Formula and Method for the Foci of an Ellipse
An ellipse is the set of all points where the sum of the distances to two fixed points — the foci — stays constant. For an ellipse in standard position, (x−h)²/a² + (y−k)²/b² = 1, centered at (h, k) with semi-axis lengths a (along x) and b (along y), this calculator finds the focal distance c and the coordinates of both foci, along with the eccentricity that describes how elongated the ellipse is.
How the calculation works
The calculator first compares a and b to find the semi-major axis — whichever value is larger, since an ellipse's foci always lie on its major axis. It then applies c = √(a² − b²), using the larger value squared minus the smaller value squared, to find c, the distance from the center to each focus. If a > b, the major axis is horizontal and the foci sit at (h − c, k) and (h + c, k). If b > a, the major axis is vertical and the foci sit at (h, k − c) and (h, k + c). When a equals b, the "ellipse" is a circle, c = 0, and both foci sit on the center. Finally, eccentricity e = c ÷ (semi-major axis) is computed: e = 0 is a perfect circle, and e approaches 1 as the ellipse grows more elongated.
Common mistakes
- Assuming a is always horizontal: in (x−h)²/a² + (y−k)²/b² = 1, a is whichever denominator sits under x² — it is only the semi-major axis if a > b. Compare a and b before deciding which axis is major.
- Using a² + b² instead of a² − b²: the focal-distance formula is c² = a² − b² (larger squared minus smaller squared), not a sum — the sum form belongs to the Pythagorean theorem for right triangles, not an ellipse's foci.
- Forgetting the center offset: if the ellipse is not centered at the origin, both foci must be shifted by (h, k) — computing them relative to (0, 0) instead gives the wrong location.
Real-world applications
- Orbital mechanics: planetary and satellite orbits are ellipses with the sun or Earth located at one focus (Kepler's first law).
- Optics and acoustics: elliptical reflectors and "whispering gallery" ceilings direct light or sound from one focus toward the other.
- Architecture and design: elliptical arches, domes, and gears are laid out using the two-foci, string-and-pins construction method.
- Analytic geometry coursework: locating the foci and computing eccentricity is a standard step in graphing and classifying conic sections.