WISKUNDE |
MATHEMATICS |
GRAAD 12 |
GRADE 12 |
NOG OEFENINGE |
MORE EXERCISES |
Analitiese meetkunde toepassings : antwoorde. |
Applications of analytical geometry : answers. |
1.1
d(AB) = 13 : (xB − xA)2
+ (yB − yA)2 = 132
1.2
B is die punt (−17 ; 8) of B(7 ; 8)
(b − (−5))2
+ (8 − 3)2 = 169
B is the point (−17 ; 8) or B(7 ; 8)
b2 + 10b − − 119 = 0
P is die punt / the point (−5 ; 8).
(b + 17)(b − 7) = 0
b = − 17 of / or b = 7
yC − yA
3 − 7
−4
1
2.1
m(AC) = ────── = ────── = ── = − ──
m(AC) X m(DB) = −1 AC ⊥ DB
xC − xA
3 − (−5)
8
2
∴ m(DB) = 2
By / At D : y − yD = m (x − xD)
y − (−2) = 2 (x − (−2))
y = 2x + 2
2.2
AC en DB sny mekaar in E. / AC and DB
2.3
By / At B: xB + xE = 2 X xE
intersect at E.
xB + (−2) = 2 X 1
By / At A : y − yA = m (x − xA)
xB = 4
1
y − 7 = − ── (x − (−5))
yB + yD = 2 X yE
2
2y = − x + 9 (1)
yB + (−2) = 2 X 4
en / and y = 2x + 2 (2)
yB = 10
(2) X 2 : 2y = 4x + 4 (3)
B is die punt / the point (4 ; 10)
(3) − (1) 0 = 5x − 5
x = 1
In / Into (2) y = 2(1) + 2
y = 4
E is die punt / the point (1 ; 4)
┏
┓
yM − yC
3 − (−5)
8
┃
xA + xB
yA + yB
┃
3.2
m(CM) = ────── = ───── = ──
┃───── ; ─────┃
3.1
M(AB) =
xM − xC
−1 − 2
−3
┃
2
2
┃
┗
┛
┏
┓
By / At C : y − yC = m (x − xC)
┃
−5 + 3
2 + 4
┃
=
┃───── ; ─────┃
= (−1 ; 3)
┃
2
2
┃
8
┗
┛
y − (−5) = − ── (x − 2)
3
X 3 : 3y + 15 = − 8x + 16
8x + 3y = 1
yC − yA
−5 − 2
−7
3.3
m(AC) = ────── = ────── = ── = −1
xC − xA
2 − (−5)
7
BP ⊥ AC : m(BP) X m(AC) = −1
∴ m(BP) = 1
By / At B : y − yB = m (x − xB)
y − 4 = 1 (x − 3)
y = x + 1
4.1
M(BD) = (0 ; −1)
4.2
ABCD is 'n reghoek as die binnehoeke regte
xB + xD = 2 X xM en / and
yB + yD = 2 X yM
hoeke is. ABCD is a rectangle if the interior
−2 + p = 2 X 0 en / and
2 + q = 2 X −1
angles are right angles.
p = 2 en / and q = −4
∠BAD = 90° as / if m(BA) X m(AD) = −1
yA − yB
yD − yA
1 − 2
−4 − 1
D is die punt (2 ; −4) / D is the point (2 ; −4)
────── X ────── = ────── X ──────
xA − xB
xD − xA
−3 − (−2)
2 − (−3)
−1
−5
= ─── X ─── = −1
−1
5
∴ ∠BAD = 90° en / and
ABCD is 'n reghoek / a rectangle.
4.3
Area(ABCD) = lb = AB X BD
———————————
———————————
= √(xB −xA)2 + (yB −yA)2 X √(xD −xB)2 + (yD −yB)2
———————————
———————————
= √(−2 −(−3))2 + (2 −1)2 X √(2 − (−2))2 + (−4 −2)2
————
——————
= √(1 + 1) X √(16 + 36)
———
= 1 X √52
——
= √52
yC − yA
−3 − 1
−4
2
4.4
m(AC) = ────── = ───── = ── = − ──
xC − xA
3 − (−3)
−6
3
2
∴ tan θ = − ──
3
verw. hoek / ref. angle = 33,69006..°
θ = 180° − 33,69006..°
= 146,31°
———————————
5.1
┏
┓
d(BD) = √(xD −xB)2 + (yD −yB)2
┃
xA + xC
yA + yC
┃
5.2
M(AC) =
┃───── ; ─────┃
┃
2
2
┃
———————————
┗
┛
= √(−1 −3)2 + (−4 −8)2
┏
┓
——————
┃
−2 + 4
3 + 1
┃
= √(16 + 144)
=
┃───── ; ─────┃
┃
2
2
┃
┗
┛
———
——
= √160
= 4 √10
= (1 ; 2)
┏
┓
yD − yC
−4 − 1
−5
┃
xB + xD
yB + yD
┃
5.4
m(CD) = ────── = ───── = ── = 1
5.3
M(BD) =
┃───── ; ─────┃
xD − xC
−1 − 4
−5
┃
2
2
┃
┗
┛
By / At C (4;1) : y − yC = m (x − xC)
┏
┓
┃
3 + −1
8 + −4
┃
=
┃───── ; ─────┃
y − 1 = 1 (x − 4)
┃
2
2
┃
┗
┛
y = x − 3
= (1 ; 2)
M is die middelpunt van AC en BD sodat hulle
5.5
m(CD) = 1 sodat / so that tan α = 1
mekaar in M halveer.
