Eccentricity Solved Examples
Here are some solved examples on the eccentricity of different conic sections.
Example 1: Calculate the Eccentricity for an Ellipse with a semi-major axis of 8 units and a distance from the centre to a focus of 5 units.
Solution:
Formula to calculate Eccentricity (e) for an Ellipse is:
e = c/a
Given values are:
Semi-major axis (a) is 8 units
Distance from the centre to a focus (c) is 5 units
Eccentricity formula is:
e = 5/8
Now, calculate the value of
e = 0.625
So, Eccentricity of the Ellipse is = 0.625
Example 2: Find the Eccentricity of the Ellipse for the given equation 16x2 + 25y2 = 400
Solution:
Given equation is: 16x2 + 25y2 = 400
General equation of Ellipse is
x2 /a2 +y2/b2= 1
To make it in standard form, divide both sides by 400
x2 /52 +y2/42= 1
So, value of semi-major axis length a = 5 and semi-minor axis length b = 4
From the formula of the Eccentricity of an Ellipse,
e = √(a2-b2)/a2
⇒ e = √(52-42)/52
⇒ e = √9/25
⇒ e = 3/5
Therefore, for given equation, the Eccentricity is 3/5.
Example 3: Find the Eccentricity of the conic section (x2/25) + (y2/16) = 1.
Solution:
Given equation is: (x2/25) + (y2/16) = 1
It can also be written as (x2/52) + (y2/42) =1
Given conic section is an Ellipse in the form of (x2/a2) + (y2/b2) = 1
Here, a = 5 and b = 4.
We know that c2 = a2-b2
⇒ c2 = 52 – 42
⇒ c2 = 25 -16 = 9
Hence, c = √9 = 3.
Formula for Eccentricity is:
e = c/a
Now, put the values of c and a, we get
e = 3/5
Hence, the Eccentricity of the given conic section (x2/25) + (y2/16) = 1 is 3/5.
Example 4: Find the Eccentricity of the hyperbola (x2/36) – (y2/9) = 1.
Solution:
Given equation is: (x2/36) – (y2/9) = 1
Given Hyperbola can be written as
(x2/62) – (y2/32) = 1
Given conic section is an Hyperbola in the form of (x2/a2) – (y2/b2) = 1
So, axis of Hyperbola is x-axis
Now, by comparing the equation, we get a = 6 and b = 3
Eccentricity formula for Hyperbola is e = √[1+(b2/a2)]
Now, put the values in the formula, we get
e = √[1+(32/62)]
⇒ e = √[1+(9/36)]
⇒ e = √(45/16)
⇒ e = 3/4√5
Therefore, 3/4√5 is the Eccentricity of the Hyperbolic equation (x2/36) – (y2/9) = 1.
Eccentricity Formula of Circle, Parabola, Ellipse, Hyperbola
Eccentricity is a non-negative real number that describes the shape of a conic section. It measures how much a conic section deviates from being circular. Generally, eccentricity measures the degree to which a conic section differs from a uniform circular shape.
Let’s discuss Eccentricity formula for circle, parabola, ellipse, and hyperbola, along with examples.
Table of Content
- Eccentricity in Geometry
- Eccentricity Formula
- Eccentricity of Circle
- Eccentricity of Parabola
- Eccentricity of Ellipse
- Eccentricity of Hyperbola
- Eccentricity of Conic Sections