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-4 -3 -2 -1 2 3 4 I -2 -1 2 3 J O ( d
ChingQuizz
:
6
exercises
a v ailable
for
Quizz
assessmen t
:
1.
Reminders :
affine
functions
E.9805
In
the
plane
pro vided
with
an
orthonormal
reference
frame
O
;
I
;
J
,
consider
the
t w o
straigh t
lines
(
d
3
)
and
(
d
4
)
admitting
the
follo wing
slop e- in tercept
forms
:
(
d
3
)
:
y
=
3
2
·
x
+
1
;
(
d
4
)
:
y
=
4
x
−
2
1
Are
the
straigh t
lines
(
d
3
)
and
(
d
4
)
parallel?
2
Determine
the
co ordinates
of
the
p oin t
of
in tersection
of
the
straigh t
lines
(
d
3
)
and
(
d
4
)
.
2.
Cartesian
e quation
of
lines
E.2111
Consider
the
line
(
d
)
with
equation
y
=3
x
−
2
and
the
follo wing
plane
p oin ts
:
A
(2
;
4 )
;
B
(
−
1
;
1 )
;
C
−
1
2
;
−
7
2
D
2
3
;
0
;
E
2
5
;
−
2
5
;
F
(
−
3
;
−
2)
1
Whic h
of
these
p oin ts
b elong
to
the
line
(
d
)
?
2
Whic h
of
these
p oin ts
v erify
the
follo wing
equation
:
y
−
3
x
+
2
=
0
3
What
observ ation
can
b e
made?
Wh y
or
wh y
not?
E.5318
In
the
plane
with
a
reference
frame
O
;
−→
i
;
−→
j
,
consider
the
straigh t
line
(
d
)
with
equation
:
2
·
x
−
y
+
5
=
0
1
Whic h
of
the
p oin ts
b elo w
b elong
to
the
line
(
d
)
:
A
(1
;
7 )
;
B
−
3
2
;
2
;
C
(
−
4
;
−
4)
Justify
y our
answ er.
2
Determine
the
co ordinates
of
the
p oin t
D
belonging
to
the
straigh t
line
(
d
)
having
abscissa
2
.
3
Determine
the
co ordinates
of
the
p oin t
E
belonging
to
the
straigh t
line
(
d
)
having
ordinate
−
1
2
.
E.7507
In
the
plane
with
a
reference
frame
O
;
−→
i
;
−→
j
,
consider
the
straigh t
line
(
d
)
with
equation
:
3
·
x
−
2
·
y
+
1
=
0
1
Whic h
of
the
p oin ts
b elo w
b elong
to
the
line
(
d
)
:
A
(3
;
5 )
;
B
−
1
2
;
−
1
8
;
C
−
2
3
;
−
1
2
Justify
y our
answ er.
2
Determine
the
co ordinates
of
the
p oin t
D
belonging
to
the
straigh t
line
(
d
)
having
abscissa
2
.
3
Determine
the
co ordinates
of
the
p oin t
E
belonging
to
the
straigh t
line
(
d
)
having
ordinate
−
3
.
E.1842
A
straigh t
line
(
d
)
passes
through
the
p oin ts
A
(
−
2.5
;
3 )
and
B
3
2
;
1
Among
the
three
Cartesian
equations,
sa y
whic h
one
corre- sp onds
to
the
line
(
d
)
:
a
2
x
+
2
y
−
1
=
0
b
−
4
x
−
3
y
+
9
=
0
c
2
x
+
4
y
−
7
=
0
E.9803
In
the
reference
frame
(
O
;
I
;
J
)
or-thonormal
b elo w,
consider
the
straigh t
line
(
d
)
shown
b elo w
:
Whic h
of
the
standard
forms
b elo w
represen ts
the
line
(
d
)
?
a
x
+
4
y
−
3
=
0
b
−
4
x
−
y
+
3
=
0
c
−
x
−
4
y
−
3
=
0
d
4
x
−
y
+
3
=
0
https:/ /c hingmath.fr
chapExoCorrec/9805 sacados/9805 chapExoCorrec/2111 sacados/2111 chapExoCorrec/5318 sacados/5318 chapExoCorrec/7507 sacados/7507 chapExoCorrec/1842 sacados/1842 chapExoCorrec/9803 sacados/9803 -4 -3 -2 -1 2 3 4 I -2 -1 2 3 J O ( d
-4 -3 -2 -1 2 3 4 I -2 -1 2 J O ( d 1 ( d 4 ( d 3 ( d 2 -4 -3 -2 -1 0 1 2 3 4 -2 -1 1 2 3 i j -4 -3 -2 -1 0 1 2 3 4 -1 1 2 3 i j
E.5334
In
the
plane
pro vided
with
a
O
;
−→
i
;
−→
j
,
w e
giv e
the
represen tation
of
the
four
straigh t
lines
(
d
1
)
,
(
d
2
)
,
(
d
3
)
and
(
d
4
)
belo w
:
Associate
one
of
the
standard
forms
sho wn
b elo w
with
eac h
of
the
straigh t
lines
b elo w
:
(
E
1
)
:
3
·
x
+
4
·
y
+
4
=
0
;
(
E
2
)
:
−
x
+
2
·
y
−
3
=
0
(
E
3
)
:
1
2
·
x
−
y
−
1
=
0
;
(
E
4
)
:
3
4
·
x
+
y
−
3
2
=
0
3.
