Foci Of Ellipse
Find the foci of an ellipse from its semi-major and semi-minor axes. Calculate focal distance (c) and eccentricity. The foci are the two fixed points that define an ellipse: any point on the ellipse has a constant sum of distances to the two foci.
Formula
c = √(a² - b²); Eccentricity = c/a
- For an ellipse with semi-major axis a and semi-minor axis b (where a ≥ b), the focal distance is c = √(a²−b²).
- The two foci are located at (−c, 0) and (+c, 0) along the major axis.
- Eccentricity e = c/a; ranges 0 (circle) to 1 (parabola). Higher eccentricity = more elongated ellipse.
Semi-major a=5, semi-minor b=3
Inputs
- Semi-Major Axis (a): 5
- Semi-Minor Axis (b): 3
c = √(5²−3²) = √(25−9) = √16 = 4. Foci at (−4,0) and (4,0). Eccentricity = 4/5 = 0.8.
Frequently asked questions
What is the definition of an ellipse?
An ellipse is the set of all points where the sum of distances to two fixed points (the foci) is constant. This property defines the ellipse's shape.
How do I interpret eccentricity?
e = 0 is a circle (foci coincide). e near 1 is very elongated (foci far apart). For an ellipse, 0 < e < 1.
What if a < b?
The calculator swaps them. The semi-major axis must be ≥ semi-minor; otherwise, b would be major.