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AUTHORTITLEINSTITUTIONPUB DATENOTEAVAILABLE FROM
EDRS PRICEDESCRIPTORS
ABSTRACT
DOCUMENT RESUME
SE 011 330
Helwig, G. Alfred; And OthersAnalytic Geometry, A Tentative Guide.Baltimore County Public Schools, Towson, Md.6753p.Baltimore County Public Schools, Office ofCurriculum Development, Towson, Maryland 21204($2.00)
EDRS Price MF-$0.65 HC-$3.29*Analytic Geometry, Curriculum, *Curriculum Guides,Geometric Concepts, Instruction, Mathematics,*Secondary School Mathematics, *Teaching Guides
This teacher's guide for a semester course inanalytic geometry is based on the text "Analytic Geometry" by W. K.Morrill. Included is a daily schedule of suggested topics andhomework assignments. Specific teaching hints are also given. Thecontent of the course includes point and plane vectors, straightlines, point and space vectors, planes, straight lines in space,circles, conics, transformation of axes, and polar coordinates.(Author/CT)
U S DEPARTMENT OF HEALTH.EDUCATION & WELFAREOFFICE OF EDUCATION
(HIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVED FROMTHE PERSON OR ORGANIZATION ORIG/NAT/NO IT POINTS OF VIEW OR OPINIONS STATED DO NOT NECESSARILYREPRESENT OFFICIAL OFFICE OF EDUCATION POSITION OR POLICY
BALTIMORE COUNTY PUBLIC SCHOOLS
A Tentative Guide
ANALYTIC GEOMETRY
TOWSON, MARYLAND
JANUARY, 1967
BALTIMORE COUNTY PUBLIC SCHOOLS
A Tentative Guide
ANALYTIC GEOMETRY
Prepared under the direction ofG. Alfred Helwig
Director of Curric'ulumB. Melvin Cole
Assistant Superintendent of Instruction
Committee Members
Elizabeth S. Hitchcock John LunnyWilliam S. Kosicki Hans. E. Mitchell
Frederick A. Schwartz, Chairman
Stanley A. Smith .
Supervisor of Senior High School Mathematics
Vincent BrantMathematics Coordinator
William S. SartoriuSuperintendent
Towson, Maryland1967
BOARD OF EDUCATION OF BALTIMORE COUNTYAigburth Manor
Towson, Maryland 21204
T. Bayard Williams, Jr.President
Mrs. John M. Crocker H. Ems lie ParksVice-President
Mrs.-Robert L. Berney Ramsay B. Thomas, M.D.H. Russell Knust Richard W. Tracey, D. V. M.
William S. SartoriusSecretary-Treasurer and Superintendent of Schools
ANALYTIC GEOMETRY
Analytic Geometry is a second semester course in theTrig-Analytics sequence. It is designed for those students who havesuccessfully completed Algebra II, Geometry and Trigonometry. Thiscourse is also prerequisite for those students desiring the Calculus inthe 12th year.
The text for the course is Analytic Geometry by W. K. Morrillwhich utilizes a vector approach to the subject. However, the use ofvector techniques in no way implies that this should become a coursein vector algebra. The page references are keyed to both the old andrevised editions.
The guide is tentative and subject to continuous teacherevaluation. The time allotment, sequence of topics and homeworkassignments are suggestive. The teacher may adjust these accordingto the needs of the students. For instance, some teachers may wish tointerchange the units on Transformation of Axes and Polar Coordinates.Teachers will want to avail themselves of appropriate overheadtransparencies in the Geometry and Calculus series, which may befound in the department files. Standardized tests are also availableat the Central Testing Office. These may be obtained through yourdepartment chairman.
The Board of Education and the Superintendent of Schoolswish to express their appreciation to the Curriculum Committee andto all mathematics teachers of Baltimore County whose efforts madepossible the development of this curriculum guide.
Special commendation is due Mrs. Betty Hagedorn,secretary to the committee, for her careful and painstaking effortsin the production of this guide.
ii
4
TABLE OF CONTENTS
FOREWORD ii
UNIT ONE The Point and Plane Vectors 1
UNIT TWO The Straight Line 11
UNIT THREE The Point and Space Vectors 18
UNIT FOUR The Plane 20
UNIT FIVE The Straight Line in Space 25
UNIT SIX The Circle 29
UNIT SEVEN The Conics 32
UNIT EIGHT Transformation of the Axes 38
UNIT NINE Polar Coordinates 45
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EPA
GE
1
UN
IT O
NE
The
Poi
nt a
nd P
lane
Vec
tors
Intr
oduc
tion
(2-1
)E
mph
asiz
e th
e fu
ndam
enta
l con
cept
of
anal
ytic
geo
met
ry:
a po
int
is a
mem
ber
of th
e gr
aph
of a
rel
atio
n if
and
onl
y if
its
coor
dina
tes
satis
fy th
e eq
uatio
n of
the
rela
tion.
The
Rec
tang
ular
Coo
rdin
ate
Syst
emN
ote
that
the
choi
ce o
f a
rect
angu
lar
syst
em o
f co
ordi
nate
s is
MR
ev19
M18
MR
ev19
-21
M18
-20
arbi
trar
y. A
dif
fere
nt s
yste
m is
oft
en m
ore
usef
ul f
or th
eso
lutio
n of
spe
cifi
c pr
oble
ms.
An
exam
ple
of a
n al
tern
ate
syst
emof
axe
s is
giv
en in
the
diag
ram
bel
ow.
Y_.
_P(
x, y
)
/x
Ox
TH
E O
BL
IQU
E S
YST
EM
OF
CO
OR
DIN
AT
ES
Em
phas
ize
the
diff
eren
ces
betw
een
a ri
ght-
hand
coo
rdin
ate
syst
eman
d a
left
-han
d co
ordi
nate
sys
tem
. Not
e th
at P
(x, y
) re
pres
ents
any
poin
t in
the
plan
e an
d no
t onl
yth
ose
in q
uadr
ant I
.
V
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E1
PAG
E
1Sy
mm
etry
(2-
3)M
Rev
... 2
2-23
(con
ed)
Sym
met
ry w
ith r
espe
ct to
a p
oint
M21
Sym
met
ry w
ith r
espe
ct to
a li
neSU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
:
The
pro
blem
s in
the
Sugg
este
d B
asic
Ass
ignm
ent a
re n
ot m
eant
to c
onst
itute
the
entir
e as
sign
men
t. T
hey
are
prob
lem
s w
hich
shou
ld b
e in
clud
ed in
ass
ignm
ent.
MR
ev...
Pp.
21,
22:
Ex.
3,
6, 8
, 10,
13,
18,
20,
23,
25,
26
MPp
. 20,
21:
Ex.
3, 6
, 8, 1
0, 1
3M
Rev
P. 2
3:E
x. 2
M.
Pp. 2
1, 2
2:E
x. 2
2Pr
ojec
tions
MR
ev...
23-
25
Stud
ents
who
hav
e st
udie
d th
e SM
SG G
eom
etry
con
side
r th
e pr
ojec
tion
of a
poi
nt o
r se
gmen
t in
a lin
e to
be
a se
t of
poin
ts.
The
text
def
ines
the
proj
ectio
n of
a d
irec
ted
segm
ent P
1P2
in a
line
L to
be
the
M.
22-2
4
dire
cted
dis
tanc
e A
B w
here
A is
the
proj
ectio
n of
P1
in L
and
B is
the
proj
ectio
n of
P2
in L
. Obs
erve
that
in S
MSG
Geo
met
ry A
B r
epre
sent
sa
segm
ent w
ith A
and
B a
s en
d po
ints
and
AB
rep
rese
nts
the
undi
rect
edle
ngth
of
AB
. How
ever
, not
e th
at in
the
text
all
segm
ents
are
dir
ecte
d-.
.seg
men
ts. P
1P2
repr
esen
ts th
e di
rect
ed s
egm
ent f
rom
P1
to P
2; A
Bre
pres
ents
the
dire
cted
dis
tanc
e fr
om I
. to
B w
here
A a
nd B
are
poi
nts
in th
e x
or y
axe
s, a
nd I
PIP2
1 re
pres
ents
the
leng
th o
f Pi
P2.
Indi
cate
the
diff
eren
ces
betw
een
the
nota
tion
used
in S
MSG
and
the
text
to a
void
con
fusi
on.
2
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E1
PAG
E
2Sc
alar
Com
pone
nts
of a
Seg
inen
t (2-
5)A
TR
ev26
-27
(con
ed)
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
ivI.
24-2
5
MR
ev...
P. 2
5:E
x. 1
, 3M
Rev
... P
. 27:
Ex.
1, 3
, 4M
Pp. 2
3, 2
4:E
x. 1
, 3M
.Pp
.25,
26:
Ex.
1, 3
, 43
Dis
tanc
e B
etw
een
Tw
o Po
ints
(2-
6)M
Rev
... 2
1-30
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
M26
-28
MR
ev...
Pp.
29,
30:
Ex.
2a,
3, 6
, 7,
8, 9
, 10,
12,
13
M..
P. 2
8:E
x. 2
a, 3
, 6, 7
, 84
Dir
ectio
n C
osin
es o
f a
Segm
ent (
2-7)
MR
ev31
-33
Not
e th
at th
e te
xt d
oes
not d
efin
e di
rect
ion
angl
es o
f a
segm
ent.
M29
-31
Thi
s sh
ould
be
done
as
follo
ws:
The
dir
ectio
n an
gles
of
P1P2
whe
re P
= (
xy
) an
d P2
= (
x2, y
2) a
re th
e an
gles
mea
sure
dI
I'1
-->
-->
,in
a c
ount
ercl
ockw
ise
dire
ctio
n fr
om tw
o ra
ys P
1A
and
P1
B
para
llel t
o th
e po
sitiv
e x
and
y ax
es r
espe
ctiv
ely
to th
e di
rect
edse
gmen
t P1P
2. T
hese
ang
les
are
deno
ted
by a
and
p
resp
ectiv
ely.
Illu
stra
te th
ese
defi
nitio
ns w
ith d
iagr
ams
sim
ilar
to th
e on
es b
elow
.
3-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EI
PAG
E
4(c
ont'd
)y /`
\
/4,B
_
/Y \,
% 2
a-,
A
.x
\x /
B 4\ 4\la
, ,
YB
*('
A
0 \17
Y
21
i
.
