Explore ellipses, parabolas, hyperbolas, and circles by adjusting parameters. Shows foci, directrix, eccentricity, and equations.
Choose a conic family and vary its defining parameters to see the curve, equation, foci, directrix or directrices, and eccentricity change together. The explorer connects symbolic standard forms with geometric landmarks, especially when comparing the closed ellipse and circle with the open parabola and two-branched hyperbola. A frequent surprise is that a directrix is not generally a line through a focus. It is a separate reference line, and a circle has no finite directrix in the usual focus-directrix description.
A concrete input and expected output from the current implementation.
Input
Select ellipse; set h = 0, k = 0, a = 5, b = 3, with a horizontal major axis.
Expected output
Equation: x^2/25 + y^2/9 = 1; foci: (-4, 0) and (4, 0); eccentricity: 0.8; directrices: x = -6.25 and x = 6.25.
The focal offset is c = sqrt(a^2 - b^2) = sqrt(25 - 9) = 4. Therefore e = c/a = 4/5 = 0.8, and the directrices are x = +/-a/e = +/-6.25.
After a conic type is chosen, changes to its geometric parameters alter the plotted locus and displayed equation. The family is tied to eccentricity: 0 for a circle, between 0 and 1 for an ellipse, 1 for a parabola, and greater than 1 for a hyperbola. These values distinguish the four families but do not determine a unique curve.
For an axis-aligned ellipse with semi-axes a and b, the focal offset satisfies c^2 = a^2 - b^2 when a is larger. The related values are e = c/a and directrix distance a/e from the center. A parabola instead has one focus and one directrix, while a hyperbola has two foci and two corresponding directrices.
A circle has eccentricity 0 and is the limiting case of an ellipse whose two foci coincide at the center. It does not have a finite directrix in the ordinary focus-directrix description. A missing or undefined circle directrix is therefore expected and does not mean that the circle equation or center is incorrect.
No. Eccentricity identifies the conic family and, for ellipses or hyperbolas, fixes a shape ratio, but it does not specify size, position, or orientation. Two ellipses can both have e = 0.8 while having different dimensions or centers. Use the displayed equation and landmarks alongside eccentricity when identifying a particular conic.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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