Eccentricity Formula of Circle, Parabola, Ellipse, Hyperbola

Eccentricity is a non-negative real number that describes the shape of a conic section. It measures how much a conic section deviates from being circular. Generally, eccentricity measures the degree to which a conic section differs from a uniform circular shape.

Let’s discuss Eccentricity formula for circle, parabola, ellipse, and hyperbola, along with examples.

Eccentricity in Conic Sections

Eccentricity in Geometry

Eccentricity of a conic section is defined as the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the closest directrix.

Simply put, the Eccentricity of a conic section is a constant value. It’s the ratio of the distance from any point on the curve to its focus and the distance between that point and its closest directrix. This constant is the same for every conic section.

• It is denoted by the letter “e” .

• Eccentricity gives a constant value for that conical shape.

• Eccentricity of a curved shape determines how flattened a circle is.

• Linear eccentricity of a conic section is the distance between the centre of the section and its focus (one or two).

• Ratio of linear eccentricity to semimajor axis is termed as eccentricity.

Eccentricity Definition

Eccentricity can be defined as actually how much deviation or variation a conic section (a Circle, Ellipse, Parabola or Hyperbola) shows from being perfectly circular or round.

A circular curve has zero eccentricity, so the Eccentricity describes how much “un-circular” or “flat” or “elongated” a conic section is. The value of Eccentricity is constant and positive for any conic section.

Eccentricity Formula

Eccentricity formula of different shapes is as follows:

Eccentricity (e) = c/a

where, 

• c = Distance between Any point on Conic Section and Focus
• a = Distance between Any point on Conic Section and Directrix

The Eccentricity formula for different shapes is tabulated below:

Eccentricity Formulas
Shape	Equation	Eccentricity
Circle	√[(x –h)2+( y–k)2]= r	e = 0
Parabola	y = a(x-h)2 + k	e = 1
Ellipse	x2 /a2 +y2/b2= 1	If a > b, e = √(a2-b2)/a2 If b > a, e = √(b2 – a2)/b2
Hyperbola	x2 /a2 – y2/b2= 1	√(a2+b2)/a2

Now let us learn the Eccentricity of different conic sections, namely Circle, Parabola, Ellipse, and Hyperbola.

Eccentricity of Circle

A circle is a set of points in a plane that are all the same distance from a fixed point known as the “centre”. The distance from the centre to any point on the circle is called the “radius”.

If the distance from the centre to focus is zero or in other way the centre of the circle is at the origin of a cartesian plane, we derive the equation of a circle. This Eccentricity presents a uniform circular shape.

Elements of Circle

• Centre: The fixed point at the middle of the circle, equidistant from all points on the circumference.
• Radius: The distance from the centre to any point on the circle, defining its size.

Eccentricity of Circle

Eccentricity of Circle Formula

We derive the equation of the circle as follows:

If “r’ is the radius of circle and C (h, k) is the centre of the circle, Then |CQ| = r.

Formula to find the distance is, √[(x –h)²+( y–k)²] = r

Taking Square on both sides, we get the equation of Circle

(x –h)²+( y – k)² = r²

Thus, the Eccentricity of the circle is zero, i.e.

e = 0

Eccentricity of Parabola

A parabola is defined as a set of points in a plane equidistant from a fixed line called the directrix and a fixed point called the focus. Put simply, the distance from the focus in the plane always has a constant ratio with the distance from the directrix in the plane.

Elements of Parabola

• Focus: A fixed point that plays a crucial role in shaping the parabolic curve.
• Directrix: A fixed straight line, the distance from which determines the parabola’s form.

Eccentricity of Parabola

General equation of a parabola is,

y = a(x-h)² + k

Eccentricity of Parabola Formula

Thus, for Parabola we get always an eccentricity 1,

e = 1

Eccentricity of Ellipse

An ellipse is a closed curve that is symmetric with respect to two perpendicular axes. It can also 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 foci) is constant.

Elements of Ellipse

• Centre: The midpoint of both major and minor axes, specifying the ellipse’s centre.
• Major Axis: The longer diameter passing through the centre, determining the ellipse’s length.
• Minor Axis: The shorter diameter perpendicular to the major axis, establishing the ellipse’s width.

Eccentricity of Ellipse

Therefore,

Eccentricity of ellipse is greater than zero but less than one i.e. (0 < e < 1).

General equation of an Ellipse is

x²/a² + y²/b²= 1

Eccentricity of Ellipse Formula

Ellipse Eccentricity Formula is

e = √(a²– b²)/a²

where,

• a is length of Semi-Major Axis
• b is Length of Semi-Minor Axis

Eccentricity of Hyperbola

A hyperbola is a conic section that is formed when a plane intersects a double right circular cone at an angle. The intersection produces two separate unbounded curves that are mirror images of each other. A hyperbola is an open curve with two branches. The plane does not have to be parallel to the axis of the cone; the hyperbola will be symmetrical in any case.

Elements of Hyperbola

• Centre: Midpoint between the two foci, determining the centre of the hyperbola.
• Transverse Axis: Line passing through the foci, establishing the major axis.
• Conjugate Axis: Perpendicular line to the transverse axis, defining the minor axis.

Eccentricity of Hyperbola

Therefore, Eccentricity of hyperbola is greater than 1, i.e.

e > 1

General equation of a Hyperbola is

x²/a² – y²/b²= 1

Eccentricity of Hyperbola Formula

Eccentricity Formula for Hyperbola is

e = √(a²+b²)/a²

For any Hyperbola, values of a and b are the lengths of the semi-major and semi-minor axis respectively.

