Ellipse | Standard Equation Formula Proof, Eccentricity, Foci, Latus Rectum,Area & Perimeter of Ellipse, Solved Examples, FAQs

Class 11 Mathematics – Written by Neeraj Anand
Published by ANAND TECHNICAL PUBLISHERS

Introduction to Ellipse

An ellipse is a type of conic section formed when a plane cuts through a cone at an angle, creating a closed curve. It is defined as the set of all points where the sum of the distances from two fixed points (called foci) is constant. This property makes ellipses unique and widely applicable in physics, astronomy, and engineering.

What is Ellipse?

An ellipse is a two-dimensional closed-plane curve which looks like an oval or a flattened circle.

Ellipse Shape

In geometry, an ellipse is a two-dimensional shape, that is defined along its axes. An ellipse is formed when a cone is intersected by a plane at an angle with respect to its base.

Ellipse Definition

An ellipse is defined as a collection of all points in a plane such that the sum of the distances from two fixed points (called the foci) is constant.

Let F₁​ and F₂​ 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:

∣PF₁​∣ + ∣PF₂​∣= 2a

Where,

• P is any point on the ellipse.
• ∣PF₁​∣ denotes the distance from P to the first focus F₁​.
• ∣PF₂​∣ denotes the distance from P to the second focus F₂​.

A circle is also an ellipse, where the foci are at the same point, which is the center of the circle.

Parts of Ellipse

Most common parts of any ellipse are:

• 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 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 of an ellipse is the shortest diameter, perpendicular to the major axis, and passing through the center of the ellipse.
• Semi-Major & Semi Minor axis : Half of major axis is called semi-major axis and half of minor axis is called semi-minor axis.
• 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.
• Conjugate Axis: Conjugate axis of an ellipse is another name for the minor axis. It is perpendicular to the transverse axis (major axis) and passes through the center of the ellipse.
• Transverse Axis: Transverse axis of an ellipse is another name for the major axis. It is the longest diameter of the ellipse and passes through the center.

Properties of Ellipse

Various properties of an ellipse are,

• Ellipse includes two focal points, also referred to as foci.
• A fixed distance is referred to as a directrix.
• The eccentricity of an ellipse ranges from 0 to 1. 0 ≤ e < 1
• The whole sum of each distance from an ellipse’s locus to its two focal points is constant.
• Ellipse has one major and one minor axis, as well as a centre.

Eccentricity of the Ellipse

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

The eccentricity of ellipse, e = c/a

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

Since c ≤ a the eccentricity is always less than 1 in the case of an ellipse.
Also,

c² = a² – b²

Therefore, eccentricity becomes:

e = √(a² – b²)/a

e = √[(a² – b²)/a²]

e = √[1-(b²/a²)]

Ellipse Equation

When the centre of the ellipse is at the origin (0,0) and the foci are on the x-axis and y-axis, then we can easily derive the ellipse equation.

The equation of the ellipse is given by;

x²/a² + y²/b² = 1

Derivation of Ellipse Equation

Now, let us see how it is derived.

The above 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 mid-point of this line segment joining the foci is known as the center, O represents the center of the ellipse in the figure given below:

The line segment passing through the foci of the ellipse is the major axis and the line segment perpendicular to the major axis and passing through the center of the ellipse is the minor axis.

The end points A and B as shown are known as the vertices which represent the intersection of major axes with the ellipse.

‘2a’ denotes the length of the major axis and ‘a’ is the length of the semi-major axis. ‘2b’ is the length of the minor axis and ‘b’ is the length of the semi-minor axis. ‘2c’ represents the distance between two foci.

Proof:

Let us consider the end points A and B on the major axis and points C and D at the end of the minor axis.