\ -P1
B I IP2
4---
,
yA
x
i
B
\L
,(7 ' (
Ip-
-.\
1
- --
----
.
--.>
A
N ---
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EJ
PAG
E
4T
he d
irec
tion
cosi
nes,
1 a
nd m
,of
PI
P2 a
re d
efin
ed to
be
the
(con
tld)
cosi
nes
of a
and
13
resp
ectiv
ely.
Em
phas
ize
the
adva
ntag
es o
fth
e co
sine
fun
ctio
n ov
er th
e ot
her
trig
onom
etri
c fu
nctio
ns.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
Pp.
32,
33:
Ex.
1, 2
, 3, 4
,5,
8M
Pp. 3
0, 3
1:E
x. 1
, 2, 3
, 4, 5
, 8
5Pl
ane
Vec
tors
(2-
8)M
Rev
... 3
3-36
Vec
tors
are
cen
tral
in th
e de
velo
pmen
tof
the
text
.T
hey
are
abst
ract
con
cept
s, b
eing
set
s of
equi
vale
nt d
irec
ted
segm
ents
.
M31
-34
Thi
s is
sim
ilar
to th
e id
ea o
f eq
uiva
lenc
ecl
asse
s. T
he a
nalo
gysh
ould
be
purs
ued
with
stu
dent
s fa
mili
arw
ith th
e co
ncep
t.St
uden
tsof
phy
sics
are
fam
iliar
with
the
vect
orco
ncep
t. H
owev
er, v
ecto
rsha
ve b
een
defi
ned
for
them
as
"qua
ntiti
es p
osse
ssin
g m
agni
tude
and
dire
ctio
n".
Thi
s is
a g
eom
etri
cal i
nter
pret
atio
n an
d it-
,w
hat o
urte
xt m
eans
by
a re
pres
enta
tive
of th
e ve
ctor
. A s
cala
r is
a r
eal
num
ber.
Exa
mpl
es o
f sc
alar
s ar
e le
ngth
, are
a,vo
lum
e, ti
me-
mas
s, a
nd te
mpe
ratu
re.
Exa
mpl
es o
f ph
ysic
al q
uant
ities
that
are
repr
esen
ted
by v
ecto
rs a
re f
orce
, vel
ocity
,an
d ac
cele
ratio
n.In
dica
te th
e si
mila
rity
bet
wee
n th
eal
gebr
a of
rea
l num
bers
and
the
alge
bra
of v
ecto
rs. H
owev
er, d
o no
t pro
veal
l the
alg
ebra
ic p
rope
rtie
sof
vec
tors
.T
he p
urpo
se o
f th
is c
ours
e is
tode
velo
p an
alyt
icge
omet
ry, n
ot v
ecto
r al
gebr
a.
EX
AM
PLE
: Exp
ress
[ 3
, 4]
in te
rms
of [
1, 0
] an
d [
0,1
] .
SOL
UT
ION
:[
3, 4
] =
[ 3
, 0 ]
+ [
0, 4
1 =
3[ 1
, 0 ]
+ 4
[ 0,
1 ]
.
5 -
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EPA
GE
5Se
vera
l exa
mpl
es li
ke th
is w
ill le
ad to
the
gene
raliz
atio
n th
at th
e(c
oned
)ve
ctor
s [
1, 0
] an
d 1
0, 1
1 a
re th
e "b
uild
ing
bloc
ks"
of th
e sy
stem
of v
ecte
rs.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
Pp.
36,
37:
Ex.
1, 2
a, 3
, 4,
5, 6
, 10,
11,
12
Om
it nu
mbe
r 8.
MP.
34:
Ex.
1, 2
a, 3
, 4, 5
, 6, 1
0 (a
nd. 1
1, 1
2 of
MR
ev. )
6A
ngle
Bet
wee
n T
wo
Vec
tors
(2-
9)iv
lRev
... 3
7-39
An
alte
rnat
e pr
oof
of th
e fo
rmul
a fo
r de
term
inin
g th
e an
gle
betw
een
two
vect
ors
is g
iven
bel
ow a
nd m
ay b
e us
ed if
the
teac
her
pref
ers
it to
the
one
in th
e te
xt.
0134
-36
-4.
-..
TH
ZO
RE
M: I
f u
= [
u11
u 2v
] an
d=
[ v1
, v2
] ar
e tw
o no
n-ze
rove
ctor
s, th
e an
gle
0 be
twee
n th
em is
giv
en b
y
cos
0 =
ul v
1+
u2
v2
lu 1
1v 1
...lb
.4.
....
PRO
OF:
Let
w=
v-u.
Thu
s,w
=1v
-u-
v -I
li1
1-2
Z
By
the
law
of
cosi
nes
(see
dia
gram
),-,
- 2
-4-
2-4
- 2
lwl
= I
ul+
Iv
1-
21u1
Iv
1 co
s 0
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EPA
GE
6
(con
ed)
--s.
-*0-
= f
Vls
V2}
U m
gIi
i ' u
2]
But
, sin
ce w
= [
v1-
ui, v
2-
u21
,
1 [
v1-
ul, v
2-
u21
12
=17
12 +
1;1
2-
2171
1;1
cos
0
(v1
- u 1
)2 +
(v2
- u
2)2
= I
.; 12
+ 1
;12
- 2
17 I
lv-4
' 1 c
os 8
22
22.
.' 1-
,-Z
-i-
D.2
(v1.
- Z
v 1u
1+
u1
) +
(v2
- 2v
2u 2
+ u
2--)
4.--
iu i
+ iv
i-.
-1-
;1.
cos
8
22
22
-2
-Y12
1711
;1(v
1+
v2)
+(u
1+
u2)
- 2(
u1v1
+ u
2v2)
= -
lu+
1 v
cos
0
1 ; I
2+
111
.4°
I
2 2
.. 2
(11
1 V
1+
u2v2
)=
171
2 +
1;12
- 2k
711;
I co
s 8
-2(u
1v
1+
u2v2
)=
-21
711;
1 co
s 0
uvi
+ u
vl
228
= c
os17
1 R
IQ
.E.D
.
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EI
PAG
E
6A
n el
egan
t fea
ture
of
the
abov
e pr
oof
is th
at th
ere
is n
o re
fere
nce
(con
t'd)
to a
ny c
oord
inat
e sy
stem
and
thus
the
proo
f ge
nera
lizes
to 3
-spa
cequ
ite e
asily
.T
he te
ache
r sh
ould
not
e th
at th
e te
xt's
def
initi
on o
f th
e do
t pro
duct
is d
iffe
rent
fro
m th
e on
e in
PSS
C P
hysi
cs.
It is
sug
gest
ed th
at th
ete
ache
r co
nsul
t the
teac
her
of p
hysi
cs to
avo
id c
onfu
sion
in th
ete
rm d
ot p
rodu
ct.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
Pp.
41,
42:
Ex.
8, 1
6, 1
7, 1
8M
......
Pp,
38,
39:
Ex.
8 (
and
16, 1
7, 1
8 of
MR
ev.)
7Pa
ralle
l and
Per
pend
icul
ar V
ecto
rs (
2-10
)M
._ 1
ev
39-4
2
The
teac
her
shou
ld s
tudy
the
conc
ept o
f pr
opor
tiona
lity
in C
hapt
er 7
,M
36-3
8
Sim
ilari
ty, o
f SM
SG G
eom
etry
with
Coo
rdin
ates
.T
his
will
cle
ar u
pth
e ap
pare
nt p
ossi
bilit
y of
div
isio
n by
zer
o in
the
disc
ussi
on o
npa
ralle
lism
of
vect
ors
on p
age
40 o
f th
e te
xt.
To
avoi
d th
isdi
ffic
ulty
, the
teac
her
may
wan
t to
adop
t the
res
ults
of
prob
lem
9on
pag
e 41
of
MR
ev a
s th
e de
fini
tion
of p
aral
lel v
ecto
rs.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
Pp.
41,
42:
Ex.
la, 2
,5,
6, 9
, 10,
11,
12,
14a
, 19
MPp
. 38,
39:
Ex.
la, 2
, 5, 6
, 7 (
and
9, 1
0, 1
1, 1
2,14
a, 1
9 of
MR
ev. )
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EI
PAG
E
8T
he C
oord
inat
es o
f a
Poin
t Tha
t Div
ides
a S
egm
ent i
n a
Giv
en R
atio
(2-
11)
ivlR
ev42
-45
The
teac
her
shou
ld e
mph
asiz
e th
e so
lutio
n of
eac
h pr
oble
m b
yus
ing
an a
ppro
pria
te v
ecto
r eq
uatio
n ra
ther
than
by
subs
titut
ion
in th
e fo
rmul
as o
n pa
ge 4
4 of
the
text
.In
eff
ect,
each
pro
blem
shou
ld b
e a
rede
velo
pmen
t of
the
form
ulas
on
page
44.
ivl.
39-4
2
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
Pp.
46,
47:
Ex.
1, 3
, 15,
16,
17,
18,
19
M.
Pp. 4
3, 4
4:E
x. 1
, 3, 1
5, 1
6, 1
7, 1
8, 1
9
9T
he M
idpo
int o
f a
Segm
ent a
nd A
naly
tic P
roof
s of
Geo
met
ric
The
orem
siv
IRev
45-4
7(2
-12)
M42
-44
The
teac
her
shou
ld s
pend
mos
t of
the
less
on s
olvi
ng p
robl
ems
such
as n
umbe
rs 9
, ii,
13, a
nd 1
4 on
pag
es 4
6 an
d 47
of
the
text
,SU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
:
1VIR
ev...
Pp.
46,
47:
Ex.
5, 7
, 9, 1
0M
......
Pp.
43,
44:
Ex.
5, 7
, 9, 1
0
10C
ompl
emen
tary
Vec
tors
(2-
13)
MR
ev48
M44
-45
Are
a of
a T
rian
gle
and
the
Bar
Pro
duct
of
Tw
o V
ecto
rs (
2-14
)1V
itley
49-5
1
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
M45
-47
MR
eve
.. Pp
. 50,
51:
Ex.
1, 2
, 3, 4
b, 5
a, 6
, 7O
mit
num
ber
9 fo
r st
uden
ts n
ot f
amili
ar w
ith d
eter
min
ants
.M
......
