Characteristics of Conics: Centers, Foci and Eccentricity

$\underline{\textbf{What are Conic Sections}}$

Conic sections are a group of shapes which can be obtained by cutting through a $\textbf{cone}$ on a $\textit{certain plane}$.

For instance, if you cut through a cone with a horizontal plane (one that is perpendicular to the axis of the cone) you will get you a circle shape. On the other hand, if you cut through a cone with a plane that is more angled you can get an ellipse or a parabola or a hyperbola. The image below gives a depiction of this:

As we see in the image, there are four types of conic sections: Circle, Ellipse, Parabola, and Hyperbola. They all share similar defining characteristics which include:

• Center
• Foci
• Eccentricity

$\underline{\textbf{Foci of Conic Sections:}}$

The $\textbf{foci}$ of a conic section is the midpoint point between the foci.

• Circles and Parabolas have only one focus

• Ellipses and Hyperbolas have two foci

$\underline{\textbf{Centers of Conic Sections:}}$

The $\textbf{center}$ of a conic section is the midpoint point between the foci. (Since a circle only has one foci, its center is located at the same spot as the focus, i.e center = focus).

$\underline{\textbf{Eccentricity of Conic Sections:}}$

$\textbf{Eccentrity}$ is the amount the conic section deviates from being perfectly circular.

• Circle has eccentricity = 0

• Ellipse has eccentricity between and not including 0 and 1

To calculate the eccentricity of a Ellipse:

• For horizontal ellipse, where $a > b$: $e = \dfrac{\sqrt{a^{2} - b^{2}}}{a}$

• For vertical ellipse, where $b > a$: $e = \dfrac{\sqrt{b^{2} - a^{2}}}{b}$

• Parabola has eccentricity = 1

• Hyperbola has eccentricity > 1

To calculate the eccentricity of a Hyperbola:

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

(Video) Eccentricity of an Ellipse

by Brian McLogan

In this video by Brian McLogan, you will learn about the eccentricity of the ellipse.

Summary

Eccentricity measures how "stretched" an ellipse is compared to a circle. It is the ratio of the distance from the center to a focus to the distance from the center to a vertex.

Characteristics of Conics

Conic sections are special shapes you get when you slice a cone in different ways. The four main types are the circle, ellipse, parabola, and hyperbola. Each of these shapes has unique features that can help us to tell them apart.

The key feature is the center or vertex of the shape. The center is like the middle point of the shape. Circles, ellipses, and hyperbolas all have a center - but parabolas don't! Parabolas have a vertex.

The focus is a special point that helps to define the shape. A circle doesn't have a focus because every point is the same distance from the center. An ellipse has two foci, and the distance between one focus and a point on the curve and then to the other focus is always the same. A parabola has only one focus that helps to shape the curve. A hyperbola also has two foci.

We can also consider eccentricity. A circle has an eccentricity of $0$, an ellipse has an eccentricity between $0$ and $1$, a parabola has an eccentricity of $1$, and a hyperbola has an eccentricity greater than $1$.