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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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concept. Use space or enter to play video.concept thumbnailBeginner

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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.Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

What is the focus of a circle?

A$\text{Center}$B$\text{Vertex}$C$\text{Axis}$D$\text{Directrix}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Practice Quiz

Single Choice QuizIntermediate

Why does a parabolic dish concentrate signals at its focus?

A$\text{Because incoming parallel rays reflect to the focus}$B$\text{Because sum of distances is constant}$C$\text{Because difference of distances is constant}$D$\text{Because it has two foci}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Analyze how the foci of an ellipse affect its shape.

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

For a hyperbola, what is constant regarding its foci?

A$\text{Difference of distances}$B$\text{Sum of distances}$C$\text{Product of distances}$D$\text{Ratio of distances}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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