The sum of distances of B from F₁ is F₁B + F₂B = F₁O + OB + F₂B (From the above figure)

⇒ c + a + a – c = 2a

The sum of distances from point C to F₁ is F₁C + F₂C

⇒ F₁C + F₂C = √(b² + c²) + √(b² + c²) = 2√(b² + c²)

By definition of ellipse;

2√(b² + c²) = 2a

⇒a = √(b² + c²)

⇒ a² = b² + c²

⇒c² = a² – b²

Special Cases:

• If c = 0 then F₁ and F₂, i.e. both foci merge together with center of ellipse. Also a² becomes equal to b², i.e. a = b so now we get a circle in this case.
• If c = a then b becomes 0 and we get a line segment F₁F₂.

Standard Equation of Ellipse

The simplest method to determine the equation of an ellipse is to assume that centre of the ellipse is at the origin (0, 0) and the foci lie either on x- axis or y-axis of the Cartesian plane as shown below:

Both the foci lie on the x- axis and center O lies at the origin.

Let us consider the figure (a) to derive the equation of an ellipse. Let the coordinates of F₁ and F₂ be (-c, 0) and (c, 0) respectively as shown. Let us consider a point P(x, y) lying on the ellipse such that P satisfies the definition i.e. the sum of distances of P from F₁ and F₂ in the plane is a constant 2a.

⇒ PF₁ + PF₂ = 2a – – – (1)

Using distance formula the distance can be written as:

Squaring and simplifying both sides we get;

Now since P lies on the ellipse it should satisfy equation 2 such that 0 < c < a.

Thus,

On simplifying,

PF₁ = a + (c/a)x

Similarly,

PF₂ = a – (c/a)x

Therefore,

PF₁ + PF₂ = 2a

Therefore the equation of the ellipse with centre at origin and major axis along the x-axis is:

where –a ≤ x ≤ a.

Similarly, the equation of the ellipse with center at origin and major axis along the y-axis is:

where –b  ≤ y ≤ b.

Area of Ellipse

Area of the circle is calculated based on its radius, but the area of the ellipse depends on the length of the minor axis and major axis. Check more here: Area of an ellipse.

Area of the circle = πr²

And,

Area of the ellipse = π x Semi-Major Axis x Semi-Minor Axis

Area of the ellipse = π.a.b

Perimeter of Ellipse

The perimeter of an ellipse is the total distance run by its outer boundary. For a circle, it is easy to find its circumference, since the distance from the center to any point of locus of circle is same. This distance is called radius.

But in the case of an ellipse, we have two axis, major and minor, that crosses through the center and intersects. Hence, we use an approximation formula to find the perimeter of an ellipse, given by:

\(\begin{array}{l}p \approx 2 \pi \sqrt{\frac{a^{2}+b^{2}}{2}}\end{array} \)

Where a and b are the length of semi-major and semi-minor axes respectively.

Latus Rectum

The line segments perpendicular to the major axis through any of the foci such that their endpoints lie on the ellipse are defined as the latus rectum.

The length of the latus rectum is 2b²/a.

L = 2b²/a

where a and b are the length of the minor axis and major axis.

General Form of Ellipse Equation

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

(x–h)²/a² + (y–k)²/b² = 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.

Types of Ellipses

Ellipses can be classified based on their orientation, size, and position. Here are some types of ellipses:

• Horizontal Ellipse
• Vertical Ellipse

Horizontal Ellipse

An ellipse where the major axis is horizontal. Its equation is

(x–h)²/a² + (y–k)²/b² = 1

Where, (h, k) is the center of ellipse.

Vertical Ellipse

An ellipse where the major axis is vertical. Its equation is

(x–h)²/b² + (y–k)²/a² = 1

Where, (h, k) is the center of ellipse.

Note: Circle is the special case of ellipse where major and minor axis are same.

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,
• t 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.

Auxiliary Circle

Auxiliary circle of an ellipse is a circle constructed using center of ellipse as center and semi major axis as radius. It is positioned so that it touches the ellipse at its endpoints of the major axis (the vertices of the ellipse). For any ellipse with a as its semi major axis, (h, k) as its center, equation of auxiliary circle is given by:

(x – h)² + (y – k)² = a²

Director Circle

Director circle is a circle that intersects the ellipse at its endpoints (vertices) and is tangent to the ellipse at its foci.

