The eccentricity of the conic section characterizes the form in which it should have a non-negative real number. In general, eccentricity is the measure of how much a curve deviates from the roundness of a particular shape. The cross-section created by the intersection of a plane and a cone is known to be called a conic section.
Different types of conic sections can be obtained, depending on the position of the intersection of the plane concerning the plane and the angle between the vertical axes of the cone. The term “eccentricity” is defined as a fixed point called the focal point, and a fixed line in the plane called the directrix. Every conic section has a point position where the distance between the point (focal point) and the line (direction) is a constant ratio.
This relationship is known as eccentricity, and the symbol “e” indicates it.
The higher the eccentricity, the smaller the curve. From this, we conclude that the curvature of these conic sections decreases with increasing eccentricity.
- The eccentricity of a circle = 0
- The eccentricity of an ellipse = between 0 and 1
- The eccentricity of a parabola = 1
- The eccentricity of a hyperbola > 1
- The eccentricity of a line = infinity
The formula for finding the eccentricity of a conic section is defined as:
Eccentricity, e = c /a, where
c = center-to-focus distance
a = center-to-vertex distance
For each conic section, the general formula quadratic form: Ax2 Bxy Cy2 Dx Ey F = 0
Eccentricity of Ellipse
Eccentricity an ellipse is defined as a set of points in a plane where the sum of the distances from two fixed points is constant. In other words, the distance from the fixed point in the plane is a constant percentage, less than the distance from the fixed point in the plane.
Therefore, the eccentricity of the ellipse is less than 1 i.e e < 1.
The general formula for the ellipse is written as:
x2/a2 y2/b2=1, and the eccentricity formula is written as √1−b2/a2.
In an ellipse, a and b are the lengths of the main and sub-half axes, respectively.
Eccentricity of Circle
A circle is defined as a set of points in a plane equidistant from a fixed point on the plane called the “center”. The term “radius” defines the distance between a center point and a point on a circle. If the center of the circle is at the origin, you can easily derive the circle equations. The circle equations are derived using the following conditions:
If “r” is the radius and C (h, k) is the center of the circle, then by definition: CP | = r.
The equation for the distance is √ [(x – h) 2 (y – k) 2] = r
If you take the squares on both sides, you will get (x – h) 2 (y – K) 2 = r2
This is a circle equation with a center point C (h, k) and a radius “r”, (x – h) 2 (y – k) 2 = r2 .
The eccentricity of the circle is also as follows. Same 0, that is, e = 0.
Eccentricity of Parabola
A parabola is defined as a set of points P whose distance from a fixed point F (focus) in the plane is equal to the distance from a fixed line l (directrix) in the plane. In other words, the distance from a fixed point in the plane is a constant proportion equal to the distance from the fixed point in the plane.
Therefore, the parabolic eccentricity is equal to 1 i.e., e = 1.
The general equation for the parabola is written as x2 = 4ay, and the eccentricity is given as 1.
Eccentricity of Hyperbola
A hyperbola is defined as a set of all points on a plane with a constant distance from two fixed points. In other words, the distance from the fixed point in the plane is a constant proportion and is greater than the distance from the fixed point in the plane.
Therefore, the eccentricity of the hyperbola is greater than 1, that is, e> 1.
The general formula for a hyperbola is x2/a2−y2/b2=1, and the eccentricity formula is written as √1 b2/a2.
The a and b of each hyperbola are the length of the semi-major axis and the semi-minor axis, respectively.
Eccentricity: Tips and Tricks
- The eccentricity of the conic section determines the curvature.
- The eccentricity of the circle is 0, and the eccentricity of the parabola is 1.
- The different eccentricities of the ellipse and the parabola are calculated using the formula e = c / a, Where c = √ a 2 b2, where a and b are the half axes of the hyperbola, and c = √ a 2 – b2.
Example 1. Find the eccentricity of the ellipse 9x2 25 y2 = 225
The equation of the ellipse in the standard form is x2/a2 y2/b2 = 1
Thus rewriting 9x2 25 y2 = 225, we get x2/25 y2/9 = 1
Comparing this with the standard equation, we get a2 = 25 and b2 = 9
⇒ a = 5 and b = 3
Here b< a. Thus e = √a2−b2/a
e = √52−32552−325
e = 4/5
Answer: The eccentricity of the ellipse x2/25 y2/9 = 1 is 4/5
Example 2. Find the eccentricity of the hyperbola whose length of the latus rectum is 8 and the length of its conjugate axis is half of the distance between its foci.
We know that,
2a = length of the transverse axis
2b = length of the conjugate axis
2ae = distance between the foci of the hyperbola in terms of eccentricity
Given LR of hyperbola = 8 ⇒ 2b2/a = 8 —–>(1)
2b = 2ae/ 2—–>(2)
e = 2b/a
Substituting the value of e in (1), we get eb = 8
b = 8/e
From (2), we know that a = 2b/e
Plugging-in b = 8/e here.
Thus a = 16/e2
We know that the eccentricity of the hyperbola, e = √a2 b2aa2 b2a
e = √256e4 16e264e2256e4 16e264e2
e = 2/√3
Answer: The eccentricity of the hyperbola = 2/√3
Example 3. What is the eccentricity of the hyperbola y2/9 – x2/16 = 1?
The standard equation of the hyperbola = y2/a2 – x2/b2 = 1
Comparing the given hyperbola with the standard form, we get
a= 3 and b = 4
We know the eccentricity of hyperbola is e = c/a
c= √(a2 b2)
c = 5
Thus the eccentricity of the given hyperbola is 5/3
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