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Eccentricity

Every focus of a conic section has a fixed line on the convex side called the directrix that's perpendicular to the axis of symmetry. The distance from the focus is directly proportional to the distance from the corresponding directrix for every point on the graph. This constant proportionality is called the eccentricity, often abbreviated as e. The formula for eccentricity is:

Eccentricity e is defined distance from a point to the focus and the distance from that same point to the directrix

In this article, we'll explore what eccentricity tells us and work out a few practice problems to develop a deeper understanding of how it works. Let's get started!

What eccentricity tells us

The value of eccentricity determines what type of conic we're looking at. When e=0 , the conic section is a perfect circle. If $e=1$ , the conic section is a parabola. If $e>1$ , the conic section is a hyperbola.

Perhaps the most interesting scenario is when $o . The resulting conic section is an ellipse, and it looks more circular the closer e is to 0. You'll need to memorize or have handy all of these rules to make optimal use of eccentricity.

Using eccentricity to determine the conic section of polar equations

Using the guidelines above, we can determine the conic section of any polar equation using just its eccentricity. Let's tackle a practice problem by identifying the conic section of the following polar equation:

$r=\frac{14}{7-7cos\left(\theta \right)}$

Recall that the polar equations of conic sections can come in two different forms:

$r=e\frac{d}{1±ecos\left(\theta \right)}$ or $r=e\frac{d}{1±esin\left(\theta \right)}$

The first step in solving this problem is converting our polar equation into one of these forms. Since it includes a cosine, let's divide everything by 7 and use the first form:

$r=\frac{14}{7-7cos\left(\theta \right)}$

$r=\frac{2}{1-cos\left(\theta \right)}$

From here, we can see that $e=1$ . The conic section is a parabola when $e=1$ , so we have our answer.

The classification of conic sections by eccentricity is as follows:

• For a circle, $e=0$
• For an ellipse, $0
• For a parabola, $e=1$
• For a hyperbola, $e>1$

Eccentricity practice problem

Given the polar equation, identify the conic section:

$r=\frac{3}{1+0.5cos\left(\theta \right)}$

First, let's examine the equation as it already resembles the standard form for a conic section in polar coordinates:

$r=e\frac{d}{1±ecos\left(\theta \right)}$

In this equation, "e" represents the eccentricity of the conic section, which determines the shape of the conic section.

Comparing our given polar equation $r=\frac{3}{1+0.5cos\left(\theta \right)}$ with the general form $r=e\frac{d}{1±ecos\left(\theta \right)}$ , we can deduce that $e=0.5$ . Since the eccentricity "e" is between 0 and 1: $\left(0 , the conic section is an ellipse.

Ellipse

Flashcards covering the Eccentricity

Precalculus Flashcards