Ellipse — Mathematics Knowledge System

Overview

An ellipse is a plane curve surrounding two focal points such that, for every point on the curve, the sum of the distances to the two foci is constant. It generalizes the circle, which is the special case where the two foci coincide. The elongation of an ellipse is measured by its eccentricity, a number strictly between 0 and 1.

Formal Definition

Definition

An ellipse centered at the origin with foci on the x-axis, in standard position (a > b > 0):

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

a = semi-major axis, b = semi-minor axis

Standard form

c^2 = a^2 - b^2

Foci located at (±c, 0)

Focal distance

E = \{P \in \mathbb{R}^2 : |PF_1| + |PF_2| = 2a\}

Sum-of-distances definition using the two foci F₁, F₂

e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}}, \quad 0 \le e < 1

Eccentricity

Properties

Vertices

(\pm a, 0) \text{ on the major axis; } (0, \pm b) \text{ on the minor axis}

Foci

(\pm c, 0), \quad c = \sqrt{a^2-b^2}

Eccentricity

e = c/a; \quad e=0 \text{ is a circle}, \quad e \to 1 \text{ is highly elongated}

Parametrization

(x,y) = (a\cos t, b\sin t), \quad 0 \le t \le 2\pi

Worked Examples

Identify a² = 25, b² = 16, so a = 5, b = 4.
$a=5,\ b=4$
Compute c using c² = a² - b².
$c^2 = 25 - 16 = 9 \implies c = 3$
Compute eccentricity e = c/a.
$e = 3/5 = 0.6$

Answer: Foci at (±3, 0); eccentricity e = 0.6.

Practice Problems

Difficulty 4/10

Find the semi-minor axis b for an ellipse with a = 10 and eccentricity e = 0.6.

Common Mistakes

Common Mistake

Using c² = a² + b² (the hyperbola relation) instead of c² = a² - b² for an ellipse.

For an ellipse the foci lie inside the curve, so c < a and c² = a² - b² (b < a). The '+' relation belongs to the hyperbola, where the foci lie outside the vertices.

Summary

An ellipse is the set of points whose distances to two fixed foci sum to a constant, 2a.
Standard form x²/a² + y²/b² = 1 with a > b > 0; foci at (±c, 0) where c² = a² - b².
Eccentricity e = c/a ranges from 0 (circle) to just under 1 (very elongated).
A circle is the degenerate ellipse case where the two foci merge into one point (c = 0).

References

WebsiteWikipedia — Ellipse