Definition: An ellipse is the set of points whose distances to two fixed points, the foci, add up to a constant; it looks like a stretched circle.
What defines an ellipse?
Pick two foci. Every point on the ellipse has the same sum of distances to them. With center at the origin and a wider-than-tall shape, the equation is x2/a2 + y2/b2 = 1 with a > b. The foci lie c units from the center, where c2 = a2 – b2.
How do you find its parts?
Worked example: x2/49 + y2/25 = 1. Here a = 7 and b = 5.
- Vertices: (±7, 0)
- Co-vertices: (0, ±5)
- c = square root of (49 – 25) = square root of 24, about 4.90, so the foci are about (±4.90, 0)
- Sum of focal distances: 2a = 14
What do students confuse?
Using c2 = a2 + b2, which belongs to hyperbolas. For an ellipse the foci stay inside the shape, so c is smaller than a. Mathbot checks that a is the larger denominator’s root.
Example: x^2/49 + y^2/25 = 1 has foci about (±4.90, 0)