Ellipse

Ellipse is an integral part of the conic section and is similar in properties to a circle. Unlike the circle, an ellipse is oval in shape. An ellipse has an eccentricity less than one, and it represents the locus of points, the sum of whose distances from the two foci is a constant value. Simple examples of the ellipse in our daily life is the shape of an egg in a two-dimensional form, the running tracking in a sports stadium, etc.

Here we shall aim at knowing the definition of an ellipse, the derivation of the equation of an ellipse, and the different standard forms of equations of the ellipse.

In mathematics, an ellipse is a closed curve that is symmetric with respect to two perpendicular axes. It can be defined as the set of all points in a plane, such that the sum of the distances from any point on the curve to two fixed points (called the foci) is constant. The fixed points are called the foci and are denoted by F and F'.

An ellipse is one of the conic sections, which is the intersection of a cone with a plane that does not intersect the cone's base.

Ellipse Definition

An ellipse is the locus of points in a plane, the sum of whose distances from two fixed points is a constant value. The two fixed points are called the foci of the ellipse.

Ellipse Equation

The general equation of an ellipse is used to algebraically represent an ellipse in the coordinate plane. The equation of an ellipse can be given as,

\(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\), where

• 'a' represents the semi-major axis (half of the length of the major axis)
• 'b' represents the semi-minor axis (half of the length of the minor axis)

Let us go through a few important terms relating to different parts of an ellipse.

• Focus: The ellipse has two foci and their coordinates are F(c, o), and F'(-c, 0). The distance between the foci is thus equal to 2c.
• Center: The midpoint of the line joining the two foci is called the center of the ellipse. It is denoted by (h, k).
• Major Axis: Major axis is the longest diameter of an ellipse. The length of the major axis of the ellipse is 2a units, and the end vertices of this major axis is (a, 0), (-a, 0) respectively.
• Minor Axis: Minor axis is the shortest diameter of an ellipse. The length of the minor axis of the ellipse is 2b units and the end vertices of the minor axis is (0, b), and (0, -b) respectively.
• Transverse Axis: The line passing through the two foci and the center of the ellipse is called the transverse axis.
• Latus Rectum: The latus rectum is a line drawn perpendicular to the transverse axis of the ellipse and is passing through the foci of the ellipse. The length of the latus rectum of the ellipse is 2b²/a.
• Conjugate Axis: The line passing through the center of the ellipse and perpendicular to the transverse axis is called the conjugate axis
• Eccentricity: (e < 1). The ratio of the distance of the focus from the center of the ellipse, and the distance of one end of the major axis from the center of the ellipse. If the distance of the focus from the center of the ellipse is 'c' and the distance of the end of the major axis from the center is 'a', then eccentricity e = c/a.

There are two standard equations of the ellipse. These equations are based on the transverse axis and the conjugate axis of each of the ellipse.

• The standard equation of the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) has the transverse axis as the x-axis and the conjugate axis as the y-axis.
• Another ellipse standard form is \(\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1\) and it has the transverse axis as the y-axis and its conjugate axis as the x-axis.

The below image shows the two standard forms of ellipse.

The first step in the process of deriving the equation of the ellipse is to derive the relationship between the semi-major axis, semi-minor axis, and the distance of the focus from the center. The aim is to find the relationship across a, b, c. The length of the major axis of the ellipse is 2a and the length of the minor axis of the ellipse is 2b. The distance between the foci is equal to 2c. Let us take a point P at one end of the major axis and aim at finding the sum of the distances of this point from each of the foci F and F'.

PF + PF' = OP - OF + OF' + OP

= a - c + c + a

PF + PF' = 2a

Now let us take another point Q at one end of the minor axis and aim at finding the sum of the distances of this point from each of the foci F and F'.

QF + QF' = \(\sqrt{b^2 + c^2}\) + \(\sqrt{b^2 + c^2}\)

QF + QF' = 2\(\sqrt{b^2 + c^2}\)

The points P and Q lie on the ellipse, and as per the definition of the ellipse for any point on the ellipse, the sum of the distances from the two foci is a constant value.

