Derivation of Eccentricity of Ellipse
Ratio of the distance between the foci of the ellipse to the length of the major axis is called eccentricity of ellipse.
Let’s consider an ellipse with a semi-major axis of length a and a semi-minor axis length b. The foci of the ellipse are located at a distance c from the center where c can be calculated using the relationship:
c = √(a2-b2)…(i)
And eccentricity is given by:
e = c/a…(ii)
Placing the value of c from (i) in (ii):
e = √(a2-b2) /a…(iii)
Squaring on both sides of (iii)
e2 = {√(a2-b2) /a}2
e2 = (a2 – b2)/a2
Rearranging the above equation:
a2e2 = a2 – b2
a2e2+b2 = a2
(a2e2 + b2)/a2 = 1
a2e2/a2 + b2/a2 = 1
e2 + b2/a2 = 1
e2 = 1 – b2/a2
e = √(1 – b2/a2)…(iv)
e = √{ 1 – ( 2b/2a )2}
e = √( 1 – (minor axis/major axis)2)
Hence we have derived the equation (iv), known as the eccentricity formula for an ellipse.
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Eccentricity of Ellipse
Eccentricity of Ellipse: eccentricity is a measure that describes how much a conic section deviates from being circular. For any point on a conic section, eccentricity is defined as the ratio of the distance to a fixed point (focus) to the distance to a fixed line (directrix).
The eccentricity of an ellipse is the ratio of the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse. It is denoted using the letter, ‘e’ and is calculated as, e = c/a where a is the length of the semi-major axis and c is the distance from the centre to the foci.
In this article, we will learn about Ellipse, Eccentricity of Ellipse, Formula for eccentricity of ellipse and others in detail.
Table of Content
- What is an Ellipse?
- Eccentricity of Ellipse
- Eccentricity of Ellipse Formula
- Eccentricity of Ellipse- Diagram
- Eccentricity of Circle
- Eccentricity of Parabola
- Eccentricity of Ellipse
- Derivation of Eccentricity of Ellipse
- Eccentricity of Ellipse Examples
- Practice Problems on Eccentricity of Ellipse