Pp.
47,
48:
Ex,
1, 2
, 3, 4
b, 5
a 6,
7
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EPA
GE
11 12
Rev
iew
for
Tes
t and
Rev
iew
Exe
rcis
esB
e su
re to
incl
ude
prob
lem
s fr
om th
e R
evie
w E
xerc
ises
at t
he
MR
ev.
. .52
-54
48-4
9M
.
end
of e
ach
chap
ter
of th
e te
xt.
Bes
ides
pro
vidi
ng g
ood
prob
lem
sfo
r re
view
, mat
eria
l is
ofte
n in
trod
uced
or
exte
nded
bey
ond
the
poin
t cov
ered
ear
lier
in th
e ch
apte
r.
Tes
t on
Cha
pter
2
-10-
.
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
E1
CO
DE
I PA
GE
UN
IT T
WO
The
Str
aigh
t Lin
e1.
z --D
irec
tion
Num
bers
and
Dir
ectio
n C
osin
es o
f a
Lin
e (3
-1)
IVZ
iev.
.. 55
-56
Em
phas
ize
that
a li
ne h
as a
n in
fini
te n
umbe
r of
pai
rs o
f di
rect
ion
M.
50-5
1
num
bers
ass
ocia
ted
with
it, w
hile
dire
ctio
n co
sine
s.it
has
only
two
pair
s of
ass
ocia
ted
EX
AM
PLE
: The
line
AB
, with
A(1
,2)
B(-
5, 6
) ha
s di
rect
ion
num
bers
:[
-6, 4
],
[ -3
, 2 ]
,[
3, -
2 ]
,[
9, -
6] e
t -...
s,
The
dir
ectio
n co
sine
sof
AB
are
:
-64
-32
I
i 2 1
11.3
'2 ,4
-1-3
i13
'A
i 13
! _1
3-2
..[
9-6
1
13L
Nff
i4-
--
13
,./ 1
3 '
313
The
Slo
pe o
f a
Lin
e (3
-2)
MR
ev57
-58
M51
-54
Alth
ough
slo
pe is
not
a n
ew c
once
pt to
the
stud
ent,
incl
inat
ion
is.
Incl
inat
ion
in th
e te
xt is
use
d in
two
cont
exts
, the
ang
le b
etw
een
the
posi
tivel
y di
rect
ed x
axi
s an
d th
e lin
e, m
easu
red
in a
cou
nter
-cl
ockw
ise
dire
ctio
n an
d as
the
mea
sure
of
that
ang
le.
SUG
GE
STE
D B
ASI
C tS
SIG
NM
EN
T:
MR
ev...
P. 5
8:E
x. la
b, 2
, 3, 4
, 5, 6
abM
.P.
54:
Ex.
lab,
2, 3
, 4, 5
, 6ab
.14
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E1
PAG
E
14 15
Para
met
ric
Equ
atio
ns o
f L
ines
(3-
3)T
his
conc
ept i
s no
t new
to th
ose
stud
ents
who
hav
e st
udie
d SM
SGG
eom
etry
. The
vec
tor
appr
oach
of
this
text
, how
ever
, sho
uld
mak
efo
r a
bette
r un
ders
tand
ing.
For
stu
dent
s w
ho d
id n
ot h
ave
SMSG
Geo
met
ry p
aram
etri
c eq
uatio
ns w
ill b
e a
new
con
cept
.St
uden
tsw
ill n
eed
this
met
hod
of w
ritin
g lin
ear
equa
tions
in th
ree
dim
ensi
ons
so th
ey s
houl
d be
com
e ad
ept i
n us
ing
this
met
hod
now
. Hav
est
uden
ts g
raph
som
e lin
es w
hose
equ
atio
ns a
re g
iven
in p
aram
etri
cfo
rm. (
Exe
rcis
e 2
at e
nd o
f se
ctio
n. )
Equ
atio
ns o
f L
ines
in D
irec
tion
Num
ber
Form
(Sy
mm
etri
c Fo
rm)
(3-4
)x
= x
1+
tax
Para
met
ric
x -
xly
y1Sy
mm
etri
c
ML
iev.
.. 59
-61
M54
-56
ka: t
ev
59-6
4M
54-
56
.
y =
y1
+ ta
yFo
rmA
UA
ayFo
rm
As
can
be s
een
abov
e th
e sy
mm
etri
c fo
rm o
f an
equ
atio
n of
a li
neis
eas
ily d
eriv
ed f
rom
the
para
met
ric
form
by
solv
ing
for
the
para
met
er. S
ymm
etri
c fo
rm is
not
usa
ble
if e
ither
Ax
or A
y is
0.
Giv
e st
uden
ts p
ract
ice
in c
onve
rtin
g fr
om p
aram
etri
c fo
rm to
sym
met
ric
form
. Hav
e th
e st
uden
ts d
o so
me
of th
e ex
erci
ses
atth
e en
d of
this
sec
tion
in s
ymm
etri
c fo
rm a
nd le
ave
thei
r an
swer
sin
that
for
m. R
ule
3-13
sho
uld
be o
mitt
ed.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
Pp.
61,
62:
Ex.
1 (
in p
aram
etri
c an
d sy
mm
etri
cfo
rm),
2, 6
ab, 9
abM
.P.
56:
Ex.
1 (
in p
aram
etri
c an
d sy
mm
etri
c fo
rm),
2, a
nd in
clud
e 6a
b 9a
b, o
f M
Rev
.
- 12
-
ti
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EPA
GE
16 17
The
Gen
eral
Lin
ear
Equ
atio
n ax
+ b
y +
c =
0(3
-5)
Hav
e th
e st
uden
ts th
orou
ghly
und
erst
and
the
two
part
pro
of o
f:
a) T
he s
et o
f po
ints
P(x
, y)
havi
ng th
e pr
oper
ty th
at P
1P[
x-
x1 ,
y -
yiI
is p
erpe
ndic
ular
to u
[ a
, b J
cont
aini
ng P
1 ha
s an
equ
atio
n ax
+ b
y =
c.
b) a
x +
by
= c
is a
n eq
uatio
n of
a li
ne w
hich
has
a v
ecto
r[
a, b
J p
erpe
ndic
ular
to it
.T
his
sect
ion
is a
n ex
citin
g an
d po
wer
ful r
esul
t of
our
prev
ious
vect
or tr
eatm
ent.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
IvIR
ev...
Pp.
64,
65:
Ex
Zab
, 3ab
, 6ab
MP.
, 59:
Ex.
Zab
, 3ab
, (an
d 6a
b of
MR
ev. )
Lin
es P
erpe
ndic
ular
to a
nd P
aral
lel t
o a
Giv
en L
ine
(3-1
1)T
his
sect
ion
and
the
exer
cise
s w
hich
fol
low
giv
e fu
rthe
r em
phas
isto
Sec
tion
(3-5
). SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
Pp.
81,
82:
Ex.
labc
d, Z
ab, 3
ab, 4
abM
.Pp
. 73,
74:
Ex.
labc
d, Z
ab, 3
ab, 4
ab
MR
ev.
.. 65
-68
M.
59-6
3
MR
ev...
80-
82M
.72
-7 3
- 13
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
ET
PA
GE
18A
ngle
Bet
wee
n T
wo
Lin
es (
3-6)
The
teac
her
shou
ld g
ive
equa
l em
phas
is to
fin
ding
the
angl
ebe
twee
n tw
o lin
es th
roug
h th
e co
sine
for
mul
a an
d th
roug
h th
eta
ngen
t for
mul
a. T
he c
osin
e m
etho
d is
alr
eady
fam
iliar
toth
e st
uden
ts f
rom
Cha
pter
2 a
nd m
ay b
e pr
efer
red
by th
em.
How
ever
, the
tang
ent m
etho
d sh
ould
be
enco
urag
ed a
s it
rein
forc
es th
e in
clin
atio
n co
ncep
t and
is u
sefu
l in
the
calc
ulus
.T
he te
ache
r sh
ould
enc
oura
ge d
raw
ings
fro
m s
tude
nts
and
give
them
exe
rcis
es w
here
a s
peci
fied
ang
le's
cos
ine
or ta
ngen
tis
req
uire
d.(P
robl
ems
8, 1
0, 1
1, 1
2 at
end
of
this
sec
tion.
)A
goo
d m
emor
y te
chni
que
for
find
ing
the
tang
ent o
f an
ang
le is
:
19
S3-
81T
an 0
-3
131
+sl
s 3
Not
e th
at th
e sy
mbo
l "01
3" m
eans
the
dire
cted
ang
le b
etw
een
1an
dL
3'SU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
.:
MR
ev.
Pp. 7
2, 7
3:E
x. la
, 3a,
5, 6
a, 8
, 10,
11,
12
MP.
65:
Ex.
la, 3
a, 5
, 6a,
8 (
and
10, 1
1, 1
2 of
MR
ev.)
Poin
t Slo
pe F
orm
of
Equ
atio
n of
Lin
e (3
-7)
MR
ev.
. .70
- 71
M63
-65
MR
ev73
-75
M65
-68
- 14
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EPA
GE
19In
terc
ept F
orm
of
Equ
atio
n of
Lin
e (3
.8)
(con
ed)
Slop
e, I
nter
cept
For
m o
f E
quat
ion
of a
Lin
e (3
-9)
The
thre
e fo
rms
abov
e sh
ould
be
a re
view
for
. the
stu
dent
s.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
Pp.
78,
79:
Ex.
labc
, 2, 4
, 13,
18,
19
M.
Pp. 7
1, 7
2:E
x. la
bc, 2
, 4 (
and
13, 1
8, 1
9 of
.MR
ev. )
20D
ista
nce
from
Lin
e to
Poi
nt (
3-12
)M
R e
v...
83-8
5
The
teac
her
shou
ld e
mph
asiz
e:
ax1
+ b
y 1+
ca.
If th
e nu
mbe
r>
0, t
hen
the
P(x
M.
74-7
6
poin
t1,
y 1)
+0"
-
lies
on th
e si
de o
f th
e lin
e ax
+ b
y +
c =
0 to
war
d w
hich
the
coef
fici
ent v
ecto
r [
a, b
] po
ints
, pro
vide
d a
> 0
.ax
1+
by
1+
cb.