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 = a²/b

The director circle’s equation is:

x² + y² = (a²)/(b²)​(x² + y²)

Examples of Ellipse

Some of the examples of ellipse are:

• Planetary Orbits: The orbits of planets around the sun and moons around planets are elliptical.
• Egg Shape: The shape of a hen’s egg closely resembles an ellipse.
• Satellite Orbits: Elliptical orbits are often used by satellites around Earth.
• Architecture: Some architectural designs, like the elliptical arches in buildings, bridges, and cathedrals, are based on ellipses.
• Sports: Tracks in sports like running, cycling, and horse racing are often elliptical in shape.

Some Other Formulas and Equations

Some other equation related to ellipse include:

• Equatio of Tangent
• Equation of Normal

Equation of Tangent to Ellipse

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

Slope Form: If a line y = mx + c touchs the ellipse x²/a² + y²/b² = 1, then equation of tangent is given by:

• y = mx ± √[a²m² + b²]

Point Form: Equation of Tangent to an ellipse x² / a² + y² / b² = 1, at a point (x₁, y₁) is:

• xx₁ / a² + yy₁ / b² = 1

Paramatric Form: The parametric equations of the tangent line passing through the point (a cos ϕ, b sin ϕ) on the ellipse are:

• x = a cosϕ + a λ sinϕ
• y = b sinϕ − b λ cosϕ

Where λ is a parameter representing the slope of the tangent line.

Implicit Form: The implicit equation of the tangent line to the ellipse at the point (a cos ϕ, b sin ϕ) is given by:

• (x sin ϕ – y cos ϕ)²/a² + (x sin ϕ + y cos ϕ)²/a² = 1

Solved Examples on Ellipse

Example 1: If the length of the semi-major axis is given as 10 cm and the semi-minor axis is 7 cm of 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: For an ellipse, the length of the semi-major axis is 8 cm and the semi-minor axis is 3 cm then find its area.

Solution: 

Given:

Length of semi-major axis of the ellipse (a) = 8 cm

Length of semi-major axis of the ellipse (b) = 3 cm

Formula for the area of the ellipse:

Area = π x a x b

Area = π x 8 x 3

Area = 24 π cm²

Example 3: 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 

FAQs on Ellipse

What is an ellipse?

When a plane, intersects a cone at an angle with respect to the base the curve formed is called an ellipse. It is the locus of points, whose sum of the distances from two foci is always constant.

What is the major and minor axis of an ellipse?

Ellipses are distinguished by two axes running along the x and y axes:

• Major Axis: The major axis is the ellipse’s longest diameter, running through the center from one end to the other at the broadest part of the ellipse. 
• Minor Axis: The minor axis is the shortest diameter of an ellipse that crosses through the center at its narrowest point. Half of the major axis is the semi-major axis, and half of the minor axis is the semi-minor axis.

What is the Area of an ellipse?

The area of the ellipse is the region covered by the shape in the two-dimensional plane. It is given by:

Area = πab

where 
a is the semi-major axis
b is the semi-minor axis

What is the equation of an ellipse?

Equation of the ellipse is given by:

(x²/a²)+(y²/b²) = 1

What are the Asymptotes of Ellipse?

Asymptotes are lines drawn parallel to the curve and it is assumed that they meet the curve at infinity. Ellipse has no asymptote.

Applications of Ellipse

• Astronomy: Planetary orbits are elliptical, with the sun at one focus.
• Engineering: Used in designing reflective properties in whispering galleries and optical instruments.
• Architecture: Elliptical domes and arches provide structural efficiency and aesthetic appeal.

---

📥 Download PDF

Students preparing for CBSE Board & JEE Mains/Advanced can download the full PDF from ANAND CLASSES for detailed explanations, solved examples, and practice problems.