Cartesian
Equations :
Constructing
the
Gr aph
E.5328
In
the
plane
pro vided
with
a
refer- ence
frame
O
;
−→
i
;
−→
j
,
consider
the
four
straigh t
lines
b elo w
defined
b y
their
standard
form
:
(
d
1
)
:
2
x
−
3
y
+
3
=
0
;
(
d
2
)
:
−
2
x
−
y
+
1
=
0
1
Determine
t w o
p oin ts
b elonging
to
eac h
of
these
t w o
straigh t
lines.
2
Dra w
eac h
of
these
straigh t
lines
in
the
reference
frame
b elo w
:
4.
Dire cting
V e ctors
E.5315
Consider
the
plane
pro vided
with
a
reference
frame
O
;
−→
i
;
−→
j
orthogonal
:
and
the
p oin ts
A
and
B
with
co ordinates
:
A
−
3
;
−
1
2
;
B
(1
;
1 )
1
Dra w
the
straigh t
line
(
AB
)
in
the
ab o v e
reference
frame.
2
Giv e
four
directing
v ectors
of
the
line
(
AB
)
at
least
one
of
whic h
has
in teger
co ordinates.
E.5319
In
the
plane
pro vided
with
a
refer- ence
frame
O
;
−→
i
;
−→
j
,
consider
the
follo wing
four
straigh t
lines
:
(
d
1
)
:
3
·
x
−
2
·
y
−
2
=
0
;
(
d
2
)
:
−
x
+
3
·
y
+
1
=
0
(
d
3
)
:
2
·
x
+
y
=
0
;
(
d
4
)
:
−
2
·
x
−
2
·
y
+
1
=
0
Give
a
directing
v ector
of
eac h
of
these
lines.
E.2112
In
the
plane
pro vided
with
a
refer- ence
frame,
w e
consider
the
standard
forms
of
the
follo wing
lines
:
(
d
1
)
:
5
x
+
y
−
2
=
0
;
(
d
2
)
:
3
x
−
y
+
5
=
0
For
eac h
of
the
lines,
giv e
an
asso ciated
direction
v ector.
https:/ /c hingmath.fr
chapExoCorrec/5334 sacados/5334 -4 -3 -2 -1 2 3 4 I -2 -1 2 J O ( d 1 ( d 4 ( d 3 ( d 2 chapExoCorrec/5328 sacados/5328 -4 -3 -2 -1 0 1 2 3 4 -2 -1 1 2 3 i j chapExoCorrec/5315 sacados/5315 -4 -3 -2 -1 0 1 2 3 4 -1 1 2 3 i j chapExoCorrec/5319 sacados/5319 chapExoCorrec/2112 sacados/2112
-4 -3 -2 -1 0 1 2 3 4 -1 1 2 3 i j -4 -3 -2 -1 0 1 2 3 4 -3 -2 -1 1 2 3 i j x x y y -4 -3 -2 -1 2 3 4 I -3 -2 -1 2 J O ( d A
E.9827
In
a
plane
with
a
reference
frame,
consider
the
standard
forms
of
the
follo wing
lines
:
(
d
1
)
:
5
x
+
2
y
+
1
=
0
;
(
d
2
)
:
2
x
+
1
2
y
−
5
=
0
For
eac h
of
the
straigh t
lines,
giv e
an
asso ciated
directing
v ector.
5.
Dire ction
ve ctors
and
line
se gments
E.9800
In
the
plane
pro vided
with
a
refer- ence
frame
O
;
−→
i
;
−→
j
,
consider
the
four
straigh t
lines
b elo w
defined
b y
their
standard
form
:
(
d
1
)
:
2
x
+
4
y
−
5
=
0
;
(
d
2
)
:
−
3
x
+
y
+
4
=
0
1
F or
eac h
of
the
straigh t
lines,
giv e
a
p oin t
and
a
directing
v ector
of
that
line.
2
Dra w
eac h
of
these
straigh t
lines
in
the
reference
frame
b elo w
:
E.11149
Dans
le
plan
m uni
d’un
rep ère
O
;
−→
i
;
−→
j
,
on
considère
les
quatre
droites
ci-dessous
définies
par
leur
équation
cartésienne
:
(
d
1
)
:
2
x
−
3
y
+
3
=
0
;
(
d
2
)
:
−
2
x
−
y
+
1
=
0
1
P our
c hacune
de
ces
droites,
déterminer
un
p oin t
de
cette
droite
et
un
co efficien t
directeur.
2
T racer
c hacune
de
ces
droites
dans
le
rep ère
ci-dessous
:
6.
Determine
a
Cartesian
e quation
E.5336
Consider
the
plane
pro vided
with
a
reference
frame
O
;
−→
i
;
−→
j
.
F or
eac h
question,
determine
a
standard
form
of
the
line
(
d
)
passing
through
the
p oin t
A
and
admitting
the
v ector
−→
u
as
directing
v ector :
a
A
(2
;
1 )
et
−→
u
(2
;
3 )
b
A
(0
;
3 )
et
−→
u
(
−
2
;
1 )
E.9836
Consider
the
plane
pro vided
with
a
(
O
;
−→
i
;
−→
j
)
orthonormal
co ordinate
system.
F or
eac h
of
the
questions,
determine
the
standard
form
of
the
line
passing
through
the
p oin t
M
and
ha ving
the
v ector
−→
u
as
its
directing
v ector :
a
M
(1
;
2 )
;
−→
u
(3
;
2 )
b
M
(
−
4
;
1 )
;
−→
u
(
−
2
;
1 )
E.9828
Consider
the
plane
pro vided
with
a
reference
frame
O
;
−→
i
;
−→
j
.
