Eccentricity

The eccentricity in the conic section uniquely characterises the shape where it should possess a non-negative real number. In general, eccentricity means a measure of how much the deviation of the curve has occurred from the circularity of the given shape. We know that the section obtained after the intersection of a plane with the cone is called the conic section. We will get different kinds of conic sections depending on the position of the intersection of the plane with respect to the plane and the angle made by the vertical axis of the cone. In terms of fixed-point called focus and the fixed-line called the directrix in the plane, the term “eccentricity” is defined. In this article, we are going to discuss the eccentric meaning in geometry, and eccentricity formula and the eccentricity of different conic sections such as parabola, ellipse and hyperbola in detail with solved examples.

Table of Contents:

Eccentric Meaning in Geometry

The eccentric meaning in geometry represents the distance from any point on the conic section to the focus divided by the perpendicular distance from that point to the nearest directrix. Generally, the eccentricity helps to determine the curvature of the shape. If the curvature decreases, the eccentricity increases. Similarly, if the curvature increases, the eccentricity decreases.

Eccentricity Definition

We know that there are different conics such as a parabola, ellipse, hyperbola and circle. The eccentricity of the conic section is defined as the distance from any point to its focus, divided by the perpendicular distance from that point to its nearest directrix. The eccentricity value is constant for any conics. The letter used to represent eccentricity is “e”.

Eccentricity Formula

The formula to find out the eccentricity of any conic section is defined as:

Eccentricity, e = c/a

Where,

c = distance from the centre to the focus

a = distance from the centre to the vertex

For any conic section, the general equation is of the quadratic form:

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

Here you can learn the eccentricity of different conic sections like parabola, ellipse and hyperbola in detail.

Eccentricity of Circle

A circle is defined as the set of points in a plane that are equidistant from a fixed point in the plane surface called “centre”. The term “radius” defines the distance from the centre and the point on the circle.  If the centre of the circle is at the origin, it will be easy to derive the equation of a circle. The equation of the circle is derived using the below-given conditions.

If “r’ is the radius and C (h, k) be the centre of the circle, by the definition, we get, | CP | = r.

We know that the formula to find the distance is,

√[(x –h)2+( y–k)2]= r

Take Square on both sides, we get

(x –h)2+( y–k)2= r2

Thus, the equation of the circle with centre C(h, k) and radius “r” is (x –h)2+( y–k)2= r2

Also, the eccentricity of the circle is equal 0, i.e. e = 0.

Eccentricity of Parabola

A parabola is defined as the set of points P in which the distances from a fixed point F (focus) in the plane are equal to their distances from a fixed-line l(directrix) in the plane. In other words, the distance from the fixed point in a plane bears a constant ratio equal to the distance from the fixed-line in a plane.

Therefore, the eccentricity of the parabola is equal 1, i.e. e = 1.

The general equation of a parabola is written as x2 = 4ay and the eccentricity is given as 1.

Eccentricity of Ellipse

An ellipse is defined as the set of points in a plane in which the sum of distances from two fixed points is constant. In other words, the distance from the fixed point in a plane bears a constant ratio less than the distance from the fixed-line in a plane.

Therefore, the eccentricity of the ellipse is less than 1, i.e. e < 1.

The general equation of an ellipse is written as:

\(\begin{array}{l}\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\end{array} \)
and the eccentricity formula is written as
\(\begin{array}{l}\sqrt{1-\frac{b^{2}}{a^{2}}}\end{array} \)

For an ellipse, a and b are the lengths of the semi-major and semi-minor axes respectively.

Eccentricity of Hyperbola

A hyperbola is defined as the set of all points in a plane in which the difference of whose distances from two fixed points is constant. In other words, the distance from the fixed point in a plane bears a constant ratio greater than the distance from the fixed-line in a plane.

Therefore, the eccentricity of the hyperbola is greater than 1, i.e. e > 1.

