Foci of an Ellipse Calculator
Find the coordinates of ellipse foci with step-by-step calculations and geometric analysis
Ellipse Parameters
Center Coordinates
Semi-axes
Length of first semi-axis
Length of second semi-axis
Ellipse Analysis
Common Ellipse Examples
Circle
Horizontal Ellipse
Vertical Ellipse
Ellipse Properties
Foci lie on the major axis inside the ellipse
Sum of distances from any point on ellipse to both foci is constant (2a)
Focal distance c = √(a² - b²)
Eccentricity e = c/a (0 ≤ e < 1)
When e = 0, the ellipse becomes a circle
Understanding Ellipse Foci
What are Foci?
The foci (plural of focus) are two special points inside an ellipse. Every point on the ellipse has the property that the sum of its distances to both foci is constant and equal to the length of the major axis (2a).
Fundamental Property
|PF₁| + |PF₂| = 2a
For any point P on the ellipse
Practical Applications
- • Planetary orbits (Kepler's laws)
- • Architectural design (domes, arches)
- • Optics and reflector design
- • Medical imaging (lithotripsy)
How to Find Foci
Step 1: Identify Semi-axes
Determine which is the major axis (a) and minor axis (b). The major axis is always the larger of the two.
Step 2: Calculate Focal Distance
c = √(a² - b²)
Step 3: Find Coordinates
For horizontal ellipses: (h±c, k)
For vertical ellipses: (h, k±c)
Special Case: Circle
When a = b, the ellipse becomes a circle and both foci coincide at the center (c = 0).