If th
e nu
mbe
r<
0, t
hen
the
poin
t P(x
1'yl
)N
ia2
+b2
lies
on th
e si
de o
f th
e lin
e op
posi
te to
that
whi
ch th
eco
effi
cien
t vec
tor
poin
ts.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev.
P. 8
5:E
x. 3
a, 4
a, 5
a, 8
, 9M
Pp. 7
6, 7
7:E
x. 3
a, 4
a, 5
a (a
nd 8
, 9 o
f iv
IR e
v. )
- 15
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E I
PA
GE
21 22
Inte
rsec
ting
Lin
es (
3 -1
3)T
his
sect
ion
give
s an
opp
ortu
nity
to in
trod
uce
the
clas
s to
dete
rmin
ants
; eva
luat
ion
of a
sec
ond
orde
r de
term
inan
t and
its
use
for
find
ing
the
poin
t of
inte
rsec
tion
of 2
line
s.If
the
clas
sfi
nds
this
eas
y or
a r
evie
w, i
t is
advi
sabl
e to
eva
luat
e th
ird
orde
r de
term
inan
ts a
s th
ey w
ill b
e us
eful
in th
e ch
apte
rs w
hich
follo
w. T
he d
iago
nal m
etho
d of
exp
ansi
on m
ay b
e us
ed in
2nd
orde
r an
d 3r
d or
der
dete
rmin
ants
. Exp
ansi
on b
y co
fact
ors
is n
eces
sary
for
the
expa
nsio
n of
all
dete
rmin
ants
bey
ond
3rd
orde
r. E
ncou
rage
stu
dent
s to
use
det
erm
inan
ts in
the
solu
tion
of p
robl
ems
3, 4
, 5 a
t the
end
of
this
sec
tion.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev.
.. P.
90:
Ex.
3, 4
, 5M
.P.
80:
Ex.
3, 4
, 5
Bis
ecto
rs o
f A
ngle
s (3
-15
)
The
set
of
poin
ts e
quid
ista
nt f
rom
the
side
s of
an
angl
e is
a r
ayw
hich
bis
ects
the
angl
e.T
he te
ache
r sh
ould
hav
e th
e st
uden
tsad
opt t
his
"loc
us"
theo
rem
to th
eir
wor
k in
Sec
tion
3 -1
5.Sp
ecia
lca
re s
houl
d be
giv
en to
mak
ing
a dr
awin
g an
d un
ders
tand
ing
whi
chsi
gn s
houl
d be
use
d in
the
prob
lem
at h
and
from
the
gene
ral
equa
tion
for
the
angl
e bi
sect
or
alx
+ b
ly +
ca 2x
+ b
2y+
c2
= +
MR
ev...
86-
88M
.78
-80
IVIR
e v
. .
.91
-93
ivi
81.-
83
2+
b 1
2la
11/a
22 +
b 2
2
- 16
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E 1
PA
GE
22N
ote:
Illu
stra
tive
exam
ple
3-26
sho
uld
read
: Fin
d an
equ
atio
n(c
ont d
)of
the
bise
ctor
of
the
angl
e su
ch th
at:
1) th
e an
gle
is f
orm
ed b
y th
e lin
es 1
1 =
3x
- 4y
+ 1
0 =
0an
d 1
2=
4x
- 3y
+ 1
2 =
0
2) th
e or
igin
is a
n in
teri
or p
oint
of
the
angl
e.SU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
:
MR
ev.
P. 9
3:E
x. 1
, 3, 6
, 8, 1
0M
P. 8
3:E
x. 1
,3
(and
6,
8, 1
0 of
MR
ev. )
23R
evie
w
Sinc
e Se
ctio
ns 3
-10,
3-1
1:, 3
-16,
3-1
7 ha
ve b
een
omitt
ed, r
evie
wex
erci
ses
10, 1
9, 2
0, 3
0, 3
5, 3
6 sh
ould
be
omitt
ed.
Z4
Tes
t
- 17
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EI
PAG
E
UN
IT T
HR
EE
The
Poi
nt a
nd S
pace
Vec
tors
25T
he P
oint
in S
pace
(9-
1)Iv
LR
ev...
297
-298
M.
277-
278
Proj
ectio
ns (
9-2)
lviR
ev29
8-29
9N
ote
the
dual
usa
ge o
f "p
roje
ctio
n of
a s
egm
ent"
as
mea
ning
bot
ha
segm
ent a
nd a
dir
ecte
d di
stan
ce.
M27
S-27
9
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
Mile
y...
P. 3
00:
Ex.
2, S
ae, 7
, 9, 1
0M
.P.
280
:E
x. 2
, Sae
, 7, 9
, 10
26Sc
alar
Com
pone
nts
of a
Seg
men
t (9-
3)M
Rev
... 3
01M
281
'
Len
gth,
or
Mag
nitu
de, o
f a
Segm
ent (
9-4)
MR
ev,
. .30
1 -3
03M
281
-283
Dir
ectio
n C
osin
es o
f a
Segm
ent (
9-5)
MR
ev30
3-30
4
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
M28
3-28
4
MR
ev...
Pp.
304
,305
: Ex.
la, 5
a, 6
, 7, 1
0M
.Pp
. 284
,285
: Ex.
la, 5
a, 6
, 7, 1
0
27Sp
ace
Vec
tors
(9-
6)IV
Ill e
v.,
. . 3
05-3
09iv
'28
5-28
8
-18-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EI
PAG
E.
27C
osin
e of
the
Ang
le B
etw
een
Tw
o V
ecto
rs (
9-7)
MR
ev...
309
-312
(con
t'd)
IA.
288-
291
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev.
.. P.
308
:E
x. 3
, 4ac
, 6a,
7,.
10a
M.
P. 2
88:
Ex.
3, 4
ac, 6
a, 7
, 10a
MR
ev...
P. 3
12: E
x, 1
, 2a9
3a,
9M
......
Pp.
291
-292
: Ex.
1, 2
a, 3
a, 9
28T
he C
oord
inat
es o
f a
Poin
t tha
t Div
ides
a S
egm
ent i
n a
Giv
en R
atio
MR
ev...
312
-314
(9"
8).i.
/1. .
. .. .
292
-294
The
Mid
poin
t of
a Se
gmen
t (9-
9)IV
IR e
v.. .
314-
315
M29
4SU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
:
MR
e-v.
.. P,
315
:E
x. 1
, 2, 5
, 7M
Pp. 2
94-2
95: E
x. 1
, 2, 5
, 7
29R
evie
w
30T
est
- 19
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EPA
GE
UN
IT F
OU
R
The
Pla
ne
31A
n E
quat
ion
of a
Pla
ne (
10-1
)M
Rev
... 3
18-3
19It
sho
uld
be s
tres
sed
that
the
equa
tion
for
a pl
ane
in s
pace
is a
linea
r eq
uatio
n in
one
or
mor
e of
the
vari
able
s x,
y, z
.T
hege
nera
l equ
atio
n of
a p
lane
is A
x +
By
+ C
x +
d =
0.
The
teac
her
shou
ld d
eriv
e ca
refu
lly th
e co
ncep
t tha
t for
this
equ
atio
n th
eno
rmal
vec
tor
[ A
, B, C
]is
_±,
to th
e pl
ane.
M.
297-
298
The
teac
her
shou
ld e
mph
asiz
e th
at:
(1)
in a
one
dim
ensi
onal
sys
tem
x =
5is
the
equa
tion
of a
poi
nt.
(2)
in a
two
dim
ensi
onal
sys
tem
x =
6Y
= 3
x +
2y
= 1
2ar
e eq
uatio
ns o
f lin
es.
(3)
in a
thre
e di
men
sion
al s
yste
mx
+ y
= 6
x +
z =
3y
+ z
= 9
x +
3y
+ 4
z =
6ar
e eq
uatio
ns o
f pl
anes
. - 20
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EI
PAG
E
SUG
GE
STE
D-B
ASI
C A
SSIG
NM
EN
T:
1VIR
ev...
P. 3
20:
Ex.
2, 4
, 6, 9
, 10,
11,
12
M. .
.. ..
P. 2
99:
Ex.
2, 4
, 6, 9
, 10,
11,
12
32.
Para
llel a
nd P
erpe
ndic
ular
Pla
nes
(10-
2)iv
iRev
321
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
M30
0
IVill
ev...
P. 3
22:
Ex.
1, 2
, 3M
.P.
301
:E
x. 1
, 2, 3
33In
terc
ept E
quat
ion
of a
Pla
ne (
10-3
)M
Rev
322-
323
The
teac
her
shou
ld g
ive
prac
tice
in th
e gr
aphi
ng o
f an
equ
atio
nby
the
inte
rcep
t met
hod.
M30
1 -3
02
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
P. 3
23:
Ex.
1, 2
MP.
302
:E
x. 1
, 2
34 -
5V
ecto
r Pr
oduc
t of
Tw
o V
ecto
rs (
10-4
)M
Rev
... 3
23-3
26M
302
The
teac
her
shou
ld s
tres
s:(1
) th
e ve
ctor
pro
duct
is a
vec
tor
and
not a
num
ber
(2)
It. X
.2.r
.is
per
pend
icul
ar to
bot
h u
and
v(3
) u
X v
is th
e no
rmal
of
the
plan
e co
ntai
ning
u a
nd v
.
The
teac
her
shou
ld d
evel
op c
aref
ully
the
theo
rem
on
page
324
(30
4)an
d be
sur
e to
incl
ude
the
area
of
para
llelo
gram
s.
- 21
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EPA
GE
36A
ltern
atin
g
The
teac
her
Prod
uct o
r Sc
alar
Tri
ple
Prod
uct o
f T
hree
Vec
tors
(10
-5)
shou
ld p
rove
in c
lass
Exa
mpl
e 8,
pag
e 32
7.(M
... p
. 305
)
u X
v
MR
ev...
326
M.
305
1 1-,-
-.I-
/--
-_-
1-
-u
Add
ition
al p
rope
rtie
s of
the
scal
ar tr
iple
pro
duct
are
:4.