2\(\sqrt{b^2 + c^2}\) = 2a

\(\sqrt{b^2 + c^2}\) = a

b² + c² = a²

c² = a² - b²

Let us now check, how to derive the equation of an ellipse. Now we consider any point S(x, y) on the ellipse and take the sum of its distances from the two foci F and F', which is equal to 2a units. If we observe the above few steps, we have already proved that the sum of the distances of any point on the ellipse from the foci is equal to 2a units.

SF + SF' = 2a

\(\sqrt{(x + c)^2 + y^2}\) + \(\sqrt{(x - c)^2 + y^2}\) = 2a

\(\sqrt{(x + c)^2 + y^2}\) = 2a - \(\sqrt{(x - c)^2 + y^2}\)

Now we need to square on both sides to solve further.

(x + c)² + y² = 4a² + (x - c)² + y² - 4a\(\sqrt{(x - c)^2 + y^2}\)

x² + c² + 2cx + y² = 4a² + x² + c² - 2cx + y² - 4a\(\sqrt{(x - c)^2 + y^2}\)

4cx - 4a² = - 4a\(\sqrt{(x - c)^2 + y^2}\)

a² - cx = a\(\sqrt{(x - c)^2 + y^2}\)

Squaring on both sides and simplifying, we have.

\(\dfrac{x^2}{a^2} - \dfrac{y^2}{c^2 - a^2} =1\)

Since we have c² = a² - b² we can substitute this in the above equation.

\(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} =1\)

This derives the standard equation of the ellipse.

There are different formulas associated with the shape ellipse. These ellipse formulas can be used to calculate the perimeter, area, equation, and other important parameters.

Perimeter of an Ellipse Formulas

Perimeter of an ellipse is defined as the total length of its boundary and is expressed in units like cm, m, ft, yd, etc. The perimeter of ellipse can be approximately calculated using the general formulas given as,
P ≈ π (a + b)

P ≈ π √[ 2 (a² + b²) ]

P ≈ π [ (3/2)(a+b) - √(ab) ]

where,

• a = length of semi-major axis
• b = length of semi-minor axis

Area of Ellipse Formula

The area of an ellipse is defined as the total area or region covered by the ellipse in two dimensions and is expressed in square units like in², cm², m², yd², ft², etc. The area of an ellipse can be calculated with the help of a general formula, given the lengths of the major and minor axis. The area of ellipse formula can be given as,

Area of ellipse = π a b
where,

• a = length of semi-major axis
• b = length of semi-minor axis

Eccentricity of an Ellipse Formula

Eccentricity of an ellise is given as the ratio of the distance of the focus from the center of the ellipse, and the distance of one end of the ellipse from the center of the ellipse

Eccentricity of an ellipse formula, e = \( \dfrac ca = \sqrt{1- \dfrac{b^2}{a^2} }\)

Latus Rectum of Ellipse Formula

Latus rectum of of an ellipse can be defined as the line drawn perpendicular to the transverse axis of the ellipse and is passing through the foci of the ellipse. The formula to find the length of latus rectum of an ellipse can be given as,

L = 2b²/a

Formula for Equation of an Ellipse

The equation of an ellipse formula helps in representing an ellipse in the algebraic form. The formula to find the equation of an ellipse can be given as,

Equation of the ellipse with centre at (0,0) : x²/a² + y²/b² = 1

Equation of the ellipse with centre at (h,k) : (x-h)² /a² + (y-k)²/ b² =1

Example: Find the area of an ellipse whose major and minor axes are 14 in and 8 in respectively.

Solution:

To find: Area of an ellipse

Given: 2a = 14 in

a = 14/2 = 7

2b = 8 in

b = 8/2 = 4

Now, applying the ellipse formula for area:

Area of ellipse = π(a)(b)

= π(7)(4)

= 28π

≈ 88 in²

Answer: Area of the ellipse = 88 in².

