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Ellipses

Lesson · Intermediate

Geometry: Coordinate Geometry

An ellipse is the set of points P in the plane such that the sum of the distances from two fixed points (the foci) is constant.

PF1+PF2=2aPF_1+PF_2=2a

Horizontal Major Axis

Ellipse with horizontal major axis showing center, vertices, foci, and a point on the ellipse
(x−h)2a2+(y−k)2b2=1,a>b>0\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1,\qquad a>b>0
c2=a2−b2,Eccentricity: e=cac^2=a^2-b^2,\qquad \text{Eccentricity: }e=\frac{c}{a}
Major Axis=2a,Minor Axis=2b\text{Major Axis}=2a,\qquad \text{Minor Axis}=2b
Area=πab\text{Area}=\pi ab

Latus Rectum

Horizontal ellipse showing a latus rectum through a focus
Latus Rectum Length=2b2a\text{Latus Rectum Length}=\frac{2b^2}{a}
Semi-Latus Rectum=b2a\text{Semi-Latus Rectum}=\frac{b^2}{a}

Vertical Major Axis

Ellipse with vertical major axis showing center, vertices, foci, and a point on the ellipse
(x−h)2b2+(y−k)2a2=1,a>b>0\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1,\qquad a>b>0
c2=a2−b2,Eccentricity: e=cac^2=a^2-b^2,\qquad \text{Eccentricity: }e=\frac{c}{a}
Major Axis=2a,Minor Axis=2b\text{Major Axis}=2a,\qquad \text{Minor Axis}=2b

Latus Rectum (Vertical Ellipse)

Vertical ellipse showing a latus rectum through a focus
Latus Rectum Length=2b2a\text{Latus Rectum Length}=\frac{2b^2}{a}

Shape and Similar Ellipses

Smaller eccentricity gives a more circular ellipse, while larger eccentricity gives a more elongated ellipse.

0≤e<10\le e<1

Two ellipses are similar if and only if they have the same eccentricity.

e1=e2e_1=e_2
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