α = 45°
M is die midpoint of both AC and BD so that
they bisect each other in M.
yD − yA
−4 − 3
−7
5.6
m(AD) = ────── = ───── = ── = −7
xD − xA
−1 − (−2)
1
yC − yA
yD − yB
m(AC) X m(BD) = ────── X ──────
xC − xA
xD − xB
tan θ = −7
1 − 3
−4 − 8
verw. hoek / ref. angle = 81,86989...°
= ────── X ──────
4 − (−2)
−1 − 3
θ = 180° − 81,86989...°
−2
−12
= ─── X ─── = −1
θ = 98,13°
6
−4
∠ADC = θ − α
∴ AC ⊥ BD en dus halveer hulle mekaar
reghoekig.
= 98,13° − 45°
∴ AC ⊥ BD and thus they bisect each other
= 53,13°
perpendicularly.
6.1
x2 + y2 = r2
6.2
x2 + y2 = r2
= 62
= 52
= 36
= 25
6.3
(x − a)2 + (y − b)2 = r2
6.4
(x − a)2 + (y − b)2 = r2
(x − 3)2 + (y − 2)2 = 72
(x − (−4)2 + (y − 6)2 = 92
x2 − 6x + 9 + y2 −4y + 4 = 49
(x + 4)2 + (y − 6)2 = 92
x2 − 6x + y2 − 4y − 36 = 0
x2 + 8x + 16 + y2 − 12y + 36 = 81
x2 + y2 + 8x − 12y = 29
6.5
(x − a)2 + (y − b)2 = r2
6.6
(x − a)2 + (y − b)2 = r2
(x − (−2))2 + (y − (−3))2 = 82
(x − 2)2 + (y − (−4))2 = 52
(x + 2)2 + (y + 3)2 = 82
(x − 2)2 + (y + 4)2 = 52
x2 + 4x + 4 + y2 + 6y + 9 − 64 = 0
x2 − 4x + 4 + y2 + 8y + 16 −25 = 0
x2 + y2 + 4x + 6y − 51 = 0
x2 + y2 − 4x + 8y − 5 = 0
6.7
(x − a)2 + (y − b)2 = r2
6.8
(x − a)2 + (y − b)2 = r2
(x − (−5))2 + (y − 6)2 = 72
(x − 7)2 + (y − (−4))2 = 42
(x + 5)2 + (y − 6)2 = 72
(x − 7)2 + (y + 4)2 = 42
x2 + 10x + 25 + y2 − 12y + 36 − 49 = 0
x2 − 14x + 49 + y2 + 8y + 16 −16 = 0
x2 + y2 + 10x − 12y + 12 = 0
x2 + y2 − 14x + 8y + 49 = 0
6.9
(x − a)2 + (y − b)2 = r2
6.10
(x − a)2 + (y − b)2 = r2
(x − (−8))2 + (y − (−6))2 = 112
(x − (−9))2 + (y − (−7))2 = 52
(x + 8)2 + (y + 6)2 = 112
(x + 9)2 + (y + 7)2 = 52
x2 + 16x + 64 + y2 + 12y + 36 = 121
x2 + 18x + 81 + y2 + 14y + 49 = 25
x2 + y2 + 16x + 12y = 21
x2 + y2 + 18x + 14y + 105 = 0
6.11
(x − a)2 + (y − b)2 = r2
6.12
(x − a)2 + (y − b)2 = r2
(x − 2,5)2 + (y − 3,5)2 = 42
(x − (−3,5))2 + (y − 8,5)2 = 122
x2 − 5x + 6,25 + y2 − 7y + 12,25 = 16
(x + 3,5)2 + (y − 8,5)2 = 122
x2 − 5x + y2 − 7y + 2,5 = 0
x2 + 7x + 12,25 + y2 − 17y + 72,25 = 144
x2 + y2 + 7x − 17y = 59,5
7.1
x2 + y2 − 4x − 6y − 3 = 0
−4
−3
x2 − 4x + (───)2 + y2 − 6y + (───)2 = 3 + (−2)2 + (−3)2
2
2
(x − 2)2 + (y − 3)2 = 16
Middelpunt / centre is (2 ; 3) en / and r = 4
7.2
x2 + y2 + 4x − 10y − 7 = 0
4
−10
x2 + 4x + (───)2 + y2 − 10y + (───)2 = 7 + (2)2 + (−5)2
2
2
(x + 2)2 + (y − 5)2 = 36
Middelpunt / centre is (−2 ; 5) en / and r = 6
7.3
x2 − 8x + y2 + 10y = 8
−8
10
