Eccentricity

What is Eccentricity?

Let’s discuss eccentricity definition, eccentricity formula, eccentricity of circle, eccentricity of parabola, eccentricity of ellipse and eccentricity of hyperbola

• Eccentricity definition – Eccentricity can be defined 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.

Different Values of Eccentricity Make Different Curves:

 At eccentricity equal to 0 We obtain a circle For 0 < eccentricity < 1 We obtain an ellipse. For the value of  eccentricity = 1 We obtain a parabola. For the value of eccentricity > 1 We obtain a hyperbola. For the value of eccentricity =∞ We obtain a line.

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

Calculating the value of eccentricity (Eccentricity Formula):

 Eccentricity of Circle: For a circle, the value of eccentricity is equal to 0. Eccentricity of Ellipse: For an ellipse, the value of eccentricity is equal to:  $\frac{\sqrt{a²-b²}}{a}$ Eccentricity of Parabola: For a parabola, the value of eccentricity is 1. Eccentricity of Hyperbola: For a hyperbola, the value of eccentricity is: $\frac{\sqrt{a²+b²}}{a}$

Eccentricity Definition

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”.

Eccentricity Formula

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 centre to the focus

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

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

Ax2 + Bxy + Cy2 + 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.

Eccentricity of Circle

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

• Now, you might think what is 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 the centre of the circle, then by the definition of circle and eccentricity , we get,

We know that the formula to find the distance is,

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

Thus, the equation of the circle with centre C (h, k) and radius equal to “r” can be written as (x –h)2+( y–k)2= r2

Also, e = 0 for a circle.

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 to1 ( e=1)

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

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 $\frac{\sqrt{a²-b²}}{a}$

For an ellipse,  the values 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 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 $\frac{\sqrt{a²+b²}}{a}$

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

Questions to be Solved:

Question 1) List down the formulas for calculating the eccentricity of hyperbola and parabola.

For a hyperbola, the value of eccentricity is: $\frac{\sqrt{a²+b²}}{a}$

For an ellipse, the value of eccentricity is equal to: $\frac{\sqrt{a²-b²}}{a}$

Q1. What is the Formula for Eccentricity?

Ans. To find the eccentricity of an ellipse. This is basically given as e = (1-b2/a2)1/2. Note that if have a given ellipse with the major and minor axes of equal length have an eccentricity of 0 and is therefore a circle. Since a is the length of the semi-major axis, a >= b and therefore 0 <= e < 1 for all the ellipses.

Q2. What is Eccentricity Science?

Ans. 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.

Q3. What Happens when Eccentricity Increases?

Ans. Generally an ellipse has an eccentricity within the range 0 < e < 1, while a circle is the special case where the value of eccentricity (e=0). Elliptical orbits with increasing eccentricity from e=0 (a circle) to eccentricity equal to 0.95. For a definite fixed value of the semi-major axis, as the value of the eccentricity increases, both the semi-minor axis and perihelion distance decreases.

Q4. Can Eccentricity be Negative?

Ans. 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. The eccentricity of a parabola is equal to1. The eccentricity of a hyperbola is greater than the value 1.