F or
eac h
question,
determine
a
standard
form
of
the
line
(
d
)
passing
through
the
p oin t
A
and
ha ving
director
v ector
−→
u
:
a
A
(3
;
−
2)
et
−→
u
1
2
;
−
1
b
A
−
2
;
1
2
et
−→
u
3
;
−
5
3
E.6508
In
the
plane
pro vided
with
a
refer- ence
frame
O
;
−→
i
;
−→
j
orthonormal,
consider
the
line
(
d
)
and
the
p oin t
A
shown
b elo w
:
1
Giv e
a
standard
form
of
the
line
(
d
)
.
2
Giv e
a
standard
form
of
the
line
(
d
)
passing
through
the
p oin t
A
and
parallel
to
the
line
(
d
)
.
https:/ /c hingmath.fr
chapExoCorrec/9827 sacados/9827 chapExoCorrec/9800 sacados/9800 -4 -3 -2 -1 0 1 2 3 4 -1 1 2 3 i j chapExoCorrec/11149 sacados/11149 -4 -3 -2 -1 0 1 2 3 4 -3 -2 -1 1 2 3 i j chapExoCorrec/5336 sacados/5336 chapExoCorrec/9836 sacados/9836 chapExoCorrec/9828 sacados/9828 chapExoCorrec/6508 sacados/6508 x x y y -4 -3 -2 -1 2 3 4 I -3 -2 -1 2 J O ( d A
-3 -2 -1 2 3 I -1 2 J O
E.5335
In
the
plane
pro vided
with
a
refer- ence
frame
O
;
−→
i
;
−→
j
,
consider
the
lines
b elo w
:
(
d
1
):
3
·
x
−
12
·
y
+
10
=
0
(
d
2
):
(1
+
2)
·
x
+
3
·
y
−
1
=
0
(
d
3
):
−
3
·
x
−
(
−
1
+
2)
·
y
+
2
=
0
(
d
4
):
(1
+
2)
·
x
+
(1
−
2)
·
y
−
1
=
0
1
Giv e
the
co ordinates
of
a
directing
v ector
of
the
line
(
d
1
)
having
its
in teger
co ordinates.
2
Giv e
the
co ordinates
of
a
directing
v ector
of
the
straigh t
lines
(
d
2
)
,
(
d
3
)
,
(
d
4
)
having
as
abscissa
an
in teger
v alue.
E.546
Consider
the
plane
equipp ed
with
an
orthonormal
co ordinate
system
(
O
;
−→
i
;
−→
j
)
.
F or
eac h
of
the
follo wing
questions,
determine
the
Cartesian
equation
of
the
line
passing
through
p oin t
M
and
ha ving
v ec- tor
−→
u
as
its
direction
v ector :
a
M
(0
;
2 )
;
−→
u
1
1
2
b
M
0
;
−
3
2
;
−→
u
2
1
7.
Linear
F unctions
and
Cartesian
Equations
E.9798
Consider
the
four
equations
b elo w,
eac h
represen ting
the
p oin ts
of
a
straigh t
line :
(
d
1
)
:
y
+
3
x
−
2
=
0
(
d
2
)
:
y
+
5
2
x
+
4
=
0
(
d
3
)
:
1
2
y
−
2
+
5
4
x
=
0
(
d
4
)
:
−
y
−
3
x
−
2
=
0
1
Giv e
the
slop e-in tercept
formrepresen ting
eac h
of
these
lines.
2
Eac h
of
these
t w o
equations
represen ts
one
of
the
straigh t
lines
presen ted
in
this
question.
Whic h
are
they?
(
E
)
:
3
y
+
9
x
−
6
=
0
;
(
F
)
:
4
y
+
10
x
+
16
=
0
E.9799
In
the
plane
pro vided
with
a
refer- ence
frame,
consider
the
line
(
d
)
admitting
as
standard
form
:
2
·
x
+
3
·
y
−
1
=
0
1
Justify
that
the
straigh t
line
(
d
)
is
the
represen tation
of
an
linear
function
2
Giv e
the
expression
of
the
linear
function
f
admitting
the
straigh t
line
(
d
)
as
its
represen tation.
8.
Affine
functions
and
dir e ction
ve ctors
E.5316
Consider
the
linear
functions
f
and
g
defined
b y
the
relation :
f
(
x
)
=
3
2
·
x
+2
;
g
(
x
)
=
−
2
x
+1
In
the
plane
pro vided
with
a
reference
frame,
note
(
d
)
and
(
d
)
the
resp ectiv e
represen tativ e
lines
of
the
functions
f
and
g
.
1
Giv e
three
directing
v ectors
of
the
line
(
d
)
.
2
Giv e
three
director
v ectors
of
the
line
(
d
)
.
E.9837
The
plane
is
pro vided
with
an
or- thonormal
co ordinate
system
(
O
;
−→
i
;
−→
j
)
:
Consider
the
line
(Δ)
whose
slop e-in tercept
formis
:
(Δ)
:
y
=
−
3
4
·
x
+
1
1
a
Dra w
the
straigh t
line
(Δ)
and
a
represen tativ e
of
the
v ector
−→
u
(4
;
−
3)
.
b
What
conjecture
can
b e
made
b et w een
the
straigh t
line
(Δ)
and
the
v ector
−→
u
.
2
Justify
that
the
v ector
−→
u
is
a
directing
v ector
of
the
line
(Δ)
.
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chapExoCorrec/5335 sacados/5335 chapExoCorrec/546 sacados/546 chapExoCorrec/9798 sacados/9798 chapExoCorrec/9799 sacados/9799 chapExoCorrec/5316 sacados/5316 chapExoCorrec/9837 sacados/9837 -3 -2 -1 2 3 I -1 2 J O
-4 -3 -2 -1 2 3 4 I -3 -2 -1 2 3 J O ( d 2 ( d 1 ( d 4 ( d 3 -3 -2 -1 2 3 4 I -1 2 J O
E.7415
Consider
the
straigh t
line
(
d
)
admit-ting
the
slop e-in tercept
form
y
=
1
2
·
x
+
3
Give
a
directing
v ector
of
the
line
(
d
)
.
