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Focus

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Focus

Study Guide

Key Definition

A focus is a point used to construct a conic section. The focus points are used differently to determine each circle, parabola, ellipse, and hyperbola.

Important Notes

A circle's focus is its center.
A parabola is determined by a focus and a directrix.
An ellipse is determined by two foci.
A hyperbola is determined by two foci.
Understanding the role of focus helps in graphing conic sections.

Mathematical Notation

$F$ denotes a focus point

$d$ denotes a directrix line

Why It Works

The focus helps in defining the geometric properties of conic sections using specific formulas: Circle: $(x-h)^2 + (y-k)^2 = r^2$ (focus at (h,k)); Parabola: $y = \frac{1}{4p}x^2$ (focus at (0,p)); Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (foci at (±c,0)); Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (foci at (±c,0)).

Remember

Each conic section relationship with its focus can be expressed through key formulas: Circle: $(x-h)^2 + (y-k)^2 = r^2$; Parabola: $y = \frac{1}{4p}x^2$; Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$; Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.

Quick Reference

Circle:$\text{Center}$

Parabola:$\text{Focus and Directrix}$

Ellipse:$\text{Two Foci}$

Hyperbola:$\text{Two Foci}$

Understanding Focus

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A circle's center is its focus. A circle is the set of all points in a plane at a given distance from the focus.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the focus of a circle?

Practice Quiz

Single Choice Quiz

Intermediate

Why does a parabolic dish concentrate signals at its focus?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Analyze how the foci of an ellipse affect its shape.

Challenge Quiz

Single Choice Quiz

Advanced

For a hyperbola, what is constant regarding its foci?

Recap

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