mathematics review - hyperbola

8/9/2019 Mathematics Review - Hyperbola

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HYPERBOLAHyperbolas don't come up much — at least not that I've noticed — in other math

classes, but if you're covering conics, you'll need to know their basics. An hyperbola

looks sort of like two mirrored parabolas, with the two "halves" being called"branches". ike an ellipse, an hyperbola has two foci and two vertices! unlike an

ellipse, the foci in an hyperbola are further from the hyperbola's center than are its

vertices

#he hyperbola is centered on a point $h, k %, which is the "center" of the hyperbola.

#he point on each branch closest to the center is that branch's "vertex". #he

vertices are some &ed distance a from the center. #he line going from one verte,through the center, and ending at the other verte is called the "transverse" ais.

#he "foci" of an hyperbola are "inside" each branch, and each focus is located some

&ed distance c from the center. $#his

means that a ( c for hyperbolas.% #he

values of a and c will vary from one

hyperbola to another, but they will be

&ed values for any given hyperbola.

)or any point on an ellipse, the sum of

the distances from that point to each of

the foci is some &ed value! for anypoint on an hyperbola, it's

the diference of the distances from the

two foci that is &ed. ooking at the

graph above and letting "the point" be

one of the vertices, this &ed distance

must be $the distance to the further

A*+-#I/0#

http://www.purplemath.com/modules/parabola.htmhttp://www.purplemath.com/modules/ellipse.htmhttp://www.purplemath.com/modules/ellipse.htmhttp://www.purplemath.com/modules/parabola.htm


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focus% less $the distance to the nearer focus%, or $a 1 c% 2 $c 2 a% 3 4a. #his &ed5

di6erence property can used for determining locations If two beacons are placed in

known and &ed positions, the di6erence in the times at which their signals are

received by, say, a ship at sea can tell the crew where they are.

As with ellipses, there is a relationship betweena, b, and c, and, as with ellipses, thecomputations are long and painful. o trust me that, for hyperbolas $where a ( c%,

the relationship is c4 2 a4 3 b4 or, which means the same thing, c43 b4 1 a4. $7es, the

8ythagorean #heorem is used to prove this relationship. 7es, these are the same

letters as are used in the 8ythagorean #heorem. 0o, this is not the same thing as

the 8ythagorean #heorem. 7es, this is very confusing. 9ust memori:e it, and move

on.%

;hen the transverse

ais is hori:ontal $in

other words, whenthe center, foci, and

vertices line up side

by side, parallel to

the x 5ais%, then

the a4 goes with

the x part of the

hyperbola's e


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ais. #his information doesn't help you graph hyperbolas, though. ?opyright @

li:abeth tapel 4=54== All -ights -eserved

)or reasons you'll learn in calculus, the graph of an hyperbola gets fairly Bat and

straight when it gets far away from its center. If you ":oom out" from the graph, it

will look very much like an "C", with maybe a little curviness near the middle. #hese"nearly straight" parts get very close to what are called the "asymptotes" of the

hyperbola. )or an hyperbola centered at $h, k % and having &ed values a andb, the

asymptotes are given by the following e


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#he measure of the amount of curvature is the "eccentricity" e, where e 3 cDa. ince

the foci are further from the center of an hyperbola than are the vertices

$so c E a for hyperbolas%, then e E =. Figger values of e correspond to the

"straighter" types of hyperbolas, while values closer to =correspond to hyperbolas

whose graphs curve


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c is not shown in the e


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always the distance between the vertices of the hyperbola and the center of the

hyperbola.

$h,k% is the center of the vertically aligned hyperbola.

a is the distance from the center of the hyperbola to each verte of the hyperbola.

ach verte of the hyperbola lies on the transverse ais of the hyperbola. #he transverse ais of a vertically aligned hyperbola is vertical.

#here is an invisible bo created between the vertices of the vertically aligned

hyperbola.

4Ka is the height of this invisible bo.

4Kb is the width of this invisible bo.

4Kc is the length of the diagonal of this invisible bo.

#he bo is not part of the hyperbola. It is a construct used to show the relationships

between a, b, and c.

c is also the distance between each foci of the hyperbola and the center of the

hyperbola.

c is not shown in the e


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becomes and

#he graph of our hori:ontally aligned hyperbola is shown below

A7/8## ) A H78-FA


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very hyperbola has asymptotes.

#he asymptotes of a hori:ontally aligned hyperbola are given by the e


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#he e


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8I?#J- ) J- H-IL0#A7 AIG0* H78-FA

A picture of the graph of our hori:ontally aligned hyperbola is shown below

)= and

)4 are

the foci

of the

hyperbo

la.

+= and

+4 are

the

vertices

of the

hyperbo

la.

? is the

center

of the

hyperbo

la.

a is the

distance from

the

center

of the

hyperbo

la to

each

verte

of the

hyperbo

la.

#his

would

be from

? to +=,

and

from ?


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to +4.

b is the

distanc

e from

thetransver

se ais

to the

top of

the bo.

#his

would

be from

+= to

8=, +4to 84,

+= to

8M, and

+4 to

8N.

c is the

distanc

e from

the

centerof the

hyperbo

la to

each

focus of

the

hyperbo

la.

#his

wouldbe from

? to )=,

and

from ?

to )4.

a is half


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the

width of

the bo.

b is half

the

heightof the

bo.

c is half

the

length

of the

diagona

l of the

bo.

In the above picture, the asymptotes are the straight lines and the hyperbola is the

curved lines.

0ote that the diagonals of the bo lie on the same line as the asymptotes of the

hyperbola.

#he transverse ais is the hori:ontal line on which the foci and vertices of the

hyperbola lie.