E.2904
Asso ciate
eac h
of
the
follo wing
equa- tions
with
a
line :
1
y
=
2
x
+
1
2
y
=
−
3
2
x
−
2
3
−
2
x
−
y
+
3
=
0
4
y
=
2
3
x
+
1
5
y
=
1
6
x
−
1
2
6
−
x
+
3
y
−
2
=
0
one
direction
v ector
from
:
a
−→
u
(3
;
2 )
b
−→
v
(
−
2
;
−
4)
c
−→
w
(
−
2
;
4 )
d
−→
r
1
2
;
1
6
e
−→
s
(6
;
1 )
f
−→
t
(
−
4
;
6 )
E.5317
In
the
plane
pro vided
with
a
refer- ence
frame
O
;
−→
i
;
−→
j
,
consider
a
straigh t
line
(
d
)
of
direction
v ector
−→
u
.
F or
eac h
question,
giv e
the
slop e
of
(
d
)
:
a
−→
u
(1
;
3 )
b
−→
u
(2
;
−
3)
c
−→
u
5
2
;
1
E.541
In
the
plane
lab eled
O
;
I
;
J
,
con- sider
the
four
lines
sho wn
b elo w
:
Assign
a
direction
v ector
to
eac h
of
these
lines
from
among
the
v ectors
listed
b elo w
:
−→
u
−
2
6
;
−→
v
2
4
;
−→
w
4
1
;
−→
r
−
1
4
;
−→
s
2
−
2
E.552
W e
equip
the
plane
with
an
orthonor- mal
co ordinate
system
(
O
;
−→
i
;
−→
j
)
:
Consider
the
p oin ts
A
(
−
2
;
75
;
−
0
;
5)
,
B
(3
;
25
;
1 )
,
and
the
v ec- tor
−→
u
1
0
;
25
.
1
a
Dra w
the
line
(
d
)
and
the
v ector
−→
u
.
b
What
conjecture
can
w e
mak e
ab out
the
line
(
AB
)
and
the
v ector
−→
u
?
2
a
Determine
the
reduced
equation
of
the
line
(
AB
)
.
b
Justify
that
the
v ector
−→
u
is
a
direction
v ector
of
the
line
(
AB
)
.
9.
Systems
of
Equations :
Intr o duction
E.11734
Déterminer
le
prix
de
c hacun
des
deux
ob jets
:
E.11735
Déterminer
le
prix
de
c hacun
des
deux
ob jets
:
E.11736
Déterminer
le
prix
de
c hacun
des
deux
ob jets
:
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chapExoCorrec/7415 sacados/7415 chapExoCorrec/2904 sacados/2904 chapExoCorrec/5317 sacados/5317 chapExoCorrec/541 sacados/541 -4 -3 -2 -1 2 3 4 I -3 -2 -1 2 3 J O ( d 2 ( d 1 ( d 4 ( d 3 chapExoCorrec/552 sacados/552 -3 -2 -1 2 3 4 I -1 2 J O chapExoCorrec/11734 sacados/11734 chapExoCorrec/11735 sacados/11735 chapExoCorrec/11736 sacados/11736
-4 -3 -2 -1 2 3 4 I 2 3 4 J O
10.
System
of
e quations :
gr aphic
solution
E.2118
W e
pro vide
the
plane
with
a
ref- erence
frame
and
consider
the
t w o
lines
(
d
1
)
and
(
d
2
)
with
resp ectiv e
standard
forms
:
3
x
−
2
y
+
2
=
0
;
x
+
4
y
−
11
=
0
1
Justify
that
the
straigh t
lines
(
d
1
)
and
(
d
2
)
are
secan t.
2
a
Dra w
in
the
b enc hmark
the
t w o
straigh t
lines
b
Graphically
deduce
the
set
of
solutions
of
the
system
of
equation
(
S
)
:
3
x
−
2
y
+
2
=
0
x
+
4
y
−
11
=
0
11.
System
of
e quations :
solving
by
line ar
c ombination
E.7368
Consider
the
system
(
S
)
defined
b y :
x
−
3
y
=
8
4
x
+
y
=
−
7
Solve
the
system
(
S
)
.
E.5544
Consider
the
system
(
S
)
defined
b y :
4
x
−
2
y
=
6
3
x
+
2
y
=
29
Solve
the
system
(
S
)
.
E.5542
Consider
the
system
(
S
)
of
equa- tions
:
2
x
+
3
y
=
14
5
x
−
2
y
=
16
Determine
the
unique
solution
pair
of
the
system
(
S
)
.
E.5541
Consider
the
system
(
S
)
defined
:
3
x
+
2
y
=
13
2
x
+
3
y
=
17
Solve
the
system
(
S
)
.
E.9831
Consider
the
system
(
S
)
of
equa- tions
:
2
x
+
3
y
=
10
5
x
+
10
y
=
20
Solve
the
(
S
)
system
of
equations.
E.7369
Solv e
the
follo wing
system
:
x
+
2
y
−
z
=
−
2
3
x
+
y
+
2
z
=
−
1
x
−
y
+
3
z
=
3
(It
wil l
b e
shown
that
this
system
admits
a
single
triplet
solu- tion)
.
12.
System
of
e quations :
solving
by
substitution
E.5547
Consider
the
system
(
S
)
of
equa- tions
:
3
x
=
y
x
+
y
=
8.4
Solve
the
system
(
S
)
.
