Hyperbola Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Hyperbola.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The set of all points in a plane where the absolute difference of the distances to two fixed points (foci) is constant. The curve has two separate branches and asymptotes.

While an ellipse keeps the SUM of distances to foci constant, a hyperbola keeps the DIFFERENCE constant. This creates two separate curves that open away from each other, each curving toward (but never reaching) a pair of asymptotic lines.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Two branches opening apart, each hugging a pair of asymptotes it never touches.

Common stuck point: The procedure for hyperbola is the easy part; the trap is using $a^2-b^2$ for the foci. Asking "Is one squared term subtracted from the other (opposite signs) with the result equaling 1?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is one squared term subtracted from the other (opposite signs) with the result equaling 1?

Worked Examples

Example 1

easy

Identify the vertices and the direction of opening for the hyperbola

\frac{x^2}{9} - \frac{y^2}{16} = 1

Answer

\text{Vertices: } (\pm 3, 0); \text{ opens left and right}

First step

The standard form

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

opens left and right (horizontally).

Full solution

2
$a^2 = 9$ , so $a = 3$ . The vertices are at $(\pm a, 0) = (\pm 3, 0)$ .
3
The transverse axis is along the $x$ -axis with vertices at $(-3, 0)$ and $(3, 0)$ .

In the standard form of a hyperbola, the positive fraction determines the direction of opening. When

x^2

is positive, the hyperbola opens horizontally; when

y^2

is positive, it opens vertically. The vertices are at distance

a

from the center along the transverse axis.

Example 2

medium

Find the equations of the asymptotes for the hyperbola

\frac{y^2}{4} - \frac{x^2}{9} = 1

Example 3

medium

Write the equation of a vertical hyperbola with vertices

(0, \pm 4)

and foci

(0, \pm 5)

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

Find the foci of the hyperbola

\frac{x^2}{25} - \frac{y^2}{144} = 1

Example 2

hard

Write the equation of the hyperbola with foci at

(0, \pm 5)

and vertices at

(0, \pm 3)

Example 3

easy

Which way does

\frac{x^2}{9} - \frac{y^2}{16} = 1

open?

Example 4

easy

\frac{x^2}{9}-\frac{y^2}{16}=1

, what are

a

and

b

Example 5

easy

Find the vertices of

\frac{x^2}{25}-\frac{y^2}{4}=1

Example 6

easy

Which way does

\frac{y^2}{4} - \frac{x^2}{9} = 1

open?

Example 7

easy

Give the center of

\frac{(x-2)^2}{16}-\frac{(y+3)^2}{9}=1

Example 8

easy

What are the asymptote slopes of

\frac{x^2}{9}-\frac{y^2}{16}=1

Example 9

easy

Find

a

and

b

for

\frac{y^2}{36}-\frac{x^2}{64}=1

Example 10

easy

Write the equation of a horizontal hyperbola centered at origin with

a=2

b=3

Example 11

medium

Find the foci of

\frac{x^2}{16}-\frac{y^2}{9}=1

Example 12

medium

Find the asymptotes of

\frac{x^2}{4}-\frac{y^2}{9}=1

Example 13

medium

Find the foci of

\frac{y^2}{9}-\frac{x^2}{16}=1

Example 14

medium

Find the eccentricity of

\frac{x^2}{16}-\frac{y^2}{9}=1

Example 15

medium

Write the equation of a hyperbola with vertices

(\pm 3,0)

and foci

(\pm 5,0)

Example 16

medium

Convert

4x^2 - 9y^2 = 36

to standard form.

Example 17

medium

Find the center and foci of

\frac{(x-1)^2}{9}-\frac{(y+2)^2}{16}=1

Example 18

challenge

Write the equation of a hyperbola with asymptotes

y=\pm\frac{2}{3}x

and a vertex at

(3,0)

Example 19

challenge

The difference of distances from a point to foci

(\pm 5,0)

is 6. Find the equation.

Example 20

challenge

Find the eccentricity of a hyperbola whose asymptotes are

y=\pm x

Example 21

medium

Find the eccentricity of

\frac{y^2}{9}-\frac{x^2}{16}=1

Example 22

medium

Find the asymptotes of

\frac{y^2}{16}-\frac{x^2}{9}=1

Example 23

easy

Find the vertices of

\frac{x^2}{36} - \frac{y^2}{49} = 1

Example 24

easy

Identify the opening direction of

\frac{y^2}{16} - \frac{x^2}{25} = 1

Example 25

easy

Find the center of

\frac{(x+4)^2}{9} - \frac{(y-1)^2}{16} = 1

Example 26

easy

Find the asymptote slopes of

\frac{x^2}{4} - \frac{y^2}{1} = 1

Example 27

easy

Write a horizontal hyperbola at origin with

a = 5

b = 12

Example 28

easy

Find

a

and

b

for

\frac{y^2}{49} - \frac{x^2}{16} = 1

Example 29

medium

Find the foci of

\frac{x^2}{9} - \frac{y^2}{40} = 1

Example 30

medium

Find the eccentricity of

\frac{x^2}{25} - \frac{y^2}{11} = 1

Example 31

medium

Find the asymptotes of

\frac{(x-2)^2}{9} - \frac{(y+1)^2}{16} = 1

Example 32

medium

Find the length of the transverse axis of

\frac{x^2}{49} - \frac{y^2}{16} = 1

Example 33

medium

Find the length of the conjugate axis of

\frac{x^2}{49} - \frac{y^2}{16} = 1

Example 34

medium

Convert

25y^2 - 16x^2 = 400

to standard form.

Example 35

medium

Find the center, vertices, and foci of

\frac{(x-3)^2}{16} - \frac{(y+2)^2}{9} = 1

Example 36

medium

Find the asymptotes of

9x^2 - 4y^2 = 36

Example 37

medium

Find the foci of

\frac{y^2}{4} - \frac{x^2}{21} = 1

Example 38

hard

Find the equation of the hyperbola with foci

(\pm 10, 0)

and asymptotes

y = \pm \frac{4}{3} x

Example 39

hard

A hyperbola has center at the origin, a vertex at

(0, 6)

, and passes through

(8, 10)

. Find its equation.

Example 40

hard

Find the eccentricity of

\frac{x^2}{16} - \frac{y^2}{20} = 1

Example 41

hard

Identify and convert

4x^2 - y^2 - 16x - 4y + 16 = 0

to standard form.

Example 42

hard

For

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

, derive the asymptote equations.

Example 43

hard

Find the equation of a hyperbola with eccentricity

\sqrt{2}

and vertex at

(3, 0)

centered at origin.

Example 44

challenge

Show that a point on the hyperbola

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

satisfies

|d_1 - d_2| = 2a

where

d_1, d_2

are distances to the foci.

Example 45

challenge

Find the points of intersection between

\frac{x^2}{16} - \frac{y^2}{9} = 1

and the line

y = x

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Related Concepts

Equation of a Circle Asymptote Conic Sections Overview

Background Knowledge

These ideas may be useful before you work through the harder examples.

equation of circleasymptote