There are different properties that help in distinguishing an ellipse from other similar shapes. These properties of an ellipse are given as,

• An ellipse is created by a plane intersecting a cone at the angle of its base.
• All ellipses have two foci or focal points. The sum of the distances from any point on the ellipse to the two focal points is a constant value.
• There is a center and a major and minor axis in an ellipse.
• The eccentricity value of an ellipse is less than one.

Let us check through three important terms relating to an ellipse.

• Auxiliary Circle: A circle drawn on the major axis of the ellipse is called the auxiliary circle. The equation of the auxiliary circle to the ellipse is x² + y² = a².
• Director Circle: The locus of the points of intersection of the perpendicular tangents drawn to the ellipse is called the director circle. The equation of the director circle of the ellipse is x² + y² = a² + b²
• Parametric Coordinates: The parametric coordinates of any point on the ellipse is (x, y) = (a cos θ, b sin θ). These coordinates represent all the points of the coordinate axes and it satisfies all the equations of the ellipse.

To draw an ellipse in math, there are certain steps to be followed. The stepwise method to draw an ellipse of given dimensions is given below.

• Decide what will be the length of the major axis, because the major axis is the longest diameter of an ellipse.
• Draw one horizontal line of the major axis' length.
• Mark the mid-point with a ruler. This can be done by taking the length of the major axis and dividing it by two.
• Construct a circle of this diameter with a compass.
• Decide what will be the length of the minor axis, because the minor axis is the shortest diameter of an ellipse.
• Now, at the mid-point of the major axis, you take the protractor and set its origin. At 90 degrees, mark the point. Then swing 180 degrees with the protractor and mark the spot. You may now draw the minor axis between or within the two marks at its midpoint.
• Draw a circle of this diameter with a compass as we did for the major axis.
• Use a compass to divide the entire circle into twelve 30 degree parts. Setting your protractor on the main axis at the origin and labeling the intervals of 30 degrees with dots will do this. Then with lines, you can link the dots through the middle.
• Draw horizontal lines (except for the major and minor axes) from the inner circle.
  They are parallel to the main axis, and from all the points where the inner circle and 30-degree lines converge, they go outward.
  Try drawing the lines a little shorter near the minor axis, but draw them a little longer as you move toward the major axis.
• Draw vertical lines (except for the major and minor axes) from the outer circle.
  These are parallel to the small axis, and from all the points where the outer circle and 30-degree lines converge, they go inward.
  Try to draw the lines a little longer near the minor axis, but when you step towards the main axis, draw them a little shorter.
  You can take a ruler and stretch it a little before drawing the vertical line if you detect that the horizontal line is too far.
• Do your best with freehand drawing to draw the curves between the points by hand.

Let us see the graphical representation of an ellipse with the help of ellipse formula. There are certain steps to be followed to graph ellipse in a cartesian plane.

Step 1: Intersection with the co-ordinate axes

The ellipse intersects the x-axis in the points A (a, 0), A'(-a, 0) and the y-axis in the points B(0,b), B'(0,-b).

Step 2 : The vertices of the ellipse are A(a, 0), A'(-a, 0), B(0,b), B'(0,-b).

Step 3 : Since the ellipse is symmetric about the coordinate axes, the ellipse has two foci S(ae, 0), S'(-ae, 0) and two directories d and d' whose equations are \(x = \frac{a}{e}\) and \(x = \frac{-a}{e}\). The origin O bisects every chord through it. Therefore, the origin O is the centre of the ellipse. Thus it is a central conic.

Step 4: The ellipse is a closed curve lying entirely within the rectangle bounded by the four lines \(x = \pm a\) and \(y = \pm b\).

Step 5: The segment \(AA'\) of length \(2a\) is called the major axis and the segment \(BB'\) of length \(2b\) is called the minor axis. The major and minor axes together are called the principal axes of the ellipse.

