Parts and Properties of an Ellipse

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.