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Eccentricity

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Eccentricity

Study Guide

Key Definition

Eccentricity $e$ is the ratio of the distance from a point on the conic section to the focus and the distance from that point to the directrix.

Important Notes

The eccentricity $e$ helps determine the type of conic section.
If $e = 0$, the conic is a circle.
If $e = 1$, the conic is a parabola.
If $e > 1$, the conic is a hyperbola.
If $0 < e < 1$, the conic is an ellipse.

Mathematical Notation

$e = \frac{PF}{PD}$ focus–directrix definition of eccentricity

$e = \frac{c}{a}$ axis-based definition for ellipses and hyperbolas

$c$ distance from center to focus

$a$ semi-major axis length

$d(P, l)$ distance from point to directrix l

Remember to use proper notation when solving problems

Why It Works

Eccentricity $e$ gives a measure of how much a conic section deviates from being circular.

Remember

Types of conic sections depend on the eccentricity $e$.

Quick Reference

Circle:$e = 0$

Ellipse:$0 < e < 1$

Parabola:$e = 1$

Hyperbola:$e > 1$

Understanding Eccentricity

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Video explanation of this concept

Beginner

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Beginner Explanation

Eccentricity e is the ratio of the distance from any point on a conic to its focus and to its directrix. It measures how 'stretched' the conic is. For a circle, this ratio is e = 0.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the eccentricity of a circle? $e = ?$

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Explain why an elliptical orbit of a satellite around Earth must have eccentricity between 0 and 1. Provide reasoning using the definition of eccentricity.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

An ellipse has a semi-major axis length a = 5 and focal distance c = 3. Compute its eccentricity e = c/a and determine the type of conic section based on this value.

Challenge Quiz

Single Choice Quiz

Advanced

Given an ellipse with eccentricity $e = \frac{3}{5}$, which statement is true?

Recap

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Review key concepts and takeaways