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Study Guide

Key Definition

A hyperbola is the set of all points P such that the absolute difference of the distances from P to two fixed points, called foci, is a constant positive value.

Important Notes

  • Each branch of a hyperbola approaches diagonal asymptotes.
  • The center of a hyperbola is the midpoint of the line segment joining its foci.
  • The transverse axis passes through the center and both vertices of the hyperbola.
  • A hyperbola with center at $(0, 0)$ and foci $(c, 0)$ and $(-c, 0)$ has the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.
  • The equations of the asymptotes are $y = \frac{b}{a}x$ and $y = -\frac{b}{a}x$.

Mathematical Notation

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is the equation of a hyperbolaRemember to use proper notation when solving problems

Why It Works

For any point P on the hyperbola, |PF1 - PF2| = 2a. By placing the foci at (±c,0) and setting that difference equal to 2a, you square both sides and simplify to arrive at the standard form $x^2/a^2 - y^2/b^2 = 1$. The asymptotes y=±(b/a)x emerge by letting the constant on the right go to zero.

Remember

The relationship $b^2 = c^2 - a^2$ helps in finding the unknowns of a hyperbola.

Quick Reference

Equation of Hyperbola:$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Understanding More on hyperbolas

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Beginner Explanation

A hyperbola is the set of points where the absolute difference of distances to two fixed foci is constant, producing two mirror-image curves opening left–right (or up–down).Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

Which equation represents a hyperbola centered at the origin with foci on the x-axis?

A$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$B$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$C$\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$D$\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

You're designing a satellite dish which is shaped like a hyperbola. Given that the distance between the foci is $10$ units and the distance between the vertices is $8$ units, find the equation of the hyperbola.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Given a hyperbola with the equation $\frac{x^2}{9} - \frac{y^2}{4} = 1$, determine the coordinates of its foci.

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

Challenge Quiz

Single Choice QuizAdvanced

Find the asymptotes for the hyperbola $\frac{x^2}{25} - \frac{y^2}{16} = 1$.

A$y = \frac{4}{5}x$ and $y = -\frac{4}{5}x$B$y = \frac{5}{4}x$ and $y = -\frac{5}{4}x$C$y = \frac{3}{4}x$ and $y = -\frac{3}{4}x$D$y = \frac{4}{3}x$ and $y = -\frac{4}{3}x$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

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