An ellipse is a smooth closed curve that looks like a stretched circle. It appears in geometry, astronomy, optics, engineering, and design because many real paths and shapes are elliptical. Planets follow nearly elliptical orbits, and whispering galleries use elliptical reflection to focus sound.
Learning the parts of an ellipse helps connect algebraic equations to precise geometric diagrams.
For an ellipse centered at the origin, the major axis is the longer direction and the minor axis is the shorter direction. The standard equation shows how far the ellipse extends horizontally and vertically, while the foci describe its special distance property. Every point on an ellipse has the same total distance to the two foci.
This links coordinate geometry, measurement, and applications such as orbital motion and reflective surfaces.
Key Facts
- Standard horizontal ellipse: x^2/a^2 + y^2/b^2 = 1, where a > b
- Standard vertical ellipse: x^2/b^2 + y^2/a^2 = 1, where a > b
- Major axis length = 2a and minor axis length = 2b
- For an ellipse centered at (0, 0), horizontal vertices are (±a, 0) and co-vertices are (0, ±b)
- Focal distance relation: c^2 = a^2 - b^2
- Focal definition: distance to focus 1 + distance to focus 2 = 2a
Vocabulary
- Ellipse
- An ellipse is the set of all points in a plane whose total distance to two fixed points is constant.
- Focus
- A focus is one of the two fixed points inside an ellipse used in its distance definition.
- Major axis
- The major axis is the longest line segment through the center of an ellipse, with endpoints at the vertices.
- Minor axis
- The minor axis is the shortest line segment through the center of an ellipse, with endpoints at the co-vertices.
- Eccentricity
- Eccentricity is the ratio e = c/a that measures how stretched an ellipse is compared with a circle.
Common Mistakes to Avoid
- Swapping a and b without checking the larger denominator is wrong because a is the semi-major axis and must correspond to the larger value in the standard ellipse equation.
- Placing the foci on the minor axis is wrong because foci always lie on the major axis, the longer direction of the ellipse.
- Using c^2 = a^2 + b^2 is wrong because ellipses use c^2 = a^2 - b^2, unlike the Pythagorean relation used for many right triangle problems.
- Forgetting to double a and b for axis lengths is wrong because a and b are semi-axis lengths, so the full major axis is 2a and the full minor axis is 2b.
Practice Questions
- 1 For the ellipse x^2/25 + y^2/9 = 1, find a, b, the vertices, the co-vertices, and the foci.
- 2 An ellipse centered at the origin has vertices at (0, ±10) and co-vertices at (±6, 0). Write its standard equation and find the coordinates of the foci.
- 3 Explain why every point on an ellipse with semi-major axis a has a total distance of 2a to the two foci, and describe how this differs from a circle.