Eccentricity

In the study of geometry, particularly coordinate geometry, conic sections hold a significant place due to their wide occurrence in nature and science. These curves are formed by slicing a cone at various angles and positions. Each resulting curve has unique geometric properties, and one such property is eccentricity, which is our topic of discussion in this article. So let’s begin!

1.0What is Eccentricity?

Eccentricity is a key geometric property used to distinguish between different conic sections—circle, ellipse, parabola, and hyperbola. Defined as a ratio involving the focus and directrix, it indicates how “stretched” a curve is. Each conic section has a unique range of eccentricity, making it essential in both theoretical geometry and real-world applications, such as astronomy and engineering. One such application of eccentricity is the eccentricity of the Earth in elliptical motion. 

2.0Eccentricity Definition

The word eccentricity comes from the Greek word “ekkentros,” meaning “out of the centre.” In mathematics, eccentricity describes how much a conic section deviates from being circular. 

In other words, the eccentricity (e) of a conic section is the ratio of the distance of any point on the curve from the focus to its perpendicular distance from the directrix.

$e=\frac{Distancefromthefocus}{Distancefromthedirectrix}$

This definition applies to all conic sections. Depending on the value of e, we can determine whether the conic section is a circle, ellipse, parabola, or hyperbola.

3.0Eccentricity of Conic Section

Eccentricity of a Circle

The circle is the most regular and symmetric of all conic sections. Every point on the circle is equidistant from the centre, and the focus and centre lie at the same point. Since all distances are equal, the curve doesn’t stretch at all. Eccentricity of a Circle is:

e=0

This is because the numerator (distance from the focus) and the denominator (distance from the directrix) do not follow the focus-directrix definition meaningfully in the usual sense, but mathematically, we define the eccentricity of a circle as zero.

A circle is a special case of an ellipse where both axes (major and minor) are equal. Since it has no elongation, its eccentricity is 0.

Eccentricity of an Ellipse

An ellipse is like a “flattened” or “stretched” circle. It has two axes: the longer one is called the major axis, and the shorter one is the minor axis. An ellipse has two foci. For every point on the ellipse, the sum of the distances to the two foci is constant. The Formula for Eccentricity of an Ellipse is:

$e=\sqrt{1-\frac{b^2}{a^2}}(Ifa>b)$

Or, 

$e=\sqrt{1-\frac{a^2}{b^2}}(Ifb>a)$

Here, 

• a= semi-major axis
• b= semi-minor axis

This formula gives a value between 0 and 1, meaning that the ellipse is more circular when e is close to 0 and more elongated when e is close to 1. Which means: 

0<e<1

Note that the greater the eccentricity, the more stretched the ellipse is. When e = 0, the ellipse becomes a circle.

Eccentricity of a Parabola

A parabola is a unique curve where each point is equidistant from a fixed point (the focus) and a fixed straight line (the directrix). This is exactly what the definition of eccentricity describes. The Eccentricity of a Parabola is:

e=1

This means that the distance from the focus and the distance from the directrix are always the same. Because of this property, a parabola represents a perfect balance between the inward and outward stretch of the curve.

Parabolas are often seen in the real world, such as in the paths of thrown objects (projectiles), the shape of satellite dishes, and certain bridges.

Eccentricity of a Hyperbola

A hyperbola is formed when the plane cuts both nappes (sides) of the cone. A hyperbola has two separate curves, each one opening in opposite directions. For any point on a hyperbola, the difference of distances from the two foci is constant. The Formula for Eccentricity of a Hyperbola:

$e=\sqrt{1+\frac{b^2}{a^2}}$

Here, 

• a: distance from the centre to the vertex (along the transverse axis)
• b: distance related to the conjugate axis (not always the vertical one)

This gives a value greater than 1, meaning the hyperbola is the most stretched of all conic sections. Mathematically, this means:

e>1

The greater the eccentricity, the more open or wider the branches of the hyperbola become.

4.0Comparison Table: Eccentricity of Conic Sections

5.0Solved Examples

Problem 1: What is the eccentricity of a circle with radius 7 cm?

Solution: A circle is a special case of an ellipse where both axes are equal and all points are equidistant from the centre.

But the eccentricity of a circle is always equal to zero. Hence, the eccentricity of a circle with radius 7 cm will also be zero. That is: e = 0 

Problem 2: A hyperbola has a semi-transverse axis 𝑎 = 4 and a semi-conjugate axis 𝑏 = 3. Calculate its eccentricity.

Solution: Given that, in a hyperbola, 𝑎 = 4 and b = 3 

Now, by using the formula:

$e=\sqrt{\frac{b^2}{a^2}}$

Substitute values:

$e=\sqrt{1+\frac{3^2}{4^2}}=\frac{5}{4}$

$e=\sqrt{1+\frac{9}{16}}=\sqrt{\frac{25}{16}}$

$e=\frac{5}{4}=1.25$

Problem 3: A satellite moves in an elliptical orbit around the Earth. The semi-major axis of the orbit is 10,000 km, and the semi-minor axis is 8,000 km. Find the eccentricity of the ellipse.

Solution: According to the question, 

Semi-major axis of orbit (a): 10,000 Km.

Semi-minor axis of orbit (b): 8,000 Km

Here a>b

Now, using the formula for eccentricity: 

$e=\sqrt{1-\frac{b^2}{a^2}}$

$e=\sqrt{1-\frac{8000^2}{10000^2}}$

$e=\sqrt{1-\frac{64}{100}}$

$e=\sqrt{\frac{36}{100}}=\sqrt{0.36}$

$e=0.6$