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Major and Minor Axes of Conics

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Major and Minor Axes of Conics

Study Guide

Key Definition

The major axis of an ellipse contains the longer line segment about which the ellipse is symmetrical, passing through the foci, center, and vertices. The minor axis contains the shorter line segment and is perpendicular to the major axis.

Important Notes

A circle has infinitely many axes of symmetry
The axis of symmetry of a parabola is perpendicular to the directrix
The major axis of an ellipse is the line through the foci
The minor axis is perpendicular to the major axis
Both axes of an ellipse are axes of symmetry

Mathematical Notation

$a$: semi-major axis length of an ellipse

$b$: semi-minor axis length of an ellipse

$c$: focal distance, where $c^2=a^2-b^2$ for ellipses and $c^2=a^2+b^2$ for hyperbolas

$e$: eccentricity, $e=c/a$

$2a$: length of major or transverse axis

$2b$: length of minor or conjugate axis

Remember to use proper notation when solving problems

Why It Works

Understanding the axes of conics helps in graphing and analyzing their properties using $equations$ and $symmetries$.

Remember

The major axis is always longer than the minor axis in an ellipse, and they intersect at the center.

Quick Reference

Major Axis of Ellipse:$2a$

Minor Axis of Ellipse:$2b$

Understanding Major and Minor Axes of Conics

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Video explanation of this concept

Beginner

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Beginner Explanation

Each conic section has specific axes of symmetry. An ellipse has a major axis (longest diameter) of length $2a$ passing through its foci and a minor axis of length $2b$ perpendicular to it. A hyperbola has a transverse axis of length $2a$ through its vertices and a conjugate axis of length $2b$ perpendicular to it. A parabola has a single axis of symmetry perpendicular to its directrix.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which axis of an ellipse is the longest?

Real-World Problem

Question Exercise

Intermediate

Elliptical Pool Design

An architect is designing an elliptical pool. The pool’s major axis is $50\ m$ long and the minor axis is $30\ m$. Calculate the area of the pool using the formula $\pi\times a \times b$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Imagine a hyperbola where the transverse axis is horizontal. Describe the orientation of the conjugate axis.

Challenge Quiz

Single Choice Quiz

Advanced

For an ellipse with $a = 6$ and $b = 4$, find the length of the major axis.

Recap

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Review key concepts and takeaways