x2 − 8x + (───)2 + y2 + 10y + (───)2 = 8 + (−4)2 + (5)2
2
2
(x − 4)2 + (y + 5)2 = 49
Middelpunt / centre is (4 ; −5) en / and r = 7
7.4
x2 + 6x + y2 + 10y = 66
6
10
x2 + 6x + (───) 2 + y2 + 10y + (───) 2 = 66 + (3) 2 + (5)2
2
2
(x + 3) 2 + (y + 5) 2 = 100
Middelpunt / centre is (−3 ; −5) en / and r = 10
7.5
x2 + y2 + 6x − 8y − 24 = 0
6
−8
x2 + 6x + (───) 2 + y2 − 8y + (───) 2 = 24 + (3) 2 + (−4) 2
2
2
(x + 3) 2 + (y − 4) 2 = 49
Middelpunt / centre is (−3 ; 4) en / and r = 7
7.6
x2 + y2 − 6x − 4y = 3
−6
−4
x2 − 6x + (───) 2 + y2 − 4y + (───) 2 = 3 + (−3)2 + (−2)2
2
2
(x − 3)2 + (y − 2)2 = 16
Middelpunt / centre is (3 ; 2) en / and r = 4
7.7
16x2 − 72x + 16y2 − 136y + 174 = 0
9
17
87
÷ 16 :
x2 − ── x + y2 − ── y = ──
2
2
8
9
−9
17
−17
87
x2 − ── x + (───) 2 + y2 − ── y + (───) 2 = − ── + (−2,25) 2 + (−4,25)2
2
4
2
4
8
(x − 2,25) 2 + (y − 4,25) 2 = 3,5 2
Middelpunt / centre is (2,25 ; 4,25) en / and r = 3,5
7.8
8x2 + 20x + 8y2 + 52y = 191
5
13
191
÷ 8 :
x2 + ── x + y2 + ── y = ───
2
2
8
5
5
13
13
191
x2 + ── x + (───) 2 + y2 + ── y + (───) 2 = ──── + (1,25) 2 + (3,25)2
2
4
2
4
8
(x + 1,25) 2 + (y + 3,25) 2 = 6 2
Middelpunt / centre is (−1,25 ; −3,25) en / and r = 6
7.9
x2 + y2 = 4y
−4
−4
x2 + y2 − 4y + (──) 2 = (──) 2
2
2
(x + 0) 2 + (y − 2) 2 = 2 2
Middelpunt is / centre at (0 ; 2) en / and r = 2
3
7.10
(x − ──) 2 + y2 = 0
2
9
x2 − 3x + (──) 2 + y2 = 0
4
−3
−3
x2 − 3x + (──) 2 + y2 = (──) 2
2
2
(x − 1,5) 2 + y2 = 1,5 2
Middelpunt is / centre at (1,5 ; 0) en / and r = 1,5
8.
Vergelyking is : x2 − 8x + y2 − 6y = 11
8.
Equation is : x2 − 8x + y2 − 6y = 11
By / At A : (−2)2 − 8(−2) + (p)2 − 6(p) = 11
By / At B : (q)2 − 8(q) + (−3)2 − 6(−3) = 11
4 + 16 + p2 − 6p −11 = 0
q2 − 8q + 9 + 18 −11 = 0
p2 − 6p + 9 = 0
q2 − 8q + 16 = 0
(p − 3) 2= 0
(q − 4) 2 = 0
p = 3
q = 4
A is die punt / the point (−2 ; 3)
B is die punt / the point (4 ; −3)
9.
Vergelyking is : x2 + y2 + 10x = 6y + 15
9.
Equation is : x2 + y2 + 10x = 6y + 15
By / At P : (2)2 + (p)2 + 10(2) − 6(p) − 15 = 0
By / At Q : (q)2 + (−4)2 + 10(q) − 6(−4) − 15 = 0
4 + p2 + 20 − 6p −15 = 0
q2 + 16 + 10q + 24 −15 = 0
p2 − 6p + 9 = 0
q2 + 10q + 25 = 0
(p − 3) 2= 0
(q + 5) 2 = 0
p = 3
q = −5
P is die punt / the point (2 ; 3)
Q is die punt / the point (−5 ; −4)
┏
┓
10.2
C en D het dieselfde x-koördinaat en dus is die
┃
xA + xB
yA + yB
┃
10.1
M(AB) =
┃───── ; ─────┃
┃
2
2
┃
afstand CD = die verskil in y-koördinate.
