Ellipse

An ellipse is a two-dimensional closed plane curve that looks like an oval or a flattened circle. Mathematically, an ellipse is defined as the set of all points in a plane such that the sum of the distances from two fixed points, called the foci (D₁ and D₂), is constant.

Let D₁ and D₂ be the foci of the ellipse, and let 2a be the length of the major axis. Then, the ellipse can be defined by the equation:

∣PD₁∣ + ∣PD₂∣= 2a

Where,

P is any point on the ellipse.
∣PD₁∣ denotes the distance from P to the first focus D₁.
∣PD₂∣ denotes the distance from P to the second focus D₂.

Shape of an Ellipse: An ellipse resembles everyday objects such as eggs, badminton rackets, and balloons. It is also the shape followed by planets as they move around the Sun in their orbits.

Parts of Ellipse

parts_of_an_ellipse — Parts of an Ellipse

Center: The midpoint of the major axis of an ellipse; the point equidistant from the foci and defining the geometric symmetry of the ellipse.

Foci: In an ellipse, the foci (singular: focus) are two fixed points used to define the shape of the ellipse. The sum of the distances from any point on the ellipse to the two foci is constant.

Vertex: Points on the ellipse where the major and minor axes intersect the ellipse. In an ellipse, there are typically four vertices.

Directrix: For any point on an ellipse, the ratio of its distance to the focus and its distance to the directrix is a constant, known as the eccentricity of the ellipse.

Major Axis: The major axis (transverse axis) of an ellipse is the longest diameter, passing through the two foci, and it is also the longest segment that can be drawn across the ellipse.

Minor Axis: The minor axis (conjugate axis) of an ellipse is the shortest diameter, perpendicular to the major axis, and passing through the center of the ellipse.

Latus Rectum: The latus rectum of an ellipse is a line segment that passes through one of the foci and is parallel to the minor axis.

Standard Form of Ellipse Equation

The ellipse equation with its center at the origin and its major axis along the x-axis is

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

Where,

"a" is the length of the semi-major axis.
b is the length of the semi-minor axis.
The major axis is along the horizontal direction if a>b.

General Form of Ellipse Equation

The general form of the equation of an ellipse in Cartesian coordinates is

\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1

Where,

(h, k) represents the coordinates of the center of the ellipse.
a and b are the lengths of the semi-major axis and semi-minor axis, respectively.

Note: When a = b, the ellipse becomes a circle.

Parametric Equation of Ellipse

The parametric equations of an ellipse in Cartesian coordinates can be written as follows:

x(t) = h + a cos (t) and y(t) = k + b sin (t)

Where,

(h, k) represents the coordinates of the center of the ellipse.
a and b are the lengths of the semi-major axis and semi-minor axis, respectively.
It is the parameter that ranges from 0 to 2π (or any multiple of 2π for multiple loops), and
(x(t), y(t)) gives the coordinates of points on the ellipse as t varies.

Ellipse Formula

For the equation \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1, if c is the distance between both foci, then the formula for the ellipse is given as

c²= a² – b²

Derivation of Equation of Ellipse

The figure represents an ellipse such that P₁F₁ + P₁F₂ = P₂F₁ + P₂F₂ = P₃F₁ + P₃F₂ is a constant. This constant is always greater than the distance between the two foci. When both the foci are joined with the help of a line segment, then the midpoint of this line segment joining the foci is known as the center.

The major axis of an ellipse is the line segment that passes through both foci. The minor axis is the line segment perpendicular to the major axis and passing through the center. The points where the major axis meets the ellipse are called the vertices. Here, (2a) is the length of the major axis, so (a) is the semi-major axis; (2b) is the length of the minor axis, so (b) is the semi-minor axis; and (2c) is the distance between the two foci.

Proof:

Let's take P and Q as the endpoints of the major axis, points R and S at the end of the minor axis, and O as the center of the ellipse.

The distance of Q from F₁ is F1Q, and Q to F₂is F2Q, and their sum is F₁Q + F2Q, and F₁Q + F₂Q = F₁O + OQ + F₂Q

c + a + a – c = 2a

The sum of distances from point R to F₁ is F₁R + F₂R

F₁R + F₂R = √(b² + c²) + √(b² + c²) = 2√(b² + c²)

By definition of the Ellipse,

2√(b² + c²) = 2a

⇒ a = √(b² + c²)

⇒ a² = b² + c²

⇒ c² = a² – b²

Properties of Ellipse

Eccentricity of Ellipse

The eccentricity of an ellipse is defined as the ratio of distances from the center of the ellipse to the semi-major axis of the ellipse.

e = c/a

Where,

c is the focal length, and
"a" is the semi-major axis length.

The eccentricity of an ellipse should be 0 < e < 1.

Furthermore, c² = a² - b²

As a result, eccentricity becomes

e = √[(a² – b²)/a²]

⇒ e = √[1 - (b²/a²)]

Auxiliary Circle

The auxiliary circle of an ellipse is a circle centered at the ellipse’s center (h, k) with radius equal to its semi-major axis a. It passes through the vertices of the ellipse (endpoints of the major axis). Its equation is:

(x - h)² + (y - k)² = a²

Director Circle

The director circle of an ellipse is the locus of points from which two perpendicular tangents can be drawn to the ellipse.

