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Eccentricity of Conic Sections

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  • 0:04 Conic Sections
  • 0:46 Eccentricity
  • 2:46 Some Examples
  • 4:47 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Conic sections have been studied for hundreds of years. In this lesson, we will look at the different conic sections and explore a characteristic of these shapes called their eccentricities.

Conic Sections

It's always fascinating to learn about mathematical concepts that originated a very long time ago. Conic sections are one of those concepts.

Conic sections are geometric shapes whose origins can be traced back to ancient Greek mathematicians working as early as 350 B.C. They observed that slicing a double cone with a plane in various ways resulted in four different shapes. The resulting shapes created by the intersections of the plane with the cone are circle, ellipse, parabola and hyperbola; these are the conic sections.


Conic Sections
ecc1


General forms of the algebraic equations for these four conic sections are shown in the image below.


Conic Section Equations
ecc2


Let's take a look at the extremely interesting concept of conic sections.

Eccentricity

A characteristic that all of the conic sections possess is eccentricity. The eccentricity of a conic section tells us how close it is to being in the shape of a circle. The farther away the eccentricity of a conic section is from 0, the less the shape looks like a circle.

Take a look at the conic shapes again. A circle is a circle, so obviously the eccentricity of a circle is 0. Which of the other three conic sections do you think would be closest to the shape of a circle?

Well, an ellipse looks like a compressed circle. When compared with the other two conic shapes, it most closely resembles a circle. Similar reasoning deduces that a parabola would be next closest, and a hyperbola the farthest from a circle in shape.

The chart below provides ranges of values of the eccentricity (e) for each of the conic sections.

Circle e = 0
Ellipse 0 < e < 1
Parabola e = 1
Hyperbola e > 1

The two conic sections with the easiest eccentricities to remember are a circle (e = 0) and a parabola (e = 1). The ellipse and hyperbola are a little trickier, but not by much. We can find the exact value of the eccentricity of these two conic shapes by using their equations.

The eccentricity of an ellipse (x - h)2 / a2 + (y - k)2 / b2 = 1 will always be between 0 and 1 and can be calculated using the following formulas:

  • When a > b, we use e = √(a2 - b2) / a.
  • When b > a, we use e = √(b2 - a2) / b.

The eccentricity of a hyperbola (x - h)2 / a2 - (y - k)2 / b2 = 1 is always greater than 1 and can be calculated using the following formula:

  • e = √(a2 + b2) / a.


Conic Section Eccentricities
ecc3


Let's put these rules to use and take a look at finding the eccentricity of a few examples.

Some Examples

Consider the following equations:

  1. (x + 1)2 / 9 - y2 / 16 = 1
  2. y = 2(x - 4)2 + 3
  3. x2 + y2 = 36
  4. x2 / 4 + (y + 7)2 / 9 = 1

First, let's identify which of the conic sections each of these equations represents. We can do this by comparing the equations to the general forms of the equations of the conic sections, as you can see in the equation solutions below.


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