E.5548
Consider
the
follo wing
system
:
(
S
)
:
2
x
+
y
=
5
3
x
+
2
y
=
8
Solve
the
system
(
S
)
.
E.1007
Consider
the
system
(
S
)
of
equa- tions
:
3
x
+
2
y
=
23
x
−
y
=
1
Solve
the
system
of
equations
(
S
)
.
13.
Paral lel
lines
and
dir e ction
ve ctors
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chapExoCorrec/2118 sacados/2118 -4 -3 -2 -1 2 3 4 I 2 3 4 J O chapExoCorrec/7368 sacados/7368 chapExoCorrec/5544 sacados/5544 chapExoCorrec/5542 sacados/5542 chapExoCorrec/5541 sacados/5541 chapExoCorrec/9831 sacados/9831 chapExoCorrec/7369 sacados/7369 chapExoCorrec/5547 sacados/5547 chapExoCorrec/5548 sacados/5548 chapExoCorrec/1007 sacados/1007
-1 2 3 4 5 6 7 8 9 10 I -1 2 3 4 J O B A
E.9801
Proposition :
let
(
d
)
and
(
d
)
be
t w o
parallel
straigh t
lines.
If
at
least
one
p oin t
on
the
line
(
d
)
does
not
b elong
to
the
line
(
d
)
then
the
lines
(
d
)
and
(
d
)
are
distinctly
parallel.
Consider
the
t w o
straigh t
lines
(
d
5
)
and
(
d
6
)
defined
b y
the
follo wing
standard
forms
:
(
d
)
:
3
x
+
6
y
−
1
=
0
;
(
d
)
:
2
x
+
4
y
+
5
=
0
1
a
Giv e
the
co ordinates
of
a
v ector
−→
u
director
of
the
line
(
d
)
and
a
v ector
−→
v
director
of
the
line
(
d
)
.
b
Justify
that
the
straigh t
lines
(
d
)
and
(
d
)
are
parallel.
2
a
Giv e
the
co ordinates
of
the
p oin t
A
belonging
to
(
d
)
and
abscissa
1
.
b
Justify
that
the
straigh t
lines
(
d
)
and
(
d
)
are
parallel
and
distinct.
E.9834
Proposition :
let
(
d
)
and
(
d
)
be
t w o
parallel
straigh t
lines.
If
at
least
one
p oin t
of
the
straigh t
line
(
d
)
belongs
to
the
straigh t
line
(
d
)
then
the
straigh t
lines
(
d
)
and
(
d
)
are
par- allel
conflated.
Consider
the
t w o
straigh t
lines
(
d
)
and
(
d
)
defined
b y
the
follo wing
standard
forms
:
(
d
)
:
2
x
+
6
y
−
8
=
0
;
(
d
)
:
−
3
x
−
9
y
+
12
=
0
1
a
Giv e
the
co ordinates
of
a
v ector
−→
u
director
of
the
line
(
d
)
and
a
v ector
−→
v
director
of
the
line
(
d
)
.
b
Justify
that
the
straigh t
lines
(
d
)
and
(
d
)
are
parallel.
2
a
Giv e
the
co ordinates
of
the
p oin t
A
belonging
to
(
d
)
and
abscissa
1
.
b
Justify
that
the
straigh t
lines
(
d
)
and
(
d
)
are
parallel
and
distinct.
E.9835
Consider
the
plane
with
a
reference
frame
O
;
−→
i
;
−→
j
and
the
straigh t
lines
(
d
)
and
(
d
)
with
standard
forms
:
(
d
):
4
x
−
6
y
+
2
=
0
;
(
d
):
x
−
3
2
·
y
+
2
=
0
1
Justify
that
the
straigh t
lines
(
d
)
and
(
d
)
are
parallel.
2
Are
the
straigh t
lines
(
d
)
and
(
d
)
distinct
parallel
or
con- fused
parallel.
E.7506
In
the
plane
with
a
reference
frame
O
;
−→
i
;
−→
j
,
consider
the
p oin ts
A
and
B
with
co ordinates
:
A
22
5
;
14
5
;
B
42
5
;
4
5
Consider
also
the
straigh t
line
(
d
)
admitting
as
standard
form
:
(
d
)
:
3
·
x
+
6
·
y
−
12
=
0
1
a
Determine
the
co ordinates
of
the
v ector
−− →
AB
.
b
Justify
that
the
segmen t
[
AB
]
has
measure
20
.
2
Justify
that
the
straigh t
lines
(
AB
)
and
(
d
)
are
parallel.
3
a
Determine
the
p oin ts
C
and
D
intersection
of
the
line
(
d
)
with
the
x-axis
and
y-axis
resp ectiv ely .
b
Justify
that
the
quadrilateral
ABC D
is
a
rhom bus.
14.
Paral lel
lines
and
system
of
e quations
E.4735
Proposition :
in
the
plane
pro vided
with
a
reference
frame,
consider
t w o
straigh t
lines
(
d
)
and
(
d
)
with
standard
forms
:
a
·
x
+
b
·
y
+
c
=
0
;
a
·
x
+
b
·
y
+
c
=
0
If
the
system
a
·
x
+
b
·
y
+
c
=
0
a
·
x
+
b
·
y
+
c
=
0
n’admet
aucune
solu- tion
alors
les
droites
(
d
)
et
(
d
)
sont
parallèles
et
distinctes.
1
Solv e
the
system
(
S
)
:
6
x
−
15
y
+
24
=
0
−
4
x
+
10
y
+
16
=
0
2
In
the
plane
pro vided
with
a
reference
frame,
oConsider
the
t w o
straigh t
lines
(
d
)
and
(
d
)
of
standard
form
:
(
d
)
:
6
x
−
15
y
+
24
=
0
;
(
d
)
:
−
4
x
+
10
y
+
16
=
0
Justify
that
the
straigh t
lines
(
d
)
and
(
d
)
are
distinctly
parallel.
