Eccentricity – Formula for Circle, Ellipse, Parabola and Hyperbola
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
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 \[\frac{\sqrt{a^2-b^2}}{a} \]
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.
FAQs on Eccentricity
1. What is the Formula for Eccentricity for an Ellipse?
To find the Eccentricity of an Ellipse formula used as \[ e = \sqrt{1-\frac{b^2}{a^2}}\]. Where a is the length of the semi-major axis and b is the length of the semi-minor axis. It should be noted that if you have an Ellipse with the major and minor axes of equal lengths then this Ellipse becomes a Circle. Thus, Eccentricity for this particular particle will be e=0. Since “a” is the length of the semi-major axis, a >= b and therefore 0 <= e < 1 for all the Ellipses.
2. What is the Use of Eccentricity in Space Technology?
Eccentricity can be defined as a measure of how an orbit deviates from circular. A perfectly circular orbit has an Eccentricity equal to zero; the higher numbers indicate more elliptical orbits. The planets Neptune, Venus, and Earth in our solar system are the planets with the least Eccentric orbits. These planets revolve in an elliptical orbit. Eccentricity calculation helps in determining the location of the planet, apogee, perigee, etc. Thus Eccentricity has a wide application in space technology. You can read more about Eccentricity by clicking here.
3. What Happens when the Eccentricity of an Ellipse Increases?
Generally, an Ellipse has an Eccentricity within the range 0 < e < 1, while a Circle is a special case where the value of Eccentricity (e=0). An increase in the Eccentricity of an Ellipse implies that the length of the semi-minor axis is nearing zero. This implies at e=1 (maximum possible value), b=0. Thus only the semi-major axis remains. This implies Ellipse converts into a straight line at e=1. Therefore, we conclude that as the Eccentricity of an Ellipse goes on increasing it approaches towards becoming a straight line.
4. Can Eccentricity be Negative?
In Mathematics, the Eccentricity of a Conic section is equal to a non-negative real number that uniquely characterizes the shape of a Conic section. The Eccentricity of an Ellipse which is not a Circle is always greater than zero but less than 1. You can find more about Ellipses from Vedantu’s website. The Eccentricity of a Parabola is equal to1. The Eccentricity of a Hyperbola is greater than the value 1. The Eccentricity of a Circle is 0, which is a special case of an Ellipse.
5. How useful is the concept of Eccentricities in Class 11 and Class 12th?
Eccentricities of Circle, Ellipse, Parabola and Hyperbola - all of these topics hold a lot of importance in Class 11 and Class 12. CBSE Class 11 and other boards as well have a specific chapter dedicated to these topics in Mathematics. All the formulae mentioned here are very useful in understanding the problems there. These concepts have a wide application in Class 11 and Class 12 Physics as well. Therefore this topic should not be taken lightly in any standard as it forms the base for many advanced and complex calculations ahead.