The general equation of a hyperbola is given as

\(\begin{array}{l}\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\end{array} \)
and the eccentricity formula is written as
\(\begin{array}{l}\sqrt{1+\frac{b^{2}}{a^{2}}}\end{array} \)

For any hyperbola, a and b are the lengths of the semi-major and semi-minor axes respectively.

Eccentricity of Conic Sections – Summary

We know that the eccentricity of the conic section increases if the curvature of the conic section decreases. The summary on the eccentricity of different conic sections is given below:

  • 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.

Solved Problems on Eccentricity

Example 1: 

Find the eccentricity of the ellipse for the given equation 9x2 + 25y2 = 225

Solution:

Given :

9x2 + 25y2 = 225

The general form of ellipse is

\(\begin{array}{l}\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\end{array} \)

To make it in general form, divide both sides by 225, we get

\(\begin{array}{l}\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\end{array} \)

So, the value of a = 5 and b = 3

From the formula of the eccentricity of an ellipse,

\(\begin{array}{l}e=\sqrt{1-\frac{b^{2}}{a^{2}}}\end{array} \)

Substituting a = 5 and b = 3,

\(\begin{array}{l}e = \sqrt{1-\frac{3^{2}}{5^{2}}} =\sqrt{\frac{25-9}{25}} =\sqrt{\frac{16}{25}}\end{array} \)

e = 4/ 5

Therefore, the eccentricity of the given ellipse is 4/5.

Example 2: 

Find the eccentricity of the conic section (x2/36) + (y2/16) = 1

Solution:

Given: (x2/36) + (y2/16) = 1

It can also be written as (x2/62) + (y2/42) =1

The given conic section is an ellipse, as it holds the form (x2/a2) + (y2/b2) = 1

Here, a = 6 and b = 4.

We know that c2 = a2-b2

c2 = 62 – 42

c2 = 36 -16 = 20

Hence, c = √20 = 2√5.

We know that the formula for eccentricity is:

e = c/a

Now, substitute the values in the formula, we get

e = (2√5)/6

e = √5/3

Hence, the eccentricity of the given conic section (x2/36) + (y2/16) = 1 is √5/3.

Example 3:

Find the eccentricity of the hyperbola (x2/16) – (y2/9) = 1.

Solution:

Given: (x2/16) – (y2/9) = 1

The given hyperbola equation can also be written as

(x2/42) – (y2/32) = 1 …(1)

Hence, the above equation is of the form: (x2/a2) – (y2/b2) = 1 …(2)

So, the axis of the hyperbola is the x-axis.

Now, by comparing the equation (1) and (2), we get a = 4 and b = 3. 

We know that the eccentricity formula for hyperbola is e = √[1+(b2/a2)]

Now, substitute the values in the formula, we get

e = √[1+(32/42)]

e = √[1+(9/16)]

e = √(25/16)

e = 5/4.

Therefore, the eccentricity of the hyperbolic equation (x2/16) – (y2/9) = 1 is 5/4.

Practice Questions on Eccentricity

  1. Determine the eccentricity of the equation (x2/4) + (y2/25) = 1.
  2. Find the eccentricity of the ellipse 36x2 + 4y= 144.
  3. Find the eccentricity of the hyperbola 5y2 – 9x= 36.

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Frequently Asked Questions on Eccentricity

Define eccentric meaning in geometry.

The eccentric meaning in geometry defines the ratio of the distance from any point on the conic section to the focus to the perpendicular distance from that point to the nearest directrix.

What is the eccentricity of parabola?

The eccentricity of the parabola is 1. (i.e) e = 1.

What is the eccentricity of Hyperbola?

The eccentricity of the hyperbola is greater than 1. (i.e) e > 1.

What is the eccentricity of a circle?

The eccentricity of a circle is 0.

Does eccentricity of the conic section increase with the decrease in the curvature of the shape?

Yes, the eccentricity of the conic section increases with the decreases in the curvature of the shape and vice versa.

Test your knowledge on Eccentricity

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