Eccentricity of Conic Sections

Eccentricity of a conic section increases if the curvature of the conic section decreases. The gist on the eccentricity of different conic sections is as follows:

• Eccentricity of Circle = 0 (i.e.) e =0.
• Eccentricity of Line = Infinity (i.e.) e =1.
• Eccentricity of Parabola = 1(i.e.) e =1.
• Eccentricity of Ellipse = Between 0 and 1 (i.e.) 0 <e <1.
• Eccentricity of Hyperbola = Greater than 1(i.e.) e > 1.

Eccentricity Solved Examples

Here are some solved examples on the eccentricity of different conic sections.

Example 1: Calculate the Eccentricity for an Ellipse with a semi-major axis of 8 units and a distance from the centre to a focus of 5 units.

Solution:

Formula to calculate Eccentricity (e) for an Ellipse is:

e = c/a

Given values are:

Semi-major axis (a) is 8 units

Distance from the centre to a focus (c) is 5 units

Eccentricity formula is:

e = 5/8

Now, calculate the value of

e = 0.625

So, Eccentricity of the Ellipse is = 0.625

Example 2: Find the Eccentricity of the Ellipse for the given equation 16x² + 25y² = 400

Solution:

Given equation is: 16x² + 25y² = 400

General equation of Ellipse is

x² /a² +y²/b²= 1

To make it in standard form, divide both sides by 400

x² /5² +y²/4²= 1

So, value of semi-major axis length a = 5 and semi-minor axis length b = 4

From the formula of the Eccentricity of an Ellipse,

e = √(a²-b²)/a²

⇒ e = √(5²-4²)/5²

⇒ e = √9/25

⇒ e = 3/5

Therefore, for given equation, the Eccentricity is 3/5.

Example 3: Find the Eccentricity of the conic section (x²/25) + (y²/16) = 1.

Solution:

Given equation is: (x²/25) + (y²/16) = 1

It can also be written as (x²/5²) + (y²/4²) =1

Given conic section is an Ellipse in the form of (x²/a²) + (y²/b²) = 1

Here, a = 5 and b = 4.

We know that c² = a²-b²

⇒ c² = 5² – 4²

⇒ c² = 25 -16 = 9

Hence, c = √9 = 3.

Formula for Eccentricity is:

e = c/a

Now, put the values of c and a, we get

e = 3/5

Hence, the Eccentricity of the given conic section (x²/25) + (y²/16) = 1 is 3/5.

Example 4: Find the Eccentricity of the hyperbola (x²/36) – (y²/9) = 1.

Solution:

Given equation is: (x²/36) – (y²/9) = 1

Given Hyperbola can be written as

(x²/6²) – (y²/3²) = 1

Given conic section is an Hyperbola in the form of (x²/a²) – (y²/b²) = 1

So, axis of Hyperbola is x-axis

Now, by comparing the equation, we get a = 6 and b = 3

Eccentricity formula for Hyperbola is e = √[1+(b²/a²)]

Now, put the values in the formula, we get

e = √[1+(3²/6²)]

⇒ e = √[1+(9/36)]

⇒ e = √(45/16)

⇒ e = 3/4√5

Therefore, 3/4√5 is the Eccentricity of the Hyperbolic equation (x²/36) – (y²/9) = 1.

Eccentricity Practice Questions

Here are some practice problems on eccentricity for you to solve, using the respective formulas:

Q1: For the given equation 16x² – 25y² = 400, find the Eccentricity of the hyperbola.

Q2: For the given equation 9x² + 25y² = 225, find out the Eccentricity of the ellipse.

Q3: Calculate the Eccentricity for an Ellipse with a semi-major axis of 36 units and a distance from the centre to a focus of 16 units.

Q4: Find out the Eccentricity of the Hyperbola y²/9 – x²/25 = 1.

Q5: What is the Eccentricity of the Hyperbola 5y² – 9x² = 25?

Eccentricity Formula- FAQs

What is Eccentricity in Simple Terms?

Eccentricity in conic section is defined as, “distance from any point on curve to its focus, divided by perpendicular distance from that point to its nearest directrix.”

Is the Eccentricity of a line 1?

No, Eccentricity of line is not 1 it is infinity. Eccentricity of parabola is 1.

Define Eccentricity in Geometry.

The eccentricity can be defined as the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the closest directrix.

Why does Circle have Zero Eccentricity?

Circle has zero eccentricity because the focus of the Circle coincides at the centre; means centre of circle is at origin.

What is Eccentricity formula for a Parabola?

To find the Eccentricity of Parabola, we use a general equation y = a(x-h)² + k. This equation gives the Eccentricity of Parabola e = 1.

What is Eccentricity formula of Hyperbola?

Eccentricity (e) of a hyperbola can be calculated using the formula:

e = √(1 + (b² / a²))

What is Eccentricity of an Ellipse?

Eccentricity of Ellipse is greater than zero but less than one i.e. (0 < e <1).

What is Eccentricity of Conjugate Hyperbola?

Eccentricity of Hyperbola is greater than 1, i.e. e > 1. The Eccentricity of a Conjugate Hyperbola can be calculated by using this formula e= √(a²+b²)/a². Here, a and b are the lengths of the semi-major and semi-minor axis respectively.

Which shape has Eccentricity equal to zero?

Circle has Eccentricity equal to Zero.

What is Formula of Eccentricity?

Eccentricity of a conic section is the distance from any point on the conic section to its focus/ the distance of the same point to its directrix. Formula of Eccentricity is e = c/a.

Can Eccentricity be Negative?

No, Eccentricity in any curved section is a unique characteristic of its shape where it does not take any non-negative real numbers.

What can be Maximum and Minimum value of Eccentricity?

Maximum value of Eccentricity is infinity. Eccentricity of a Hyperbola is always greater than 1. Minimum value of Eccentricity is zero. A Circle has zero Eccentricity.