E.9802
Proposition :
in
the
plane
pro vided
with
a
reference
frame,
consider
t w o
straigh t
lines
(
d
)
and
(
d
)
with
standard
forms
:
a
·
x
+
b
·
y
+
c
=
0
;
a
·
x
+
b
·
y
+
c
=
0
If
the
system
a
·
x
+
b
·
y
+
c
=
0
a
·
x
+
b
·
y
+
c
=
0
admet
une
infinité
de
solutions
alors
les
droites
(
d
)
et
(
d
)
sont
parallèles
et
confondues.
1
Solv e
the
system
(
S
)
:
5
x
−
4
y
+
5
=
0
4
x
−
16
5
y
+
4
=
0
2
In
the
plane
pro vided
with
a
reference
frame,
consider
the
t w o
straigh t
lines
(
d
)
and
(
d
)
defined
b y
the
follo w- ing
standard
forms
:
(
d
)
:
5
x
−
4
y
+
5
=
0
;
(
d
)
:
4
x
−
16
5
y
+
4
=
0
Justify
that
the
straigh t
lines
(
d
)
and
(
d
)
are
parallel
and
coinciden t.
https:/ /c hingmath.fr
chapExoCorrec/9801 sacados/9801 chapExoCorrec/9834 sacados/9834 chapExoCorrec/9835 sacados/9835 chapExoCorrec/7506 sacados/7506 -1 2 3 4 5 6 7 8 9 10 I -1 2 3 4 J O B A chapExoCorrec/4735 sacados/4735 chapExoCorrec/9802 sacados/9802
-4 -3 -2 -1 2 3 4 I -2 -1 2 J O A B ( d x x y y -4 -3 -2 -1 2 3 4 I -2 -1 2 J O ( d
E.9829
1
Solv e
the
system
of
equations
:
6
x
−
3
y
+
9
=
0
−
4
x
+
2
y
−
6
=
0
2
In
the
plane
pro vided
with
a
reference
frame,
consider
the
t w o
straigh t
lines
(
d
)
and
(
d
)
defined
b y
the
stan- dard
forms
:
(
d
)
:
6
x
−
3
y
+
9
=
0
;
(
d
)
:
−
4
x
+
2
y
−
6
=
0
Specify
the
relativ e
p osition
of
the
straigh t
lines
(
d
)
and
(
d
)
.
E.9830
1
Solv e
the
system
(
S
)
:
2
x
+
6
y
−
7
=
0
−
3
x
−
9
y
+
12
=
0
2
In
the
plane
pro vided
with
a
reference
frame,
consider
the
t w o
straigh t
lines
(Δ)
and
(Δ
)
defined
b y
their
stan- dard
forms
:
(Δ)
:
2
x
+
6
y
−
7
=
0
;
(Δ
)
:
−
3
x
−
9
y
+
12
=
0
Specify
the
relativ e
p osition
of
the
straigh t
lines
(Δ)
and
(Δ
)
.
15.
Intersection
of
lines
with
Cartesian
e quations
E.5745
Consider
the
plane
pro vided
with
a
reference
frame
O
;
I
;
J
in
whic h
the
straigh t
line
(
d
)
passes
through
the
p oin ts
A
(
−
3
;
−
1)
and
B
(2
;
0 )
:
1
Determine
a
standard
form
of
the
line
(
d
)
.
Consider
the
straigh t
line
(Δ)
with
standard
form
:
x
+
2
y
−
3
=
0
2
a
Justify
that
the
straigh t
lines
(
d
)
and
(Δ)
are
secan t.
b
Dra w
the
straigh t
line
(Δ)
in
the
ab o v e
reference
frame.y
3
Determine
the
co ordinates
of
the
p oin t
of
in tersection
of
the
straigh t
lines
(
d
)
and
(Δ)
.
E.5823
In
the
plane
pro vided
with
a
ref- erence
frame
O
;
I
;
J
,
consider
the
straigh t
line
(
d
)
shown
b elo w
:
1
Determine
a
standard
form
of
the
line
(
d
)
.
Consider
the
line
(Δ)
with
standard
form
:
(Δ)
:
5
x
+
6
y
−
6
=
0
2
a
Justify
that
the
straigh t
lines
(
d
)
and
(Δ)
are
secan t.
b
Dra w
the
line
(Δ)
in
the
reference
frame
b elo w.
3
Algebraically ,
determine
the
co ordinates
of
the
p oin t
of
in tersection
of
the
lines
(
d
)
and
(Δ)
.
https:/ /c hingmath.fr
chapExoCorrec/9829 sacados/9829 chapExoCorrec/9830 sacados/9830 chapExoCorrec/5745 sacados/5745 -4 -3 -2 -1 2 3 4 I -2 -1 2 J O A B ( d chapExoCorrec/5823 sacados/5823 x x y y -4 -3 -2 -1 2 3 4 I -2 -1 2 J O ( d
-4 -3 -2 -1 2 3 4 I -1 2 3 4 J O
-8 -7 -6 -5 -4 -3 -2 -1 2 I -3 -2 -1 2 3 J O A B C
E.5395
Consider
the
plane
pro vided
with
a
reference
frame
O
;
I
;
J
and
the
t w o
straigh t
lines
(
d
1
)
and
(
d
2
)
admitting
as
standard
forms
:
(
d
1
)
:
x
−
2
y
+
3
=
0
;
(
d
2
)
:
3
x
+
4
y
−
13
=
0
1
Justify
that
the
straigh t
lines
(
d
1
)
and
(
d
2
)
are
secan t.
