Eccentricit

High School

Definition

Denoted by e or \(ε\), eccentricity tells you how drawn out a conic section is. In other words, it shows how far the conic section is from being circular. When you have a circle, its eccentricity is 0. The eccentricity of a line, on the other hand, is infinite.

Worked examples

\(e = 0\) (circle), \(0 < e < 1\) (ellipse), \(e = 1\) (parabola), \(e > 1\) (hyperbola)

Eccentricity increases as the conic becomes more elongated, from circle to hyperbola.

\(e = \frac{c}{a}\) for an ellipse, where \(c\) is the distance from center to focus and \(a\) is the semi-major axis

A larger ratio means the foci are farther from the center, making the ellipse more stretched.

Common mistakes

\(e = 1\) for a circle → \(e = 0\) for a circle A circle is perfectly round, so its eccentricity is zero, not one.
\(e = \frac{a}{c}\) → \(e = \frac{c}{a}\) Eccentricity is the focal distance divided by the semi-major axis, not the other way around.
All ellipses have the same eccentricity → Ellipses have \(0 < e < 1\); rounder ones have smaller \(e\) Eccentricity varies — a flatter ellipse has higher eccentricity closer to 1.

Where you'll use it next

You'll use eccentricity when classifying and graphing conic sections in precalculus, analyzing planetary orbits in physics and astronomy, and solving applications involving satellite paths and reflective properties.

Found in 1 StudyPug lesson

Conics - Ellipse

Grade 12 Math

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