410
.1(
1 )
Ll
v, a
nd w
are
cop
lana
r if
and
onl
y if
u(v
X w
) =
0
(2)
the
valu
e of
the
scal
ar tr
iple
pro
duct
is n
ot c
hang
ed b
ya
cycl
ic p
erm
utat
ion
of th
e el
emen
ts w
_hic
h_pr
ai th
epr
oduc
t. u
(v x
u)
= w
(u x
v)
= v
(w x
u)
(3)
the
dot a
nd c
ross
in a
sca
lar
trip
le p
rodu
ct c
an b
ein
terc
hang
ed w
ithou
t aff
ectin
g th
e va
lue
of th
e pr
oduc
t.u
(v X
w)
= (
u X
v)
w
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev.
.. P
p. 3
26,3
27: E
x. 2
, 4, 5
,6,
7M
.P.
305
:E
x. 2
, 4, 5
, 6, 7
-22-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E 1
PA
GE
I
37D
eter
min
ing
a Pl
ane
Satis
fyin
g T
hree
Con
ditio
ns (
10-6
)M
R e
v32
8
The
illu
stra
tions
in th
e te
xt a
re b
ased
on
the
theo
rem
at t
he b
otto
mof
the
page
328
. An
alte
rnat
e m
etho
d is
to f
ind
the
cros
s pr
oduc
tof
two
of th
e ve
ctor
s w
hich
res
ults
in th
e no
rmal
of
the
plan
e of
thes
e tw
o ve
ctor
s. F
ollo
w E
xam
ple
10-1
on
page
319
.
M30
5
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
P. 3
30:
Ex.
1, 2
, 3, 4
MPp
. 307
88:
Ex.
1, 2
, 3, 4
38D
ista
nce
From
a P
lane
to a
Poi
nt (
10-7
)iV
IR e
v...
331
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
ivl
308
MR
ev...
P. 3
33:
Ex.
1, 2
, 3M
P. 3
10:
Ex.
1, 2
, 3
39A
ngle
Bet
wee
n T
wo
Plan
es (
10-8
)M
R e
v. .
. 334
The
mag
nitu
de o
f th
e fo
ur d
ihed
ral a
ngle
s fo
rmed
by
two
inte
rsec
ting
plan
es a
re n
umer
ical
ly e
qual
, res
pect
ivel
y, to
the
four
cor
resp
ond-
ing
angl
es f
orm
ed b
y th
e tw
o lin
es th
at c
an b
e dr
awn
thro
ugh
any
poin
tin
spa
ce p
erpe
ndic
ular
to th
e gi
ven
plan
es.
M31
0
- 23
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E1
PAG
E
39(c
oned
)B
---.
0,,.
.._0
.--
A
The
teac
her
shou
ld s
tres
s th
at th
e an
gle
form
ed b
y tw
o pl
anes
isfo
und
by c
ompu
ting
the
angl
e be
twee
n th
e tw
o no
rmal
s an
d is
sim
ilar
to th
e th
eore
m in
Geo
met
ry w
hich
sta
tes
"tw
o an
gles
who
se s
ides
are
res
pect
ivel
y pe
rpen
dicu
lar
are
equa
l."SU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
:
MR
evP.
336
:E
x. 1
MP.
312
:E
x. 1
40R
evie
w
41T
est
- 24
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E1
PAG
E
UN
IT F
IVE
Stra
ight
Lin
e in
Spa
ce
42D
irec
tion
Num
bers
and
Dir
ectio
n C
osin
es (
11-1
)M
Rev
338
Sinc
e th
e an
alog
y to
Sec
tions
9-3
and
9-5
is s
o ve
ry m
arke
d, o
nly
a br
ief
disc
ussi
on o
f th
is s
ectio
n sh
ould
be
nece
ssar
y. N
ever
thel
ess
a di
scus
sion
sho
uld
ensu
e fo
r re
view
and
rei
nfor
cem
ent p
urpo
ses.
M31
5
Para
met
ric
Equ
atio
ns o
f a
Lin
e (1
1-2)
Mae
v34
0
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
M31
7
MR
ev...
P. 3
39:
Ex.
I a
, lb,
Za,
3, 4
M.
P. 3
16:
Ex,
la, l
b, 2
a, 2
b, 3
, 4M
Rev
.P.
342
: Ex.
la, l
c, 2
aM
P. 3
20:
Ex.
la, l
c, 2
a
43Sy
mm
etri
c E
quat
ions
of
a L
ine
(11-
3)M
Rev
... 3
41M
.31
8
The
Gen
eral
Equ
atio
ns o
f a
Lin
e (1
1-4)
MR
ev...
341
The
mos
t com
mon
err
or m
ade
by s
tude
nts
is th
at o
f th
inki
ng th
atsi
nce
a fi
rst d
egre
e eq
uatio
n in
two
vari
able
s re
pres
ents
a li
ne in
two
dim
ensi
ons,
then
a f
irst
deg
ree
equa
tion
in th
ree
vari
able
sre
pres
ents
a s
trai
ght l
ine
in s
pace
.G
reat
em
phas
is s
houl
d be
plac
ed o
n th
e fa
ct th
at th
is is
not
true
. Em
phas
ize
that
a li
ne in
spac
e ca
nnot
be
desc
ribe
d by
less
than
two
firs
t deg
ree
equa
tions
.
M31
8
Inde
ed, i
t sho
uld
be a
ppar
ent t
hat e
ven
whe
n tw
o eq
uatio
ns a
re
- 25
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E I
PAG
E
43(c
oned
)
44
used
, thi
s do
es n
ot r
epre
sent
a u
niqu
e de
scri
ptio
n.Si
nce
anin
fini
te n
umbe
r of
pla
nes
pass
thro
ugh
a gi
ven
line,
any
two
such
plan
es c
onsi
dere
d to
geth
er c
an b
e us
ed to
rep
rese
nt th
e lin
e.
Show
that
the
cros
s pr
oduc
t of
the
coef
fici
ent v
ecto
rs o
f tw
o pl
anes
who
se in
ters
ectio
n de
scri
bes
the
line
dete
rmin
es a
trip
le o
fdi
rect
ion
num
bers
for
that
line
.
Stud
ents
sho
uld
be f
acile
in c
hang
ing
from
the
gene
ral t
o th
esy
mm
etri
c an
d pa
ram
etri
c fo
rms.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
P. 3
43:
Ex.
3ad
, 4a,
5, 6
, 7, 8
, 9, 1
5bM
.P.
340
:E
x. 3
ad, 4
a, 5
, 6, 7
, 8, 9
, 15b
Inte
r se
ctin
g Pl
anes
(11
-5)
The
dev
elop
men
t and
pra
ctic
e in
spa
tial p
erce
ptio
n is
part
icul
arly
evi
dent
in th
is s
ectio
n.St
uden
ts s
houl
d be
ask
ed to
subm
it sk
etch
es s
how
ing
the
diff
eren
t pos
sibi
litie
s in
whi
ch th
ree
plan
es in
spa
ce c
an in
ters
ect. --
--7/
MR
ev.
345
Ni
322
7.Z
/--
- 26
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EPA
GE
' 44(c
ont'd
)/ \
-7\
/-X
\
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
IVIR
ev...
Pe
347:
Ex.
1, 2
, 3, 4
M...
.0. P
. 324
:E
x. 1
, 2, 3
, 4
45R
evie
w
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
IvIR
ev...
Rev
iew
Exe
rcis
es P
. 352
:E
x. 1
, 2, 3
, 4, 5
, 9ab
M.
Rev
iew
Exe
rcis
es P
. 328
:E
x. 1
, 2, 3
, 4,
5, 9
ab
The
se p
robl
ems
mak
e us
e of
the
alge
brai
c re
pres
enta
tions
of
both
line
s an
d pl
anes
.G
reat
cau
tion
mus
t be
exer
cise
d by
the
stud
ent i
n ke
epin
g th
e tw
o di
stin
ct.
- 27
-
LE
SSO
NR
EFE
RE
NC
ESC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SC
OD
EPA
GE
45T
he r
esul
ts in
the
answ
er s
ectio
n do
not
alw
ays
mak
e us
e of
the
(con
ed)
obvi
ous,
giv
en p
oint
in w
ritin
g th
e eq
uatio
ns f
or a
line
.N
ever
thel
ess
the
resu
lts a
re c
orre
ct a
nd th
is a
ppar
ent d
iscr
epan
cyill
ustr
ates
aga
in th
e w
ide
choi
ce o
f ex
pres
sion
s w
hich
one
has
avai
labl
e to
des
crib
e a
line
in s
pace
.St
uden
ts m
ay w
ish
to w
ork
for
the
answ
er g
iven
by
the
auth
or, b
ut it
sho
uld
be u
nder
stoo
d th
atth
is is
not
obl
igat
ory.
46T
est
- 28
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E I
PA
GE
UN
IT S
IX
The
Cir
cle
Intr
oduc
tion
By
som
e de
fini
tions
a c
ircl
e is
a c
onic
sec
tion;
by
othe
rs, i
t is
riot
.R
efer
to S
ectio
ns 5
-1 a
nd 5
-2.
The
wor
d "r
adiu
s" is
use
d in
this
text
bot
h as
a li
ne s
egm
ent
and
as a
dis
tanc
e.
47St
anda
rd E
quat
ion
(4-1
)M
Rev
103-
104
M91
-93
The
Gen
eral
Equ
atio
n of
a C
ircl
e (4
-2)
MR
ev10
6-10
7
In p
revi
ous
Geo
met
ry c
ours
es th
e st
uden
ts h
ave
had
som
ew
ork
with
wri
ting
equa
tions
of
circ
les
whe
re th
e ce
nter
and
rad
ius
isgi
ven.
The
y w
ill a
lso
be f
amili
ar w
ith f
indi
ng th
e ce
nter
and
radi
usof
a c
ircl
e w
hen
the
equa
tion
of a
cir
cle
is g
iven
. A d
rill
wou
ldhe
lp th
e te
ache
r di
scov
er th
e st
uden
ts' b
ackg
roun
d on
this
subj
ect.
M93
-95
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
P. 1
05:
Ex.
lf, 2
c, 4
c, 5
ac, 6
c, 8
M..,
P. 9
3:E
x. lf
, 2c,
4c,
5ac
, 6c
(and
8 o
f M
Rev
.)M
Rev
... P
. 108
:E
x. 1
, 11,
13,
16
M..
. Pp.
95,
96:
Ex.
1, 1
1, 1
3 (a
nd 1
6 of
.MR
ev.)
