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Latus Rectum

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Latus Rectum

Study Guide

Key Definition

The latus rectum of a conic section is the chord that passes through the focus, is perpendicular to the major axis, and has endpoints on the curve. For a parabola, its length is $4p$, where $p$ is the focal length.

Important Notes

For a parabola: $4p$
For an ellipse: $\frac{2b^2}{a}$
For a circle: a circle has no latus rectum (no analogue)
For a hyperbola: $\frac{2a^2}{b}$
For ellipses, the major axis is always larger than the minor axis

Mathematical Notation

$\times$ represents multiplication

$\div$ represents division

$\sqrt{x}$ represents square root

$\angle$ represents angle

$\pi$ equals approximately 3.14159

Remember to use proper notation when solving problems

Why It Works

The latus rectum provides a measure of the width of the conic section at the focus, helping to understand the shape and size of the curve.

Remember

For a parabola, the length of the latus rectum is $4p$, where $p$ is the distance from the vertex to the focus.

Quick Reference

Parabola:$4p$

Ellipse:$\frac{2b^2}{a}$

Circle:Diameter $2r$

Hyperbola:$\frac{2a^2}{b}$

Understanding Latus Rectum

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Video explanation of this concept

Beginner

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Beginner Explanation

The latus rectum of a conic is the chord through the focus perpendicular to the major axis. For a parabola $y^2=4px$, its length is $4p$. For an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, length is $2b^2/a$. For a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, length is $2a^2/b$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the length of the latus rectum of a parabola with focal length $p$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A satellite dish is shaped like a parabola. If the focus is 5 meters from the vertex, what is the length of the latus rectum?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If an ellipse has a minor axis of 4 and a major axis of 6, find the length of the latus rectum.

Challenge Quiz

Single Choice Quiz

Advanced

For a hyperbola with transverse axis length $a = 3$ and conjugate axis length $b = 4$, what is the latus rectum?

Recap

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Review key concepts and takeaways