Eccentricity in Conic Sections Explained Clearly

Let’s discuss Eccentricity definition, Eccentricity formula, the Eccentricity of the Circle, Eccentricity of Parabola, the Eccentricity of Ellipse and Eccentricity of Hyperbola

• Eccentricity Definition - Eccentricity can be defined by how much a Conic section (a Circle, Ellipse, Parabola or Hyperbola) actually varies from being circular.

• A Circle has an Eccentricity equal to zero, so the Eccentricity shows you how un - circular the given curve is. Bigger Eccentricities are less curved.

Eccentricity Formula

In Mathematics, for any Conic section, there is a locus of a point in which the distances to the point (Focus) and the line (known as the directrix) are in a constant ratio. This ratio is referred to as Eccentricity and it is denoted by the symbol “e”.

The formula to find out the Eccentricity of any Conic section can be defined as

Eccentricity, Denoted by \[e = \frac{c}{a}\]

Where,

c is equal to the distance from the center to the Focus

a  is equal to the distance from the center to the vertex

So we can say that for any Conic section, the general equation is of the quadratic form:

\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F\] and this equation equals zero.

Now let us discuss the Eccentricity of different Conic sections namely Parabola, Ellipse and Hyperbola in detail.

Eccentricities of Circle, Parabola, Ellipse and Hyperbola

The Eccentricity of Circle

• A Circle can be defined as the set of points in a plane that are equidistant from a fixed point in the plane surface which is known as the “centre”. 

• Now, you might think about what is the radius. The term “radius” is used to define the distance from the centre and the point on the Circle. 

•  If the centre of the Circle is at the origin, it becomes easy to derive the equation of a Circle. 

We can derive the equation of the Circle is derived using the below-given conditions.

In a given Circle if “r” is equal to the radius and C (h, k) is equal to the centre of the Circle, then by the definition of Circle and Eccentricity, we get,

| CP | = radius(r)

We know that the formula to find the distance is,

\[\sqrt{(x-h)^2+(y-k)^2}\]= radius(r)

Taking Square on both the sides, we get the following equation,

\[(x-h)^2+(y-k)^2 = radius ^2\]

Thus, the equation of the Circle with center C (h, k) and radius equal to “r” can be written as \[(x –h)^2+( y–k)^2= r^2\]

Also, e = 0 for a Circle.

The Eccentricity of Parabola

• A Parabola in Mathematics 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, we can say that 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 always equal to 1 ( e=1)

The general equation of a Parabola can be written as x2 = 4ay and the Eccentricity is always given as 1.

Fun Fact: Whenever a projectile is launched it covers a Parabolic path. This is because along with a horizontal component it is accompanied by a vertical component as well. The projectile covers the longest horizontal distance when it is thrown at 45 degrees. So next time when you throw a ball and want to impress your friends, make sure you launch the projectile at 45 degrees and see the Parabolic motion in front of your eyes!

The Eccentricity of Ellipse

• An Ellipse can be defined as the set of points in a plane in which the sum of distances from two fixed points is constant. 

• In simple 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 denoted as \[\frac{\sqrt{a^2-b^2}}{a} \]

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

Fun Fact: You might find it interesting to know that Ellipse and its related concepts have a very wide application in the field of space sciences. Because the majority of the celestial bodies like planets, satellites, comets, man-made satellites, etc. tend to travel in elliptical orbits. Apart from the concept of Eccentricity, other concepts like apogee and perigee come into the picture while understanding the positions on these bodies. This makes the study of Ellipses and related concepts even more interesting. 

The Eccentricity of Hyperbola

• A Hyperbola is defined as the set of all points in a plane where the difference of whose distances from two fixed points is constant. 

• In simpler 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 always greater than 1. i.e., e > 1

The general equation of a Hyperbola is denoted as \[\frac{\sqrt{a^2+b^2}}{a} \]

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

Fun Fact: Scientists use the concepts related to Hyperbola to position radio stations. This ensures optimization of the area covered by the signals from a station. This enables people to locate objects over a wide area. This application played an important role in world war two.

Different Values of Eccentricity Make Different Curves:

Eccentricity is often represented as the letter (Keep in mind you don't confuse this with Euler's number E, they are different).

Calculating the Value of Eccentricity (Eccentricity Formula):

Questions to be Solved

1.  List down the formulas for calculating the Eccentricity of Hyperbola and Ellipse.

Ans: For a Hyperbola, the value of Eccentricity is: \[\frac{\sqrt{a^2+b^2}}{a} \]

For an Ellipse, the value of Eccentricity is equal to \[\frac{\sqrt{a^2-b^2}}{a} \]

2. List down the formulas for calculating the Eccentricity of Parabola and Circle.

Ans: For a Parabola, the value of Eccentricity is 1

For a Circle, the value of Eccentricity = 0. Because for a Circle a=b

Where, a is the semi-major axis and b is the semi-minor axis for a given Ellipse in the question

Eccentricity from Vedantu’s Website

All the content related to Eccentricities of Parabola, Circle, Hyperbola and Ellipse on this website are prepared by subject matter experts of Vedantu. These experts have years of expertise in the field of Mathematics. They have closely monitored the past question papers over the years for various exams such as Class 11, Class 12 board exams, IIT-JEE exams, State CET exams, etc. and only after thorough research and analysis the content on this page has been made available to you. 

To leverage these efforts which are taken on your behalf by the Vedantu experts you should refer to the concept of Eccentricity only from the Vedantu website. This will ensure you get an edge over others and perform very well in all exams where questions related to this topic are asked.

Other Concepts Explained by Vedantu

Apart from Eccentric, Vedantu has explained many other Mathematical concepts on its website. These can help students in making Math easy and fun for them. All the concepts are explained by the subject matter experts of Vedantu only. Other related topics that you may find interesting are as follows:

These concepts along with the concept of Eccentricity will help you solidify your concepts in geometry and assist you in coming out with flying colours.