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Section 8.3 Hyperbolas

Definition: A hyperbola is the set of all points, the difference of whose distances from two fixed points is constant. Each fixed point is called a focus (plural = foci). The focal axis is the line passing through the foci.

Visit this link and use the interactive tool to create various hyperbolas: https://www.intmath.com/plane-analytic-geometry/hyperbola-interactive.php

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Basic “vertical” hyperbola:

Equation: 2 2

2 21

y x

a b

Asymptotes: ay x

b

Foci: (0, )c , where 2 2 2c a b Vertices: (0, )a

Eccentricity: )1( a

c

a

-a

(0, )c

ay x

b

ay x

b

(0, )c

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Basic “horizontal” hyperbola:

Equation: 2 2

2 21

x y

a b

Asymptotes: by x

a

Foci: ( ,0)c , where 2 2 2c a b Vertices: ( ,0)a

Eccentricity: )1( a

c

Note: The transverse axis is the line segment joining the two vertices. The conjugate axis is the line segment perpendicular to the transverse axis, passing through the center and extending a distance b on either side of the center. (These terms will make more sense after we do the graphing examples.)

a ( ,0)c

by x

a

by x

a

( ,0)c -a

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Graphing hyperbolas: To graph a hyperbola with center at the origin:

Rearrange into the form 2 2

2 21

x y

a b or

2 2

2 21

y x

a b .

Decide if it’s a “horizontal” or “vertical” hyperbola.

o if 2x comes first, it’s horizontal (vertices are on x-axis). o If 2y comes first, it’s vertical (vertices are on y-axis).

Use the square root of the number under x2 to determine how far to measure in x-direction.

Use the square root of the number under y2 to determine how far to

measure in y-direction. Draw a box with these measurements.

Draw diagonals through the box. These are the asymptotes. Use the

dimensions of the box to determine the slope and write the equations of the asymptotes.

Put the vertices at the edge of the box on the correct axis. Then draw a

hyperbola, making sure it approaches the asymptotes smoothly.

2 2 2c a b where 2a and 2b are the denominators.

The foci are located c units from the center, on the same axis as the vertices. When graphing hyperbolas, you will need to find the orientation, center, values for a, b and c, lengths of transverse and converse axes, vertices, foci, equations of the asymptotes, and eccentricity.

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Example 1: Find the orientation, vertices and asymptotes for the following hyperbola:

2 2

116 49

y x

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Example 2: Find all relevant information and

graph 1436

22

yx.

Vertices: Foci: Eccentricity: Length of transverse axis: Length of conjugate axis: Slant Asymptotes:

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Example 3: Find all relevant information and

graph 2 2

14 9

y x .

Vertices: Foci: Eccentricity: Length of transverse axis: Length of conjugate axis: Slant Asymptotes:

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Popper for Section 8.3

Question#1: Find the vertices of this hyperbola: 2 2

116 25

x y

a) 4,0 , 4,0

b) 5,0 , 5,0

c) 0,4 , 0, 4

d) 0,5 , 0, 5

e) 4,5 , 4, 5

f) None of these

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The equation of a hyperbola with center not at the origin: Center: (h, k)

2 2

2 2

( ) ( )1

x h y k

a b

or

2 2

2 2

( ) ( )1

y k x h

a b

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Example 4: Write the following equation in standard form and find the center of this hyperbola:

2 29 4 18 16 43x y x y

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To graph a hyperbola with center not at the origin:

Rearrange (complete the square if necessary) to look like

2 2

2 2

( ) ( )1

x h y k

a b

or

2 2

2 2

( ) ( )1

y k x h

a b

.

Start at the center ( , )h k and then graph it as before. To write down the equations of the asymptotes, start with the equations

of the asymptotes for the similar hyperbola with center at the origin. Then replace x with x h and replace y with y k .

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Example 5: Write the equation in standard form, find all relevant information and graph:

2 21 3

116 9

x y

Center: Vertices: Length of transverse axis: Length of conjugate axis: Slant Asymptotes:

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Here’s the actual graph:

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Popper for Section 8.3

Question#2: Find the center for this hyperbola: 2 2

2 11

16 25

x y

a) (2,1) b) (2,-1) c) (1,2) d) (-1,2) e) (-2,1) f) None of these

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Example 6: Write an equation of the hyperbola with center at (-2, 3), one vertex is at (-2, -2) and eccentricity is 2.

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Example 7: Write an equation of the hyperbola if the vertices are (4, 0) and (-4, 0) and the asymptotes have slopes 1 .

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Popper for Section 8.3 Question#3: Which of the following is the equation of a hyperbola with center (2,3)?

a) 2 2

2 31

16 4

x y

b) 2 2

2 31

16 4

y x

c) 2 2

2 31

16 4

y x

d) 2 2

2 31

16 4

x y

e) 2 2

2 31

16 4

x y

f) None of these