Parts and Properties of an Ellipse

Study Notes

Parts of an Ellipse

An ellipse has two foci which are always located on the major axis.
The center of an ellipse is the midpoint between the two foci.
The major axis is the distance between the two end vertices. The center divides the major axis into two equal halves, each called the semi-major axis or major radius, represented by 'a'.
The minor axis is the distance between the two end co-vertices. The center divides the minor axis into two equal halves, each called the semi-minor axis or minor radius, represented by 'b'.
A vertex is a point where the ellipse intersects the major axis.
A co-vertex is a point where the ellipse intersects the minor axis.

Properties of an Ellipse

Ellipses always have two foci.
Ellipses have a center, a major axis, and a minor axis.
The sum of the distances between any point on the ellipse and the two foci is constant and equal to the total length of the major axis.
The eccentricity of all ellipses is always less than one (e < 1).

Eccentricity of an Ellipse

Eccentricity (e) is calculated using the formula: e = c/a
c = the distance between the two foci
a = the semi-major axis

How To Find the Eccentricity Of An Ellipse

The center of the ellipse is at the origin, with foci at F₁(-c, 0) and F₂(c, 0)
A and B are the vertices.
AB = 2a (the length of the major axis).
c = half the distance between the two foci.
a = the semi-major axis.
PF₁ + PF₂ = 2a, where P is a point on the ellipse.
OF₂ = c, where O is the center of the ellipse.
P(0,b) is a point on the ellipse.
Eccentricity = 0 (for a circle).
0 ≤ e < 1 for all ellipses.

Parabolic Segment

The area of a parabolic segment is calculated using the formula: (2ab)/3

Parts of an Ellipse

An ellipse has two foci, which lie on the major axis.
The center of an ellipse is the midpoint between the two foci.
The major axis is the longest diameter of the ellipse, passing through both foci and the center.
- The distance between the end vertices is the major axis.
- Half of the major axis is the semi-major axis or major radius denoted by 'a'.
The minor axis is the shortest diameter of the ellipse, perpendicular to the major axis and passing through the center.
- The distance between the end co-vertices is the minor axis.
- Half of the minor axis is the semi-minor axis or minor radius denoted by 'b'.
The vertices are the points where the ellipse intersects the major axis.
The co-vertices are the points where the ellipse intersects the minor axis.

Properties of an Ellipse

The sum of the distances between any point on the ellipse and the two foci is constant, and equal to the total length of the major axis: PF₁ + PF₂ = 2a
The eccentricity of an ellipse is always less than one (e < 1).

Eccentricity of an Ellipse

Eccentricity (e) is calculated as the ratio of the distance between the two foci (c) to the semi-major axis (a): e = c/a
When the eccentricity of an ellipse is 0, it becomes a circle.
The eccentricity of an ellipse is always between 0 and 1 (0 ≤ e < 1).

How To Find the Eccentricity Of An Ellipse

The center of the ellipse is denoted as O.
The foci are denoted as F₁(-c, 0) and F₂(c, 0).
The vertices are denoted as A and B.
The distance between the two vertices (AB) is equal to 2a.
'c' is half the distance between the two foci.
'a' is the semi-major axis.
PF₁ + PF₂ = 2a
OF₂ = c
P(0,b) is a point on the ellipse.

Parabolic Segment

The area of a parabolic segment is given by the formula: Area = (2ab)/3.

Description

This quiz covers essential concepts regarding the parts and properties of an ellipse, including definitions of the foci, axes, vertices, and co-vertices. Test your understanding of how these elements interact and the fundamental characteristics that define an ellipse.