#he bo is not part of the hyperbola. It is a construct used to show the relationship

between the variables a, b, and c, and the asymptotes of the hyperbola.

)-/JA )- #H *I#A0? )-/ #H ?0#- ) #H H78-FA # A?H

)?J ) #H H78-FA

c is the distance from the center of the hyperbola to each focus of the hyperbola.

c is also half the length of the diagonal of the bo.

If you look at the picture, you will see that the length of a is e


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a is the length of the hori:ontal leg of this right triangle $line segment ?+4%

b is the length of the vertical leg of this right triangle $line segment +484%.

Fy the 8ythagorean )ormula

#hat is the relationship between a, b, and c.

#he e


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d= 3 distance between 8O and )=

d4 3 distance between 8O and )4

8O is at the point $=,4.MMMMMMMM%

)= is at the point $,5M%

)4 is at the point $=,5M%

d= 3 3

3 3 ==.MMMMMMMM

d4 3 3

3 3 O.MMMMMMMM

Qd=5d4Q 3 Q==.MMMMMMMM 5 O.MMMMMMMMQ 3 QPQ 3 P

dM 3 distance between 8P and )=

dN 3 distance between 8P and )4

8P is at the point $54,5==.NM4RNN4R%

)= is at the point $,5M%

)4 is at the point $=,5M%

dM 3

3 3 3 S.PPPPPPPR

dN 3

3 3 3 =N.PPPPPPPR

QdM5dNQ 3 QS.PPPPPPPR 5 =N.PPPPPPPRQ 3 Q5PQ 3 P

A picture of these points and their relationship to each other is shown below

8O is at the top right, 8P is at the bottom left, )= is at the middle left, and )4 is at

the middle right.


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G-A8H ) A +-#I?A7 AIG0* H78-FA

#he general e


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positions of each term. #he term is now the positive term, and the

term is now the negative term.

#he term is now e


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A7/8## ) A H78-FA

#he asymptotes of a hori:ontally aligned hyperbola are given by the e


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A7/8## ) J- +-#I?A7 AIG0* H78-FA

)or our vertically aligned hyperbola that we >ust graphed, the e


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and

olving for k, we get

k 3 5M in both e


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)= and

)4 are

the foci

of the

hyperbola.

+= and

+4 are

the

vertices

of the

hyperbo

la.

? is the

centerof the

hyperbo

la.

a is the

distanc

e from

the

center

of the

hyperbola to

each

verte

of the

hyperbo

la.

#his

would

be from

? to +=,and

from ?

to +4.

b is the

distanc

e from


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the

transver

se ais

to the

sides of

the bo. #his

would

be from

+= to

8=, +=

to 84,

+4 to

8M, and

+4 to

8N.

c is the

distanc

e from

the

center

of the

hyperbo

la to

each

focus of the

hyperbo

la.

#his

would

be from

? to )=,

and

from ?

to )4.

a is half

the

height

of the

bo.

b is half


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the

width of

the bo.

c is half

the

lengthof the

diagona

l of the

bo.

In the above picture, the asymptotes are the straight lines and the hyperbola is the

curved lines.

0ote that the diagonals of the bo lie on the same line as the asymptotes of thehyperbola.

#he transverse ais is the vertical line on which the foci and vertices of the

hyperbola lie.

#he bo is not part of the hyperbola. It is a construct used to show the relationship

between the variables a, b, and c, and the asymptotes of the hyperbola.

)-/JA )- #H *I#A0? )-/ #H ?0#- ) #H H78-FA # A?H

)?J ) #H H78-FA

c is the distance from the center of the hyperbola to each focus of the hyperbola.

c is also half the length of the diagonal of the bo.

If you look at the picture, you will see that the length of a is e


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#hat is the relationship between a, b, and c.

#he e


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)4 is at the point $O,5S%

d= 3 3

3 3 R.=SMMTSSS

d4 3 3

3 3 =O.=SMMTSSS

Qd=5d4Q 3 QR.=SMMTSSS 5 =O.=SMMTSSSQ 3 Q5SQ 3 S

dM 3 distance between 8P and )=

dN 3 distance between 8P and )4

8P is at the point $4,5S.POPSON4NT%

)= is at the point $O,4%

)4 is at the point $O,5S%

dM 3 3

3 3 ==.R=PRS=4

dN 3 3

3 3 M.R=PRS=4

QdM5dNQ 3 Q==.R=PRS=4 5 M.R=PRS=4Q 3 QSQ 3 S

A picture of these points and their relationship to each other is shown below

8O is at the top right, 8P is at the bottom left, )= is at the middle top, and )4 is at

the middle bottom.


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??0#-I?I#7 -A#I ) A H78-FA

#he eccentricity ratio of a hyperbola is determined by the e


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G-A8H A0* 8I?#J- ) H78-FA #HA# HA A )A##- ?J-+ $??0#-I?I#7

-A#I I HIGH-%

#he graph and picture is based on the e


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A picture of the graph of this e


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G-A8H A0* 8I?#J- ) H78-FA #HA# HA A HA-8- ?J-+ $??0#-I?I#7

-A#I I ;-%

#he graph and picture is based on the e


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#o graph this e


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#he higher e resulted in a hyperbola that had a Batter curve. #his means the

branches of the hyperbola curved away from each other at a very slow rate.

#he lower e resulted in a hyperbola that had a sharper curve. #his means the

branches of the hyperbola curved away from each other at a very high rate.

uestions and ?omments may be referred to me via email at

theoptsadcUyahoo.com

7ou may also check out my website at

httpDDtheo.=hosting.com
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mathematics review - hyperbola

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