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 that 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 depends 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.

Definition

For any conic section, there is a locus of a point in which the distances to the point(focus) and the line(directrix) are in the constant ratio. That ratio is known as eccentricity and it is denoted by the symbol “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 we will discuss the eccentricity of different conic sections like parabola, ellipse and hyperbola in detail.

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

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

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

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

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

Solved Problem

Question:

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

Solution:

Given :

9x2 + 25y2 = 225

The general form of ellipse is

\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\)

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

\(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\)

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

From the formula of the eccentricity of an ellipse, \(\sqrt{1-\frac{b^{2}}{a^{2}}}\) \(e = \sqrt{1-\frac{3^{2}}{5^{2}}} =\sqrt{\frac{25-9}{25}} =\sqrt{\frac{19}{25}}\)

e = 4/ 5

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

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