2
Represen t
in
the
graph
b elo w
the
t w o
straigh t
lines
(
d
1
)
and
(
d
2
)
.
3
Determine
the
co ordinates
of
the
p oin t
of
in tersection
of
the
t w o
straigh t
lines
(
d
1
)
and
(
d
2
)
.
E.2911
In
the
plane
pro vided
with
a
ref- erence
frame
O
;
I
;
J
,
consider
the
p oin ts
A
(
−
2
;
3 )
and
B
(1
;
−
1)
and
the
straigh t
line
(
d
)
whose
Cartesian
equation
is
giv en
b elo w
:
(
d
)
:
2
x
−
y
+
3
=
0
1
Justify
that
the
line
(
AB
)
admits
the
equation
b elo w
as
Cartesian
equation
:
4
x
+
3
y
−
1
=
0
2
Justify
that
the
straigh t
lines
(
AB
)
and
(
d
)
are
secan t.
3
a
Solv e
the
follo wing
system
of
equations
:
2
x
−
y
+
3
=
0
4
x
+
3
y
−
1
=
0
b
What
do es
the
co ordinate
p oin t
(
x
;
y
)
solution
of
the
previous
system
represen t,
graphically .
E.2916
In
a
plane
with
a
reference
frame,
consider
the
t w o
straigh t
lines
(
d
)
and
(
d
)
with
standard
forms
:
(
d
)
:
2
x
−
y
+
1
=
0
;
(
d
)
:
3
x
+
y
−
2
=
0
1
What
are
the
relativ e
p ositions
of
the
straigh t
lines
(
d
)
and
(
d
)
?
2
a
Solv e
the
system
:
2
x
−
y
+
1
=
0
3
x
+
y
−
2
=
0
b
In terpret
graphically
the
set
of
solutions
of
the
previ- ous
system.
16.
Relative
p osition
of
line
and
ge ometry
E.5338
In
the
plane
pro vided
with
a
refer- ence
frame
O
;
−→
i
;
−→
j
,
consider
the
four
p oin ts
A
,
B
,
C
,
D
with
co ordinates
:
A
(
−
2
;
1 )
;
B
(4
;
3 )
;
C
(3
;
−
4)
;
D
(
−
2
;
−
2)
1
a
Determine
a
standard
form
of
the
line
(
AC
)
.
b
Determine
a
standard
form
of
the
line
(
BD
)
.
2
Determine
the
co ordinates
of
the
p oin t
of
in tersection
of
the
diagonals
of
the
quadrilateral
ABC D
.
E.5396
In
the
plane
pro vided
with
a
refer- ence
frame
O
;
−→
i
;
−→
j
,
consider
the
follo wing
three
p oin ts
:
A
(
−
3
;
−
2)
;
B
(1
;
1 )
;
C
(
−
2
;
2 )
1
Determine
a
standard
form
of
the
line
(
AB
)
.
2
Determine
a
standard
form
of
the
line
(
d
)
passing
through
the
p oin t
C
and
parallel
to
the
line
(
AB
)
.
3
a
Determine
the
co ordinates
of
p oin t
M
midpoin t
of
segmen t
[
AC
]
.
b
Determine
a
standard
form
of
the
line
(
BM
)
c
Determine
the
co ordinates
of
the
p oin t
D
intersection
of
the
straigh t
lines
(
BM
)
and
(
d
)
.
d
What
is
the
nature
of
the
quadrilateral
ABC D
?
Jus- tify
y our
answ er.
https:/ /c hingmath.fr
chapExoCorrec/5395 sacados/5395 -4 -3 -2 -1 2 3 4 I -1 2 3 4 J O chapExoCorrec/2911 sacados/2911 chapExoCorrec/2916 sacados/2916 chapExoCorrec/5338 sacados/5338 chapExoCorrec/5396 sacados/5396 -8 -7 -6 -5 -4 -3 -2 -1 2 I -3 -2 -1 2 3 J O A B C
-4 -3 -2 -1 2 3 4 I -2 -1 2 3 4 J O
O B A − i − j M ¸ ¸
E.6663
Consider
the
plane
pro vided
with
a
reference
frame
O
;
I
;
J
orthonormal,
the
t w o
p oin ts
A
and
B
:
A
(
−
1
;
1 )
;
B
1
;
7
3
and
the
straigh t
line
(Δ)
admitting
as
standard
form
:
(Δ)
:
3
·
x
+
2
·
y
−
10
3
=
0
1
Determine
a
standard
form
of
the
line
(
AB
)
.
2
a
Justify
that
the
straigh t
lines
(
AB
)
and
(Δ)
are
se- can t.
b
Determine
the
co ordinates
of
the
p oin t
N
intersection
of
the
straigh t
lines
(
AB
)
and
(Δ)
.
3
a
Justify
that
the
p oin t
M
2
;
−
4
3
belongs
to
the
line
(Δ)
.
b
Justify
that
the
line
(Δ)
is
the
p erp endicular
bisector
of
the
segmen t
[
AB
]
.
The
b enc hmark
b elo w
is
giv en
as
a
guide
....
E.7505
Consider
the
billiard
table
sho wn
b elo w
où
the
white
ball
is
mo deled
b y
the
p oin t
A
and
w e
wish
to
determine
the
p osition
of
the
p oin t
M
of
con tact
on
the
left-hand
cushion
so
that
the
ball
joins
in
one
strip
the
hole
mo deled
b y
the
p oin t
B
.