For an ellipse with semi-major axis a and semi-minor axis b, the director circle has its center at the origin, and its radius r is given by

r = \sqrt{a^2 + b^2}

The director circle's equation is

x²+ y²= a² + b²

Formula for Ellipse

Area of Ellipse

As we know, the area of the circle is calculated using its radius, whereas the area of the ellipse depends on the length of the minor axis and major axis.

Area of circle = πr²
Area of ellipse = π × Semi-Major Axis × Semi-Minor Axis

Area of ellipse = πab
Where,
a is the semi-major axis
b is the semi-minor axis

Perimeter of Ellipse

The perimeter of an ellipse is the total length of the curve boundary. An ellipse has two axes, the major axis and the minor axis; both axes cross through the center and intersect each other. The approximate formula to find the perimeter of an ellipse is

P = 2\pi \sqrt{\frac{a^2 + b^2}{2}}
Where,
a is the semi-major axis
b is the semi-minor axis.

Latus Rectum Formula

Latus rectum is defined as the line segments perpendicular to the major axis of the ellipse and passing through any of the foci in such a manner that their endpoints always lie on the ellipse.

The length of the latus rectum is defined in the diagram given below.

The length of the latus rectum for the ellipse is,

L = 2b²/a
Where,
a is the semi-major axis
b is the semi-minor axis.

Equation of Tangent to Ellipse

There are various forms of equations of tangents to an ellipse:

Slope Form: If a line y = mx + c touches the ellipse \frac{x^2}{a ^2} + \frac{y^2}{b^2} = 1 then the equation of the tangent is given by:

y = mx ± √[a²m² + b²]

Point Form: The equation of the tangent to an ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, at a point (x₁, y₁), is

\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1

Equation of Normal to Ellipse

If a line y = mx + c touches the ellipse x² / a² + y² / b² = 1, then the equation of the normal is given by

y = mx - \frac{a^2 - b^2}{\sqrt{a^2 + b^2m^2}}

Key Features

Various key features related to ellipses are mentioned in the table below:

Aspect	Formula/Equation
General Equation	\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
Parametric Equation	x = a cos(θ), y = b sin(θ)
Eccentricity	e = \sqrt{1 - \frac{b^2}{a^2}}
Foci	(±ae, 0)
Center	(h, k)
Major Axis Length	2a
Minor Axis Length	2b
Focus-Directrix Relation	PF₁ + PF₂ = 2a
Latus Rectum Length	2b²/a
Area	A = πab
Perimeter	P ≈ 4aE(e), where E(e) is the complete elliptic integral of the second kind.
Slope of Tangent Line	m = −\frac{b^2x}{a^2y}

Parabola
Circle
Hyperbola

Solved Examples

Example 1: If the length of the semi-major axis is given as 10 cm and the semi-minor axis is 7 cm for an ellipse. Find its area.

Solution:

Given, the length of the semi-major axis of an ellipse, a = 10 cm
Length of the semi-minor axis of an ellipse, b = 7 cm
We know the area of an ellipse using the formula;
Area = π x a x b
Area = π x 10 x 7
Area = 70 x π
Therefore Area = 219.91 cm²

Example 2: Find the lengths for the major axis and minor axis of equation 7x²+3y²=21.

Solution:

Given equation is 7x²+3y²= 21
Dividing both sides by 21, we get
x²/3 + y²/7 = 1
We know that, Standard Equation of Ellipse
x²/b²+y²/a² = 1
As the foci lies on y-axis, for the above equation , the ellipse is centered at origin and major axis on y-axis then;
b² = 3
b = 1.73
a²= 7
a = 2.64
Thus,
Length of Major Axis = 2a
= 5.28
Length of Minor Axis = 2b
= 3.46

Practice Questions

Question 1: Find the area of an ellipse where the semi-major axis is 14 cm and the semi-minor axis is 6 cm.

Question 2: Find the lengths of the major and minor axes for the ellipse with the equation \frac{x^2}{9} + \frac{y^2}{16} = 1.

Question 3: Find the area of an ellipse with a semi-major axis of 18 cm and a semi-minor axis of 7 cm.

Question 4: For the ellipse represented by the equation \frac{x^2}{64} + \frac{y^2}{25} = 1, find the lengths of the major and minor axes.

Parts of Ellipse

Standard Form of Ellipse Equation

General Form of Ellipse Equation

Parametric Equation of Ellipse

Ellipse Formula

Derivation of Equation of Ellipse

Properties of Ellipse

Eccentricity of Ellipse

Auxiliary Circle

Director Circle

Formula for Ellipse

Area of Ellipse

Perimeter of Ellipse

Latus Rectum Formula

Equation of Tangent to Ellipse

Equation of Normal to Ellipse

Key Features

Related Articles:

Solved Examples

Practice Questions

Explore