- 29
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E 1
PA
GE
48 49
Cir
cle
Det
erm
ined
by
Thr
ee C
ondi
tions
(4-
3)
An
alte
rnat
e m
etho
d to
the
dete
rmin
ant s
olut
ion
give
n in
the
text
is s
ugge
sted
as
follo
ws:
(1)
To
find
the
equa
tion
of a
cir
cle
whi
ch p
asse
s th
roug
h th
ree
give
n po
ints
, fin
d th
e eq
uatio
ns o
f th
e pe
rpen
dicu
lar
bise
ctor
s of
two
segm
ents
det
erm
ined
by
the
poin
ts.
(2)
The
poi
nts
of in
ters
ectio
n of
the
perp
endi
cula
r bi
sect
ors
is th
e ce
nter
of
the
circ
le.
(3)
The
rad
ius
of th
e ci
rcle
will
be
the
dist
ance
fro
m th
ece
nter
to o
ne o
f th
e gi
ven
poin
ts.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
evP.
111
:E
x. la
h, 4
ahM
.Pp
. 98,
99:
Ex.
lah,
4ah
Sym
met
ry (
4-4)
Tan
gent
s to
a C
ircl
e fr
om a
n E
xter
nal P
oint
(4-
5)
Len
gth
of a
Tan
gent
fro
m a
Poi
nt O
utsi
de a
Cir
cle
to th
e C
ircl
e (4
-6)
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
evP.
117
:E
x. 1
, 2, 3
, 4, 5
MPp
. 104
, 105
:E
x. 1
, 2, 3
, 4, 5
MR
ev...
108
-111
M.
96-9
9
iVIR
ev...
112
-114
M99
-100
MR
ev11
4-11
5iv
l.10
0-10
1
iviR
ev.
. .11
5-11
6iv
'10
2-10
3
- 30
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
Er
CO
DE
1
PAG
E
50R
adic
al A
xis
(4-7
)M
Rev
117-
121
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
M.
104-
108
MR
ev...
P. 1
21:
Ex.
lac,
2ab
, 3ac
M.
P. 1
08:
Ex.
lac,
2ab
,, 3a
c
51E
quat
ion
of th
e T
ange
nt to
the
Cir
cle
x2+
y2
= r
2at
a P
oint
on
MR
ev12
2-12
3th
e C
ircl
e (4
-8)
M.
109-
110
Equ
atio
n of
the
Tan
gent
toth
e C
ircl
ex2
+ y
2 =
r2
whe
n th
e Sl
ope
MR
ev12
4-12
6of
the
Tan
gent
is K
now
n(4
-9)
M11
1-11
3
The
'stu
dent
sho
uld
not m
emor
ize
any
spec
ial f
orm
ulas
to d
o th
epr
oble
ms
in th
is s
ectio
n.In
stea
d, e
ach
prob
lem
sho
uld
beso
lved
usi
ng v
ecto
r m
etho
ds.
Para
met
ric
Equ
atio
ns o
f th
e C
ircl
e (4
-11)
MR
ev13
2-13
3
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
M11
8-12
0
MR
ev...
Pp.
123
, 124
:E
x. la
f, 3
af, 5
afM
.P.
110
: Ex.
laf,
3af
, 5a.
fM
Rev
... P
. 126
:E
x. la
, 2a,
3a
MP.
113
:E
x. la
, 2a,
3a
52R
evie
w
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
evPp
. 133
, 135
:E
x. 1
(a-
h in
clus
ive)
, 3, 5
, 11,
16
MPp
. 120
, 121
:E
x. 1
(a-
h in
clus
ive)
, 3,
5, 1
1, 1
6
53T
est
- 31
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E i
PAG
E
UN
IT S
EV
EN
The
Con
ics
54T
he C
onic
as
a Se
ctio
n of
a C
one
(5-1
)M
Rev
... 1
38-1
39G
ener
al D
efin
ition
of
a C
onic
(5-
2)M
.12
4-12
7
The
teac
her
shou
ld b
e aw
are
that
whe
n e
= 0
the
defi
nitio
n fa
ils.
But
whe
n e
appr
oach
es z
ero,
the
ellip
se a
ppro
ache
s a
circ
le a
sa
limit.
Hen
ce, a
cir
cle
is a
spe
cial
kin
d of
elli
pse.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
P. 1
41: E
x. 1
, 2, 3
, 4, 9
, 10,
12
M.
P. 1
27:
Ex.
1, 2
, 3, 4
, 9, 1
0, 1
255
The
Par
abol
aIv
Lie
v...
142-
145
Exp
lana
tion
of T
erm
s (5
-3)
M12
8-13
1
A G
eom
etri
c C
onst
ruct
ion
of th
e Pa
rabo
la (
5-4)
Sim
ple
Equ
atio
ns o
f th
e Pa
rabo
la (
5-5)
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
P. 1
46:
Ex.
4, 5
, 6, 7
a, 8
aM
P. 1
32:
Ex.
4, 5
, 6, 7
a, 8
a
56T
he L
atus
Rec
tum
(5-
6)/v
ia e
v.. .
146-
147
M13
2-13
3
- 32
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EPA
GE
56SU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
:(c
ont'd
)M
R e
v...
Pp. 1
47, 1
48:
Ex.
3, 4
M.
Pp. 1
33, 1
34:
Ex,
3, 4
57Pa
rabo
lic A
rch
(5-7
)M
Rev
148-
151
Para
met
ric
Equ
atio
ns o
f th
e Pa
rabo
la (
5-8)
M13
4-13
7
App
licat
ions
of
Para
bolic
Cur
ves
(5-9
)SU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
:
MR
ev...
P. 1
51:
Ex.
1, 2
, 3M
P. 1
37:
Ex.
1, 2
, 3
58Si
mpl
e E
quat
ions
of
the
Elli
pse
(5-1
0)iV
aR e
v.15
3-15
4
Com
pare
the
defi
nitio
n of
the
ellip
se w
ith th
at o
f th
e pa
rabo
la,
notin
g th
e va
lue
of "
e".
M13
9-14
0
59Si
mpl
e E
quat
ions
of
the
Elli
pse
(con
tinue
d)M
R e
v. .
. 154
-156
Dev
elop
the
sim
ple
equa
tion
of th
e el
lipse
.(T
he s
et o
f ca
lcul
uspr
ojec
tual
s co
ntai
ns a
tran
spar
ency
on
this
topi
c.H
owev
er, i
tis
not
bas
ed o
n th
e de
fini
tion
of th
e el
lipse
whi
ch is
use
d in
our
text
. )
M14
0-14
3
App
ly b
2=
a2
- c2
and
c =
ae
to th
e w
ritin
g of
sim
ple
equa
tions
of th
e el
lipse
.
- 33
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E 1
PAG
E
59SU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
:(c
ont'd
)M
Rev
... P
. 157
:E
x. 1
, 4, 5
ace
M.
Pp. 1
42, 1
43:
Ex.
1, 4
, 5ac
e60
Exp
lana
tion
of T
erm
s fo
r th
e E
llips
e (5
-11)
.MR
ev.
. .15
7-15
8M
143-
144
The
Foc
al R
adii
(5-1
2)M
R e
v.. .
159
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
M.
145
MR
ev...
Pp.
158
, 159
:E
x. 1
,3,
5, 7
, 9M
P. 1
44:
Ex.
1, 3
,5,
7, 9
MR
evP.
159
:E
x. 2
M.
P. 1
45:
Ex.
2
61D
iscu
ssio
n of
the
Equ
atio
n A
x2+
Cy2
+ F
= 0
whe
re A
and
C a
re1V
IRev
... 1
60-1
62of
Lik
e Si
gn a
nd a
re N
ot E
qual
to Z
ero
(5-1
3)M
.14
5-14
7
Cla
rify
the
mea
ning
of
"dis
cuss
ion"
of a
n eq
uatio
n.(R
efer
toE
x. 5
-5, M
Rev
... P
. 161
, M.
P 14
7. )
Def
ine
poin
t elli
pse
and
imag
inar
y el
lipse
.Sp
ecia
l not
e sh
ould
also
be
mad
e at
this
tim
e co
ncer
ning
the
circ
le a
s a
spec
ial
ellip
se.
(Ref
er to
MR
ev...
P. 1
61, M
P 14
6.)
The
Lat
us R
ectu
m o
f an
Elli
pse
(5-1
4)M
Rev
... 1
62-1
63
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
M14
8
IVIR
ev...
Pp.
163
, 164
:E
x. la
b, 2
ace,
3ac
e, 4
a, 5
M.
P. 1
49:
Ex.
lab,
2ac
e, 3
ace,
4a,
5
- 34
-
eso
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E1
PAG
E
62 63
Con
stru
ctio
n of
an
Elli
pse
(5-1
5)T
he C
onst
ruct
ion
in M
etho
d 1
is a
n A
pplic
atio
n of
5-1
2
Para
met
ric
Equ
atio
ns o
f th
e E
llips
e (5
-16)
The
se f
ollo
w d
irec
tly f
rom
con
stru
ctio
n m
etho
d 2.
App
licat
ions
of
the
Elli
pse
(5-1
7)SU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
:
MR
evPp
. 163
, 164
: Ex.
ld, 2
f, 3
h, 4
cM
.P.
149
:E
x. ld
, 2f,
3h,
4c
MR
ev.
P. 1
67: E
x 1
acM
P. 1
52:
Ex.
lac
Sim
ple
Equ
atio
n of
the
Hyp
erbo
la (
5-18
)U
se th
e de
fini
tion
of 5
-2 to
der
ive
the
sim
ple
equa
tion
of th
ehy
perb
ola.
Exp
lana
tion
of T
erm
s fo
r th
e H
yper
bola
(5-
19)
Stud
ents
sho
uld
beco
me
adep
t at t
he d
iscu
ssio
n of
the
sim
ple
hype
rbol
a.SU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
:
MR
ev...
P. 1
70:
Ex.
4M
.P.
154
:E
x. 4
MR
ev...
P. 1
75:
Ex.
lbe,
2ac
eiM
.P.
159
:E
x. lb
e, 2
acei
MR
ev...
165
-167
M15
0-15
1
ivIR
ev16
7M
152
MR
ev16
8M
152
MR
ev...
168
-170
M15
2-15
4
MR
ev...
