Ellipse Equation: Formula, Properties, and Graphing

An ellipse is a special geometric shape that appears in many practical scenarios. Do you know that planets revolve around the sun in an elliptical path? The speciality of an ellipse is that every point on its curve maintains a constant total distance from two fixed focal points. 

This unique property makes ellipses essential in physics, engineering, and astronomy. Understanding the ellipse equation and ellipse formula helps students explore and connect mathematical concepts to real-world applications more interestingly.

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What is an Ellipse?

An ellipse is an oval-shaped geometrical shape drawn on a two-dimensional plane. It looks like an elongated circle with two diameters of different lengths. Apart from the centre, there are two more fixed points called foci. 

Take any point on the boundary of the ellipse, and calculate the sum of its distances from both foci. Every time, you will get the same value.

Definition of Ellipse

An ellipse is defined as the path of a point moving in a plane such that the combined distances of the point from two other fixed points are constant.

 For example, for any point P on the curved shape, the sum of the distances from two fixed points, F and G, is constant. In other words, f + g = constant.

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Different Terms Related to Ellipse

The different terms related to an ellipse are given below, with an explanation.

• Focus: The ellipse has two points called foci, and their coordinates are F(c, 0) and F'(-c, 0). The distance between the two foci is equal to 2c.

• Centre: The midpoint of the line joining the two foci is called the centre of the ellipse.

• Major Axis: It is the longer diameter of the ellipse. If the endpoints of this major axis are (a, 0), (-a, 0), respectively, the length of the ellipse's major axis is 2a units.

• Minor Axis: It is the shorter diameter of the ellipse. If the end vertices of the minor axis are (0, b) and (0, -b), respectively, then the length of the ellipse's minor axis is 2b units.

• Vertex: The vertices of the ellipse are the two points of intersection of the ellipse with the major axis.  The given figure's vertices are (a, 0) and _-a, 0).

• The Latus Rectum is a line drawn through the foci and perpendicular to the major axis. The length of the latus rectum of the ellipse is 2b2/a.

• Transverse Axis: The line passing through the ellipse's centre and the two foci is called the transverse axis.

• Conjugate Axis: The line passing through the ellipse's centre and perpendicular to the major axis is called the conjugate axis.

• Eccentricity is the ratio of the distance of the focus from the centre of the ellipse to the distance of one end of the major axis from the centre of the ellipse. If the distance of the focus from the centre of the ellipse is 'c' and the distance of the end of the major axis from the centre is a, then eccentricity e = c/a.

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General Equation of an Ellipse

For an ellipse with center (h, k) and vertices as (a, 0), (-a, 0), (b, 0) and (-b, 0), the general equation of an ellipse is given as follows:

(x - h)2/a2 + (y - k)2/b2 = 1

Standard Equation of an Ellipse

If we take the coordinate of the center of the ellipse as (0, 0) and the major axis as the x-axis and the conjugate axis as the y-axis, we get the equation as follows:

x2/a2 + y2/b2 = 1

This is the equation of a horizontal ellipse.

If we take the coordinate of the center of an ellipse as (0, 0) and the major axis as the y-axis and the conjugate axis as the x-axis, we get the equation as follows:

x2/b2 + y2/a2 = 1

This is the equation of the vertical ellipse.

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Formula of Eccentricity of an Ellipse

The eccentricity is an important parameter of an ellipse that defines its shape. Take the distance between one of the foci and center of the ellipse. Divide it by the distance between the center and the vertex of that side of the ellipse. The value of this ratio gives the eccentricity of the ellipse.

The formula for eccentricity is:

e = c/a

where ‘c’ is the distance of focus (0, c) from the centre (0,0) and ‘a’ is the distance of the vertex (a, 0) from the centre (0, 0) of the ellipse.

Graphing of an Ellipse

The graphical representation of an ellipse involves the following steps:

• Take the centre of the ellipse as O (0, 0).

• Take the x-axis as the major, and the vertices are (-a, 0) and (a, 0)

• Take the y-axis as a minor axis, and the vertices are (0, -b) and 0, b)

• The ellipse is the closed curve enclosed within a rectangle whose four sides are given by the equations x = a, y = b, x = -a, y = -b. 

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Ellipse Equation Solved Examples

Write the equation of an ellipse with a major axis along the x-axis and centre O (0,0) and passing through the points (8,0), (-8, 0), (0, 4) and (0, -4)

Solution:

The equation of the ellipse is:

x2/a2 + y2/b2 = 1

Here, a = 8 and b =4

So, the equation of the ellipse is

x2/ (8)2 + y2/ (4)2 = 1

or, x2/64 + y2/16 = 1

or, (x2 + 4y2) /64 = 1

or, x2 + 4y2 = 64

Find the eccentricity of the ellipse with centre (0,0), foci (3, 0) and (-3, 0) and vertices (6, 0) and (-6, 0)

Solution:

The eccentricity of an ellipse is:

e = c/a

or, e = 3/6

or, e = ½

The definition of an ellipse and its equation represent an interesting concept in geometry, describing a set of points that maintain the same sum of distances from two foci. Understanding an ellipse's general equations and formulas helps improve mathematical insight and solve various mathematical problems related to physics and engineering.

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