Conic Sections (Ellipse): Definition, Equation, Properties, Eccentricity and More

An ellipse is the locus of all the points in a plane whose distances from two fixed points in the plane are constant. The fixed points, which are surrounded by the curve, are known as foci (singular focus).

The ellipse has an oval shape, and its area is defined by its major and minor axes. The ellipse is related to other parts of the conic section, such as the parabola and hyperbola, which are open and unbounded in shape. Even the standard equation of hyperbola and ellipse i.e., x2/a2 – y2/b2 = 1 and x2/a2 + y2/b2 = 1 are closely related, the only difference being of a negative sign.

Let’s learn about the definition, properties, eccentricity, and standard equation of ellipse with examples for better understanding.

Definition of an Ellipse

An ellipse is the set of all points on an XY plane whose distance from two fixed points (called foci) adds up to a fixed value in terms of locus.

When a plane cuts a cone at an angle to the base, one of the conic sections formed is the ellipse.

Ellipse Shape

An ellipse is a two-dimensional shape defined along its axes in geometry. When a plane intersects a cone at an angle to the base of the cone, an ellipse is formed.

There are two focal points. For all places on the curve, the sum of the two distances to the focal point is always constant.

A circle is also an ellipse in which the foci are at the same point, which is the circle’s centre.

Major and Minor Axis

Ellipse is characterised by its two axes along the x and y axes:

Major axis

Minor Axis

The major axis is the ellipse’s longest diameter (typically represented by ‘a’), which runs through the centre from one end to the other, at the broadest part of the ellipse. The minor axis is the ellipse’s shortest diameter (denoted by ‘b’), passing through the centre at its narrowest part.

Properties of an Ellipse

  • Ellipse has two focal points or foci.
  • The fixed line parallel to the minor axis at distance (d) from the centre is known as the ellipse’s directrix.
  • The ellipse’s eccentricity varies from 0 to 1.
  • The total of each distance between an ellipse’s locus and its two focal points is constant.
  • Ellipse has one major and minor axis and a centre.

Eccentricity of the Ellipse

The ratio of distances between an ellipse’s centre and semi-major axis is known as the eccentricity of the ellipse.

The ellipse’s eccentricity, e = c/a

Where c is the focal length and an is the semi-major axis length.

Since c ≤ a the eccentricity of an ellipse is always larger than 1.

Also,

c2 = a2 – b2

As a result, eccentricity becomes:

e = √(a2 – b2)/a

   = √[(a2 – b2)/a2]

   = √[1-(b2/a2)]

Standard Equation of an Ellipse

When the ellipse’s centre is at the origin (0,0) and the foci are on the x- and y-axes, we can simply calculate the ellipse equation.

The standard equation of an ellipse is written as;

x2/a2 + y2/b2 = 1

Ellipse Formula

An ellipse, as we know, is a closed-shape structure in a two-dimensional plane. As a result, it covers a region in a 2D plane. So the ellipse’s area is defined as its limited region. Since the ellipse differs from the circle in shape, the formula for its area differs as well.

Area of the ellipse = π.a.b

where,

a=Semi-Major Axis

b=Semi-Minor Axis

Circumference of an ellipse is given by,

p ≈ 2π √a2+b2/2

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