┗
┛
C and D have the same x-coordinates and thus
┏
┓
┃
−7 + 1
5 + 3
┃
=
┃───── ; ─────┃
the distance CD = the difference in y-coordinates.
┃
2
2
┃
┗
┛
CD = yC − yD = 4 − 0 = 4
C is die punt / is the point (−3 ; 4)
10.3
(x − a) 2 + (y − b) 2 = r2
yD − yB
3 − 0
3
10.4
m(BD) = ────── = ───── = ──
xD − xB
1 − (−3)
4
(x − (−3)) 2 + (y − 4) 2 = 42
3
x2 + 6x + 9 + y2 − 8y + 16 = 16
tan ∠BDX = ───
4
x2 + 6x + y2 − 8y + 9 = 0
∠BDX = 36,8698..°
∠BDX = ∠BAD ... raaklyn-koord / tanchord
∠BAD = 36,87°
11.1
−2 + x = 2 X 2 en/and −2 + y = 2 X −5
11.2
NM2 = (xM − xN) 2 + (yM − yN) 2
x = 4 + 2 en/and y = −10 + 2
= (2 − 2) 2 + (−5 − 0) 2
x = 6 en/and y = −8
= 0 + 25
L is die punt / is the point (6 ; −8)
NM = 5
r = 5
yK − yN
0 − (−2)
2
1
11.3
m(KN) = ────── = ───── = ── = ──
xK − xN
2 − (−2)
4
2
1
tan KNO = ───
2
∠KNO = 26,56505..°
∠KLN = ∠KNO ... raaklyn-koord / tan-chord
= 26,57°
—————————————
12.1
d(AB) = √(xB − xA) 2 + (yB − yA) 2
12.2
A : (x − a)2 + (y − b)2 = r2
—————————————
——
= √(−1 − (−2)) 2 + (−2 − 1) 2
(x − (−2))2 + (y − (−1))2 = (√10)2
————
= √(1 + 9)
x2 + 4x + 4 + y2 + 2y + 1 = 10
——
= √10
x2 + 4x + y2 + 2y = 5
yB − yA
−2 − 1
12.3
m(AB) = ────── = ───────
12.4
E (y = 0) : x2 + 4x + (0)2 + 2(0) − 5 = 0
xB − xA
−1 − (−2)
x2 + 4x − 5 = 0
−3
= ─── = −3
(x + 5)(x − 1) = 0
1
1
m(⊥) X m(AB) = −1 : m(⊥) = ───
x = −5 of / or x = 1
3
E is die punt / is the point (1 ; 0)
BD : y − yB = m(x −xB)
C (x = 0) : (0)2 + 4(0) + y2 + 2y − 5 = 0
1
y − (−2) = ── (x −(−1))
y2 + 2y − 5 = 0
3
—————————
− (2) ± √((2)2 − 4(1)(−5))
X 3 : 3y + 6 = x +1
y = ─────────────────
2(1)
y = 3,45 of / or y = −1,45
C is die punt / is the point (0 ; 3,45)
yQ − yP
9 − 6
3
13.1
m(PQ) = ────── = ────── = ──
13.2
By / At S y = 0 : 3x − 4(0) + 3 = 0
xQ − xP
11 − 7
4
y − yP = m(x − xP)
3x = −3
3
y − 6 = ── (x − 7)
x = −1
4
X 4 : y − 24 = 3x − 21
3x − 4y + 3 = 0
┏
┓
┃
xP + xS
yP + yS
┃
13.4
r2 = PR2 = (xR − xR) 2 + (yR − yR) 2
13.3
M(PS) =
┃───── ; ─────┃
┃
2
2
┃
= (3 − 7) 2 + (3 − 6) 2
┗
┛
┏
┓
= 16 + 9 = 25
┃
7 + (−1)
6 + 0
┃
=
┃───── ; ─────┃
┃
2
2
┃
(x − a) 2 + (y − b) 2 = r2
┗
┛
(x − 3) 2 + (y − 3) 2 = 25
= (3 ; 3)
x2 − 6x + 9 + y2 − 6y + 9 = 25
x2 + y2 = 6x + 6y + 7
13.5
m(⊥) X m(RP) = −1
13.6
Stel / Put y = 0 in x2 + y2 = 6x + 6y + 7
3
m(⊥) X ── = −1
x2 + (0)2 = 6x + 6(0) + 7
4
4
m(raaklyn / tangent) = − ──
x2 − 6x − 7 = 0
3
Raaklyn / Tangent : y − yP = m(x − xP)
x = 7 OF / OR x = −1
4
y − 6 = − ── (x − 7)
T is die punt / the point (7 ; 0)
3
X −3 : −3y + 18 = 4x − 28
4x + 3y = 46