171
-173
M15
5-15
7
- 35
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E I
PA
GE
64A
sym
ptot
es o
f H
yper
bola
(5-
20)
MR
ev17
3-17
5
The
def
initi
on o
f th
e as
ympt
ote
as g
iven
in th
e te
xt s
houl
d be
stre
ssed
in p
refe
renc
e to
suc
h in
form
al a
nd in
corr
ect p
hras
es a
sM
157-
159
"asy
mpt
ote
neve
r to
uche
s th
e cu
rves
.T
his
may
be
true
for
the
asym
ptot
es o
f a
hype
rbol
a; h
owev
er, i
t is
not t
rue
for
asym
ptot
esof
suc
h cu
rves
as
y =
ex
sin
x.SU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
:
MR
ev...
P. 1
75: E
x. 1
df,
2fl
m, 3
M.
P. 1
59:
Ex.
ldf,
2f1
m, 3
65T
he F
ocal
Rad
ii (5
-21)
MR
ev17
6-17
7
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
M16
0-16
1
MR
ev...
P. 1
78:
Ex.
1, 2
M.
P. 1
61:
Ex.
1, 2
66D
iscu
ssio
n of
the
Equ
atio
n A
x2
+ C
y2+
F =
0 w
here
A +
C +
0, a
ndIV
IRev
178-
179
A a
nd C
are
Opp
osite
in S
ign
(5-2
2)M
161-
163
The
Lat
us R
ectu
m o
f th
e H
yper
bola
MR
ev.
179-
130
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
M16
3-16
4
MR
ev...
P. 1
80:
Ex.
2, 3
MP
164:
Ex.
2, 3
67C
onju
gate
Hyp
erbo
las
(5-2
4)M
Rev
... 1
81M
... ,
164
Equ
ilate
ral H
yper
bola
s (5
-25)
MR
ev.
181
M16
4-16
5
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EI
PAG
E
67SU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
:(c
ont'd
)M
R e
v.. .
P. 1
81:
Ex.
labd
, 2ab
ceoq
txM
.P.
165
:E
x. la
bd, 2
abce
oqtx
68C
onst
ruct
ion
of th
e H
yper
bola
(5-
26)
MR
ev18
3-18
4M
166-
167
App
licat
ions
of
the
Hyp
erbo
la (
5-27
)M
Rev
184
M16
7
Para
met
ric
Equ
atio
ns o
f a
Hyp
erbo
la (
5-23
)M
R e
v. .
184
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
M16
7
MR
ev...
P. 1
85:
Ex.
lad,
tad
MP.
168
:E
x. la
d, ta
d69
Sum
mar
y an
d R
evie
w (
5-29
)iV
iRev
... 1
85-1
88
All
the
prob
lem
s in
the
revi
ew e
xerc
ises
are
goo
d ex
ampl
es o
fth
e co
ncep
ts o
f th
is u
nit.
The
ass
ignm
ent b
elow
rep
rese
nts
only
a m
inim
um.
M16
8-16
9
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
IviR
ev...
P. 1
86:
Ex.
labd
, 2ab
ceg,
4, 9
M.
P. 1
69:
Ex.
labd
, 2ab
ceg,
4, 9
70T
est
- 37
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EPA
GE
71In
trod
uctio
n
Tra
nsla
tion
ofT
he te
xt u
ses
tran
slat
ion
is u
sed
whe
nde
rive
d.fo
llow
ing
Y A
(6-1
)
the
Axe
s
a ve
ctor
of th
e ax
es.
the
form
ulas
For
cons
iste
ncy,
trad
ition
alyc
leri
vatio
n
UN
IT E
IGH
T
Tra
nsfo
rmat
ion
of th
e A
xes
(6-2
)pr
oof
to o
btai
n th
e fo
rmul
as f
or th
eH
owev
er, t
he tr
aditi
onal
sca
lar
proo
ffo
r th
e ro
tatio
n of
the
axes
are
the
teac
her
may
wis
h to
use
the
of th
e tr
ansl
atio
n fo
rmul
as.
P(x,
y)
iNxi
s IT
')
1
ivIR
ev...
192
M.
174
MR
ev...
192
-195
174-
176
M.
01(h
, k)
I-0
B(h
, 0)
A(x
, 0)
tv
- 38
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E I
PAG
E
71T
HE
OR
EM
: Let
x-y
be
a re
ctan
gula
r co
ordi
nate
sys
tem
and
let
(con
ed)
x'-y
' be
a se
cond
rec
tang
ular
coor
dina
te s
yste
m w
hich
is a
tran
slat
ion
of th
e fi
rst s
yste
m.
If th
e co
ordi
nate
s of
a p
oint
Pre
lativ
e to
the
x-y
syst
em a
re P
(x, y
) an
d th
e co
ordi
nate
sof
Pre
lativ
e to
the
x' -
y' s
yste
m a
re P
(xl,y
1), t
hen
x =
x' +
han
dy
= y
' +k
whe
re01
(h,k
) ar
e th
e co
ordi
nate
s of
the
cent
erof
the
x'-y
'sy
stem
with
res
pect
toth
e x-
y sy
stem
.
PRO
OF:
Pro
ject
P(x
, y)
onto
the
x-ax
is.
Let
A b
e th
e im
age
of P
in th
e x-
axis
whe
re A
has
coo
rdin
ates
(x,
0).
Sim
ilarl
y,th
e im
age
of 0
' in
the
x-ax
is is
B w
ith c
oord
inat
es (
h, 0
), b
yth
e de
fini
tion
of a
tran
slat
ion.
Thu
s,
OA
= O
B +
BA
-
x =
h +
xl
Or
X1
= X
- h
.
Sim
ilarl
y,y'
= y
- k
.Q
. E. D
.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
Pp.
194
, 195
:E
x. la
ce, 2
aeg,
3ac
deM
.Pp
. 176
,177
: Ex.
lace
, 2ae
g,3a
cde
- 39
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EI
PAG
E
72 73
Sim
plif
icat
ion
of th
e E
quat
ion
Ax2
+ C
y2+
Dx
+ E
y +
F =
0,
Whe
re B
oth
A a
nd C
are
Not
Zer
o (6
-3)
The
for
mul
as d
evel
oped
in th
is s
ectio
n sh
ould
not
be
mem
oriz
ed.
IVIc
lev
195-
200
M17
7-18
2
MR
ev20
0-20
5M
182-
187
The
stu
dent
sho
uld
lear
n th
e ge
nera
l pro
cedu
re a
nd a
pply
it in
each
exa
mpl
e. F
or in
stan
ce, t
o el
imin
ate
the
linea
r te
rms
in2
24x
- y
+ 2
4x +
2y
+36
=0
The
stu
dent
sho
uld
repl
ace
x by
x' +
h a
nd y
by
y' +
k a
ndde
term
ine
the
valu
es o
f h
and
k w
hich
tran
sfor
ms
the
equa
tion
to o
ne o
f th
e fo
rm
Ax'
z+
Cyl
z+
F =
0.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
P. 1
99: E
x. Z
, 4, 8
, 15
MP.
181
:E
x. -
2, 4
, 8, 1
5
The
Sta
ndar
d E
quat
ions
of
the
Con
ics
(6-4
)
The
teac
her
shou
ld d
evel
op in
cla
ss:
(1)
the
stan
dard
equ
atio
ns f
or th
e pa
rabo
la(2
)ei
ther
the
stan
dard
equ
atio
n of
the
ellip
se o
r th
est
anda
rd e
quat
ion
of th
e hy
perb
ola.
Em
phas
ize
that
the
poin
t C w
ith c
oord
inat
es (
h, k
) is
the
cent
er o
fth
e el
lipse
or
hype
rbol
a, b
ut th
at p
oint
C w
ith c
oord
inat
es (
h, k
) is
the
vert
ex o
f th
e pa
rabo
la in
sta
ndar
d fo
rm. T
he s
tude
nts
shou
ldbe
abl
e to
com
plet
e ch
arts
sim
ilar
to th
e fo
llow
ing
for
each
con
ic.
- 40
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E1
PAG
E
73(c
one
d)A
DIS
CU
SSIO
N O
F T
HE
HY
PER
BO
LA
IN
ST
AN
DA
RD
FO
RM
Equ
atio
nC
ente
rT
rans
vers
eA
xis
Ecc
entr
icity
Ver
tices
(x-h
)2(y
-k)2
c(h,
k)
=x
hc
e:a
or Nia
+b2
V1
(h+
a, k
)
(11-
a, k
)U
2
+a2
b2
= 1
e =
a
Axi
s of
Sym
met
ryE
nd P
oint
s of
Min
or A
xis
Foci
Dir
ectr
ices
y =
kB
(h,
k+
b)1
B (
h, k
-b)
2
F (h
+aE
, k)
1
F (h
-a, k
)2
ax
= h
+e a
x =
h -
E
Asy
mpt
otes
End
Poi
nts
of L
ater
a R
ecta
x-h
y-k
0L
I(k+
aE,k
+b2/
a)
L2,
(k
- a
E ,
k +
bz /a
)
+=
ab
- 41
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EPA
GE
73T
he s
tude
nt is
not
exp
ecte
d to
fill
out
all
colu
mns
for
the
gene
ral
(con
t'd)
equa
tion
form
of
the
hype
rbol
a. H
owev
er, i
n an
y gi
ven
prob
lem
,al
l ent
ries
sho
uld
be c
ompl
eted
for
the
disc
ussi
on.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
P. 2
05: E
x. 2
aceg
ikm
oqsu
MP.
187
:E
x. 2
aceg
ikm
oqsu
74R
otat
ion
of th
e A
xes
(6-5
)M
Rev
... 2
05-2
10
Stud
ents
fam
iliar
with
mat
rice
s m
ay w
ish
to w
rite
for
mul
asM
187-
192
(6-1
7) to
(6-
20)
as, Ix
1
[cos
0-s
ine
0rx
iI
1y
Isi
n 0
cos
0_.
,
i IY
I
Ref
erT
heto
app
endi
x in
the
text
.m
nem
onic
dev
ice xl
Yi
xco
s 0
-sin
0
Ysi
n 0
cos
0
is u
sefu
lin
rem
embe
ring
line
ar e
quat
ions
(6-1
5, 1
4, 1
5, 1
6)
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
1VIR
ev.
