
6. Why can the focus not lie on the directrix?
A parabola needs a positive distance between the focus and directrix. If the focus lies on the directrix, the required curve becomes invalid.
7. What does the discriminant show?
For a parabola in general conic form, the discriminant B² - 4AC equals zero. This calculator displays that value as a check.
8. Can I save the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a formatted report with the equation and main measurements.