The length of semi-major axis is \(a\) and semi-minor axis is b.

☛Related Topics:

• Coordinate Geometry
• Conics in Real Life
• Cartesian Coordinates
• Parabola
• Hyperbola

What is the Definition of Ellipse?

An ellipse is the locus of a point whose sum of the distances from two fixed points is a constant value. The two fixed points are called the foci of the ellipse, and the equation of the ellipse is \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\). Here

• a is called the semi-major axis
• b is called the semi-minor axis of the ellipse.

What is the Equation of Ellipse?

The equation of the ellipse is \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\). Here a is called the semi-major axis and b is the semi-minor axis. For this equation,

• the origin is the centre of the ellipse
• the x-axis is the transverse axis, and
• the y-axis is the conjugate axis.

How to Find the Equation of an Ellipse?

The equation of the ellipse can be derived from the basic definition of the ellipse: An ellipse is the locus of a point whose sum of the distances from two fixed points is a constant value. Let the fixed point be P(x, y), the foci are F and F'. Then the condition is PF + PF' = 2a. On further substitutions and simplification, we have the equation of the ellipse as \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\).

What are the Properties of Ellipse?

The different properties of an ellipse are as given below,

• When a plane intersects a cone at the angle of its base, an ellipse is formed.
• All ellipses have two foci, a center, and a major and minor axis.
• The sum of the distances from any point on the ellipse to the two foci gives a constant value.
• The value of eccentricity for all ellipses is less than one.

What is the Eccentricity of Ellipse?

The eccentricity of the ellipse refers to the measure of the curved feature of the ellipse. For an ellipse, the eccentric is always greater than one. (e < 1). Eccentricity is the ratio of the distance of the focus and one end of the ellipse, from the center of the ellipse. If the distance of the focus from the center of the ellipse is 'c' and the distance of the end of the ellipse from the center is 'a', then eccentricity e = c/a.

What are the Foci of an Ellipse?

The ellipse has two foci, F and F'. The midpoint of the two foci of the ellipse is the center of the ellipse. All the measurements of the ellipse are with reference to these two foci of the ellipse. As per the definition of an ellipse, an ellipse includes all the points whose sum of the distances from the two foci is a constant value.

What is the General Equation of Ellipse?

The general equation of ellipse is given as:

• With centre at (0,0) : x²/a² + y²/b² = 1
• With centre at (h,k) : (x-h)² /a² + (y-k)²/ b² =1

where, a is length of semi-major axis and b is length of semi-minor axis.

What is the Standard Equation of an Ellipse?

The standard equation of ellipse is used to represent a general ellipse algebraically in its standard form. The standard equations of an ellipse are given as,

• \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\), for the ellipse having the transverse axis as the x-axis and the conjugate axis as the y-axis.
• \(\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1\), for the ellipse having transverse axis as the y-axis and its conjugate axis as the x-axis.

What are the Vertices of an Ellipse?

There are four vertices of the ellipse. The length of the major axis of the ellipse is 2a and the endpoints of the major axis are (a, 0), and (-a, 0). The length of the minor axis of the ellipse is 2b and the endpoints of the minor axis are (0, b), and (0, -b).

What is the Conjugate Axis of an Ellipse?

The axis passing through the center of the ellipse, and which is perpendicular to the line joining the two foci of the ellipse is called the conjugate axis of the ellipse. The standard equation of ellipse is \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\), its minor axis is y-axis, and it is the conjugate axis.

What are Asymptotes of Ellipse?

The ellipse does not have any asymptotes. Asymptotes are the lines drawn parallel to a curve and are assumed to meet the curve at infinity. We can draw asymptotes for a hyperbola.

How to Find the Transverse Axis of an Ellipse?

The line passing through the two foci and the center of the ellipse is called the transverse axis of the ellipse. The major axis of the ellipse falls on the transverse axis of the ellipse. For an ellipse having the center and the foci on the x-axis, the transverse axis is the x-axis of the coordinate system.