.. P
p. 2
09,2
10: E
x, la
cd, l
ace
M.,
..Pp
. 191
,192
:E
x. la
cd, l
ace
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E I
PA
GE
75 76
Sim
plif
icat
ion
of th
e E
quat
ion
Ax2
+ B
xy +
Cy2
+ D
x +
Ey
+ F
= 0
Whe
re B
+ 0
(6-
6)T
he te
ache
r sh
ould
dev
elop
equ
atio
n (6
-26)
ver
y ca
refu
lly in
cla
ss.
Mak
e su
re th
e st
uden
ts u
nder
stan
d th
e si
gn c
onve
ntio
n fo
r th
isfo
rmul
a th
orou
ghly
. Pro
ve in
cla
ss th
at c
hoos
ing
the
sign
s of
cos
26 a
nd c
tn 2
6 id
entic
ally
gua
rant
ees
that
the
xy-t
erm
can
alw
ays
be e
limin
ated
by
a ro
tatio
n th
roug
h a
posi
tive
acut
e an
glf.
..
MR
ev21
0-21
2M
.19
2-19
4
iviR
ev.,
212-
215
M19
4-19
7
The
stu
dent
sho
uld
not m
emor
ize
form
ula
(6-2
6), b
ut s
houl
dkn
owA
-C20
cot
=B
and
j(1
- co
s 26
)1
(1 +
cos
26)
9N
Isi
n 0
=an
d co
s=
.2
2
The
teac
her
shou
ld d
evel
op m
etho
ds f
or s
impl
ifyi
ng th
e ge
nera
lqu
adra
tic e
quat
ion
in tw
o va
riab
les
for
only
thos
e ca
ses
whi
chha
ve a
n xy
term
but
no
linea
r te
rm.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
P. 2
15:
Ex.
1, a
lso
elim
inat
e th
e xy
-ter
m a
nddi
scus
s th
e eq
uatio
n xy
= 1
MP.
197
:E
x. 1
, als
o el
imin
ate
xy-t
erm
and
dis
cuss
the
equa
tion
xy =
1
Sim
plif
icat
ion
of th
e E
quat
ion
Ax2
+ B
xy +
Cy
2+
Dx
+ E
y +
F -
-z 0
Whe
re B
4 0
(6-
6)(C
ontin
ued
from
Les
son
75)
- 43
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EI
PAG
E
76D
evel
op m
etho
ds f
or s
impl
ifyi
ng th
e ge
nera
l qua
drat
ic e
quat
ion
(con
t'd)
in tw
o va
riab
les
in w
hich
bot
h th
e xy
-ter
m a
nd li
near
term
sap
pear
.SU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
:
MR
ev.
P. 2
15:
Ex.
7M
P. 1
97:
Ex.
7
77In
vari
ants
of
the
Seco
nd D
egre
e E
quat
ion
(6-7
)M
Rev
... Z
16 -
218
M19
8-20
0
78T
he C
hara
cter
istic
4A
C -
B2
and
the
Dis
crim
inan
t A (
6-8)
MR
ev21
8-22
0M
.20
0-20
2SU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
:
MR
ev...
Pp.
221
,122
2:E
x. 1
, 3, 4
, 5, 6
, 9M
.P.
202
:E
x. 1
, 3, 4
,5,
6, 9
79R
evie
w
80T
est o
n C
hapt
er 6
- 44
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
Ei P
AG
E
UN
IT N
INE
Pola
r C
oord
inat
es
81Po
lar
Coo
rdin
ates
of
a Po
int (
7-2)
Mae
v22
8-23
0
Em
phas
ize
that
the
poin
t P(r
, 0)
also
has
the
coor
dina
tes
M20
9-21
1
P(r,
0 +
2k,
r) a
nd P
(-t,
0 +
(2k
+ 1
) w
), w
here
k is
inte
gral
.H
ence
, the
re d
oes
not e
xist
a 1
-1 c
orre
spon
denc
e be
twee
n th
ese
t of
poin
ts in
the
plan
e an
d th
e or
dere
d pa
irs
of r
eal
num
bers
thou
ght a
s po
lar
coor
dina
tes.
(Ref
er to
top
of p
age
230.
) T
his
lead
s to
str
ange
occ
urre
nces
(se
e Se
ctio
n 7-
12).
The
Rel
atio
n B
etw
een
Pola
r C
oord
inat
es a
nd th
e. C
arte
sian
MR
ev
230-
234
Coo
rdin
ates
(7-
3)M
211-
215
The
teac
her
shou
ld n
ote
care
fully
that
the
prob
lem
of
chan
ging
the
form
of
an e
quat
ion
from
pol
ar c
oord
inat
es to
rec
tang
ular
coor
dina
tes,
or
conv
erse
ly.
Thi
s is
equ
ival
ent t
o pr
ovin
g th
atth
e se
t of
poin
ts d
efin
ed b
y po
lar
coor
dina
tes
is e
qual
to th
e se
tof
poi
nts
defi
ned
by c
arte
sian
coo
rdin
ates
.Si
nce
A =
B if
and
only
if A
C B
and
BC
A, t
he s
tude
nt m
ust s
how
bot
h th
ere
ctan
gula
r fo
rm a
nd p
olar
for
m o
f an
equ
atio
n ca
n be
cha
nged
into
the
othe
r ex
actly
.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
P. 2
30: E
x. 2
abef
MP.
211
:E
x. 2
abef
MR
evP.
234
: Ex.
laeg
, 2ab
cgM
P. 2
15:
Ex.
laeg
, 2ab
cg
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
EPA
GE
82 83
The
Lin
e (7
-4)
Con
side
r on
ly th
e ca
ses
of th
e ho
rizo
ntal
line
, ver
tical
line
,an
d a
line
thro
ugh
the
pole
.T
here
is n
o ne
ed to
dis
cuss
the
gene
ral l
ine
in p
olar
coo
rdin
ates
.
The
Cir
cle
(7-5
)D
iscu
ss th
e fo
llow
ing
case
s:(1
)a
circ
le w
ith c
ente
r at
the
pole
and
rad
ius
r(2
)a
circ
le w
ith c
ente
r on
the
pola
r ax
is a
nd p
assi
ngth
roug
h th
e po
le(3
)a
circ
le w
ith c
ente
r on
the
90°
axis
and
pas
sing
thro
ugh
the
pole
.D
o no
t hol
d st
uden
ts r
espo
nsib
le f
or th
e ge
nera
l cas
e of
the
circ
le.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
evP.
236
:E
x. 1
, 2, 6
, 15
MP.
217
: Ex.
1, 2
, 6M
Rev
Pp. 2
38, 2
39:
Ex.
2ac
, 4ab
d, 5
abc
M.
Pp. 2
18, 2
19: E
x. 2
ac, 4
abd,
5ab
c
Inte
rcep
t Poi
nts
(7 -
6)
Sym
met
ry (
7-7)
Obs
erve
that
in r
ecta
ngul
ar c
oord
inat
es th
e gr
aph
of y
= f
(x)
issy
mm
etri
cal w
ith r
espe
ct to
the
y-ax
is if
and
onl
y if
f(-
x)-=
'f(x)
.
MR
ev...
234
-236
M.
215-
217
MR
ev...
236
-239
M21
7-21
9
1v1R
ev23
9M
219-
220
MR
ev23
9-24
3M
. .22
0-22
3,
How
ever
, in
pola
r co
ordi
nate
s, th
is c
ondi
tion
is n
ot s
uffi
cien
t.
- 46
-
LE
SSO
NSC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SR
EFE
RE
NC
EC
OD
E I
PAG
E
83(c
oned
)Fo
r ex
ampl
e, if
f(r
, 0)
+ f
(r, -
0 +
21m
), s
ymm
etry
with
res
pect
toth
e po
lar
axis
may
exi
st.
Furt
her
test
s ar
e ne
cess
ary.
We
mus
tch
eck
to s
ee w
heth
er o
r no
t f(r
, 0)
= f
(-r,
-0
+ [
2k
+ 1
17r
).If
this
is th
e ca
se, t
hen
the
grap
h of
f(r
, 0)
= 0
is in
deed
sym
met
rica
lw
ith r
espe
ct to
the
pola
r ax
is.
Thi
s is
one
of
the
cons
eque
nces
of
the
fact
that
ther
e do
es n
ot e
xist
a 1
-1 c
orre
spon
denc
e be
twee
nth
e po
ints
in th
e pl
ane
and
pair
s of
pol
ar c
oord
inat
es.
Sim
ilar
rem
arks
app
ly to
sym
met
ry w
ith r
espe
ct to
the
IzT
.-a
xis
and
with
resp
ect t
o th
e po
le.
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
MR
ev...
Pp.
247
, 248
: Dis
cuss
the
sym
met
ry o
f th
e gr
aphs
of th
e eq
uatio
ns in
exe
rcis
es la
dg(a
h)M
.Pp
. 227
,228
: Dis
cuss
the
sym
met
ry o
f th
e gr
aphs
of th
e eq
uatio
ns in
exe
rcis
es la
dg(a
h)
84E
xten
t (7-
8)M
Rev
.. 24
3M
223-
224
Sket
chin
g C
urve
s R
epre
sent
ing
p =
f(0
)(7
-9)
MR
ev24
4
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
-M
224
227
MR
ev...
P. 2
48:
Ex.
lace
MP.
228
:E
x. la
ce85
The
Con
ic (
7-10
)M
Rev
.. 24
9SU
GG
EST
ED
BA
SIC
ASS
IGN
ME
NT
:M
......
229
-231
MR
evP.
251
:E
x. la
, 2a
M. ,
, ,,
P. 2
31: E
x, la
, Za - 47
-
LE
SSO
NR
EFE
RE
NC
ESC
OPE
AN
D T
EA
CH
ING
SU
GG
EST
ION
SC
OD
Ei P
AG
E
86L
ocus
Pro
blem
s (7
-11)
Mile
y...
251
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
M23
2-23
3
MR
ev...
P. 2
53: E
x. 2
MP.
233
:E
x. 2
87In
ters
ectio
n of
Cur
ves
in P
olar
Coo
rdin
ates
(7-
12)
MR
ev...
253
SUG
GE
STE
D B
ASI
C A
SSIG
NM
EN
T:
M23
3-23
8
MR
ev...
P. 2
58:
Ex.
1, 3
, 5M
P. 2
39:
Ex.
1, 3
, 5
88R
evie
wM
Rev
... 2
59M
239
89T
est
- 48
-