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More on hyperbolas

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 <a href="asymptotes">$asymptotes$</a>.
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 hyperbola

Remember 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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Video explanation of this concept

Beginner

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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).

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

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

Real-World Problem

Question Exercise

Intermediate

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.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

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

Challenge Quiz

Single Choice Quiz

Advanced

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

Recap

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Review key concepts and takeaways