Using
the
O
;
−→
i
;
−→
j
où
the
b ottom
and
left
bands
are
co- inciden t
with
the
abscissa
and
ordinate
axes
resp ectiv ely ,
w e
ha v e
the
co ordinates
:
A
(5
;
1 )
;
B
(8
;
4 )
We
note
M
the
con tact
p oin t
of
the
reb ound
on
the
left
band.
Assuming
p erfect
reb ound
on
the
strip,
w e
admit
that,
if
the
straigh t
line
(
AM
)
admits
the
v ector
−→
u
(1
;
a
)
,
où
a
∈
R
,
then
the
v ector
−→
v
(
−
1
;
a
)
is
a
director
v ector
of
the
line
(
MB
)
.
Determine
the
co ordinates
of
the
M
.
Any
trace
of
researc h
or
initiativ e,
ho w ev er
incomplete,
will
b e
tak en
in to
accoun t
in
the
assessmen t.
17.
Proje cte d
ortho gonal
E.4596
In
the
plane
pro vided
with
a
refer- ence
frame
O
;
I
;
J
orthonormal,
consider
the
t w o
p oin ts
:
A
(2
;
2 )
;
B
(0
;
−
1)
The
line
(
d
)
is
the
line
with
equation
:
y
=
1
2
·
x
−
1
Consider
a
p oin t
M
on
the
line
(
d
)
with
abscissa
x
.
W e
wish
to
determine
the
p osition
of
the
p oin t
M
so
that
the
distance
AM
is
minimal ;
w e
admit
that
this
p oin t
is
the
orthogonal
pro ject
of
the
p oin t
A
on
the
straigh t
line
(
d
)
.
1
a
Justify
that
the
p oin t
B
is
a
p oin t
on
the
straigh t
line
(
d
)
.
b
The
p oin t
M
having
abscissa
x
and
b elonging
to
the
line
(
d
)
,
giv e
the
co ordinates
of
the
p oin t
M
.
2
Sho w
that
the
length
of
segmen t
[
AM
]
is
:
AM
2
=
5
4
·
x
2
−
7
·
x
+
13
3
W e
no w
consider
the
p oin t
M
realizing
the
situation
:
ˇ
AM
is
minimal
ı
a
Sho w
that
the
abscissa
of
the
p oin t
M
verifies
the
equa- tion
:
5
2
·
x
2
−
7
·
x
=
0
b
Deduce
the
co ordinates
of
p oin t
M
.
18.
Unclassified
exer cises
E.5974
Consider
the
plane
pro vided
with
a
reference
frame
O
;
I
;
J
.
Let
A
and
B
be
the
p oin ts
with
coordinates
(
−
3
;
2 )
and
(3
;
0 )
respectiv ely
1
a
Giv e
the
co ordinates
of
a
v ector
−→
u
director
of
the
https:/ /c hingmath.fr
chapExoCorrec/6663 sacados/6663 -4 -3 -2 -1 2 3 4 I -2 -1 2 3 4 J O chapExoCorrec/7505 sacados/7505 O B A − i − j M ¸ ¸ chapExoCorrec/4596 sacados/4596 chapExoCorrec/5974 sacados/5974
line
(
AB
)
.
b
Determine
the
standard
form
of
the
line
(
AB
)
.
2
Consider
the
p oin t
C
(
−
1
;
−
2)
and
a
v ector
−→
v
with
co- ordinates
:
−→
v
(2
;
1 )
.
a
Justify
that
all
p oin ts
on
the
line
(
AB
)
hav e
co ordi- nates
x
;
−
1
3
·
x
+1
.
b
Determine
the
co ordinates
of
the
p oin t
D
belonging
to
the
line
(
AB
)
such
that
the
v ectors
−− →
CD
and
−→
v
are
collinear.
3
Consider
the
straigh t
line
(
d
)
admitting
the
follo wing
equation
as
standard
form
:
(
d
)
:
x
−
y
+
2
=
0
Determine
the
co ordinates
of
the
in tersection
p oin ts
of
the
lines
(
AB
)
and
(
d
)
.
E.4727
In
a
plane
with
an
orthonormal
co- ordinate
system
O
;
I
;
J
,
consider
the
follo wing
four
p oin ts.
A
(
−
2
;
3 )
;
B
(2
;
1 )
;
C
(2
;
−
4)
;
D
(5
;
2 )
The
aim
of
the
exercise
is
to
sho w
that
the
straigh t
lines
(
AB
)
and
(
CD
)
are
p erp endicular.
1
Justify
that
the
straigh t
lines
(
AB
)
and
(
CD
)
are
not
parallel.
2
Determine
the
co ordinates
of
the
p oin t
M
intersection
of
the
straigh t
lines
(
AB
)
and
(
CD
)
.
3
Determine
the
co ordinates
of
the
single
B
verifying
that
the
p oin t
M
is
the
midp oin t
of
the
segmen t
[
BB
]
.
4
Justify
that
the
straigh t
line
(
CD
)
is
the
p erp endicular
bisector
of
the
segmen t
[
BB
]
.
5
Conclude
on
the
relativ e
p osition
of
the
straigh t
lines
(
AB
)
and
(
CD
)
.
E.2271
Consider
the
plane
pro vided
with
a
refer- ence
frame
O
;
I
;
J
orthonormal
and
Consider
the
follo wing
four
p oin ts
in
the
plane
:
C
(3
;
2 )
;
D
(
−
1
;
1 )
;
E
2
;
−
5
2
;
F
0
;
11
2
Show
the
line
(
EF
)
is
the
p erp endicular
bisector
of
segmen t
[
CD
]
.
https:/ /c hingmath.fr
chapExoCorrec/4727 sacados/4727 chapExoCorrec/2271 sacados/2271