Calculator Form
Example Data Table
| Case | Center | a | b | Orientation | Foci | Directrices |
|---|---|---|---|---|---|---|
| Example 1 | (0, 0) | 5 | 3 | Horizontal | (-4, 0), (4, 0) | x = -6.25, x = 6.25 |
| Example 2 | (2, -1) | 5 | 4 | Vertical | (2, -4), (2, 2) | y = -9.333333, y = 7.333333 |
| Example 3 | (-3, 1.5) | 6 | 2 | Horizontal | (-8.656854, 1.5), (2.656854, 1.5) | x = -9.363961, x = 3.363961 |
Formula Used
For a standard ellipse, the larger semi-axis is a and the smaller semi-axis is b.
The linear eccentricity is c = √(a² - b²).
The eccentricity is e = c / a.
For a horizontal ellipse, the foci are (h ± c, k).
For a vertical ellipse, the foci are (h, k ± c).
The directrices are x = h ± a/e for a horizontal ellipse.
The directrices are y = k ± a/e for a vertical ellipse.
The standard equation is ((x - h)² / a²) + ((y - k)² / b²) = 1 for a horizontal major axis.
It becomes ((x - h)² / b²) + ((y - k)² / a²) = 1 for a vertical major axis.
How to Use This Calculator
Choose whether you want to enter semi-axis values or full axis lengths.
Enter the ellipse center as h and k.
Select whether the major axis is horizontal or vertical.
Type the axis values carefully. The major value must be larger.
Press the calculate button to show the focus points and directrix lines.
Review the vertices, co-vertices, eccentricity, area, and equation.
Use the export buttons to save your result table as CSV or PDF.
Ellipse Focus and Directrix Guide
Understand the ellipse clearly
An ellipse is a smooth closed curve. It has two fixed points called foci. Every point on the curve keeps the same sum of distances to those foci. This property helps define the shape. The directrix gives another useful definition. A point on the ellipse has a distance ratio to the focus and directrix equal to the eccentricity. That ratio stays constant. Because of this, focus and directrix data matter in coordinate geometry.
Why students calculate focus and directrix
Students often know the center and axis lengths first. From there, they need the foci, directrices, vertices, and equation. These values appear in algebra, analytic geometry, conic section homework, and exam questions. A good ellipse focus and directrix calculator removes repetitive work. It also reduces sign mistakes. This is helpful when the center is shifted or the major axis is vertical.
What this calculator returns
This tool computes the center, semi-major axis, semi-minor axis, linear eccentricity, and eccentricity. It also shows both focus points and both directrix equations. The result includes vertices and co-vertices. The standard ellipse equation is displayed in shifted form. Extra outputs include area, latus rectum length, and an approximate circumference. These added measures make the tool useful for class practice and revision sheets.
How the formulas connect
The main relation is c² = a² - b². Once c is known, the focus locations follow from the ellipse orientation. Eccentricity is e = c/a. The directrix distance is found with a/e, which is also a²/c. Horizontal ellipses use x-based directrix equations. Vertical ellipses use y-based directrix equations. When you understand these links, conic section problems become easier to solve and explain.
FAQs
1. What is the focus of an ellipse?
The focus is a fixed point inside the ellipse. Every point on the ellipse has a distance relationship with the foci. Two foci always exist for a true ellipse.
2. What is a directrix in an ellipse?
A directrix is a fixed line used with a focus. The ratio of a point’s distance from the focus to its distance from the directrix equals the eccentricity.
3. Why must the major axis be larger?
The major axis defines the longest diameter of the ellipse. If both semi-axis values are equal, the graph becomes a circle and the standard ellipse focus-directrix setup changes.
4. How do I know if the ellipse is horizontal or vertical?
If the longer radius runs left to right, the ellipse is horizontal. If the longer radius runs up and down, the ellipse is vertical.
5. Can I enter full axis lengths instead of semi-axis values?
Yes. This calculator lets you choose full major and minor lengths. It automatically converts them into semi-major and semi-minor values before solving.
6. What is eccentricity?
Eccentricity measures how stretched the ellipse is. It equals c divided by a. Its value stays between zero and one for every ellipse.
7. Are the directrices always outside the ellipse?
Yes. For a standard ellipse, each directrix lies outside the curve. Their positions depend on the center, the major axis, and the eccentricity.
8. Can this calculator help with homework checking?
Yes. It is useful for checking algebra steps, shifted equations, focus positions, and directrix lines. It also helps verify orientation and axis values quickly.