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8/9/2019 Mathematics Review - Hyperbola
1/30
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
8/9/2019 Mathematics Review - Hyperbola
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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
8/9/2019 Mathematics Review - Hyperbola
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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
8/9/2019 Mathematics Review - Hyperbola
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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
8/9/2019 Mathematics Review - Hyperbola
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c is not shown in the e
8/9/2019 Mathematics Review - Hyperbola
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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
8/9/2019 Mathematics Review - Hyperbola
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becomes and
#he graph of our hori:ontally aligned hyperbola is shown below
A7/8## ) A H78-FA
8/9/2019 Mathematics Review - Hyperbola
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very hyperbola has asymptotes.
#he asymptotes of a hori:ontally aligned hyperbola are given by the e
8/9/2019 Mathematics Review - Hyperbola
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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 ?
8/9/2019 Mathematics Review - Hyperbola
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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
8/9/2019 Mathematics Review - Hyperbola
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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
8/9/2019 Mathematics Review - Hyperbola
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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
8/9/2019 Mathematics Review - Hyperbola
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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
8/9/2019 Mathematics Review - Hyperbola
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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
8/9/2019 Mathematics Review - Hyperbola
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A7/8## ) J- +-#I?A7 AIG0* H78-FA
)or our vertically aligned hyperbola that we >ust graphed, the e
8/9/2019 Mathematics Review - Hyperbola
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and
olving for k, we get
k 3 5M in both e
8/9/2019 Mathematics Review - Hyperbola
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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
8/9/2019 Mathematics Review - Hyperbola
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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
8/9/2019 Mathematics Review - Hyperbola
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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
8/9/2019 Mathematics Review - Hyperbola
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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
8/9/2019 Mathematics Review - Hyperbola
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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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