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. .2542 ( 2) . .2545 10
28
( 2547)
(Acceleration Program)
5
6
-
( )
1 1.
2.
3.
4.
10
6
24
38
.
4
2 5.
6.
7.
8.
38
30
48
6
200
1 9.
10.
11. 2 3
12.
27
20
36
24
5
2 13.
14.
15.
15
38
40
.
200
6
1
16.
17.
18.
(12 )
(25 )
(13 )
30
20
50
100
1
1
1.1 3
1.1 6
2
8
1.2 10
1.2 12
3
14
1.3 16
1.3 18
4
20
1.4 22
1.4 24
5
28
1.5 30
1.5 34
6
38
1.6 40
1.6 42
7
44
1.7 46
1.7 50
6
10
( ) 91
1.10 94
1.10 105
11
( ) 115
1.11 118
1.11 134
8
( ) 55
1.8 57
1.8 65
9
( ) 70
1.9 72
1.9 81
⌦
11
⌦
1
4
2
1.
2.
1.
1.
2.
3.
2. ( )
3.
X P(x1, y1) Q(x2, y1) ( )
PQ = |x1 - x2|
Y P(x1, y1) Q(x1, y2) ( )
PQ = |y1 - y2|
X Y P(x1, y1) Q(x2, y2)
PQ = 2
212
21 yyxx )()(
4.
1.
2. 4 - 5
3. 1.1
4.
5. 1.1
⌫ ⌫ ⌦
⌫
225.
1. 1.1
2. 1.1
3.
4.
6.
1.
2.
3.
1.
2.
1. 1.1
2.
1. 95 %
2. 95 %
90 %
90 %
7.
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33
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P Q 2 P Q |PQ|
PQ P Q
1. X P(x1, y1) Q(x2, y1) ( )
PQ = |x1 - x2| PQ = |x2 - x1|
2. Y P(x1, y1) Q(x1, y2) ( )
PQ = |y1 - y2| PQ = |y2 - y1|
3. X Y P(x1, y1) Q(x2, y2)
PQR
PQ2 = |x1- x2|
2 + |y1- y2|
2
PQ = 2
212
21 |yy||xx|
PQ = 2
212
21 yyxx )()(
P(x1, y1) Q(x2, y2) 2
P Q PQ = 2
212
21 yyxx )()(
1 A(-3, 2) B(5, 2)
A(-3, 2) B(5, 2) X
AB = |x1 - x2|
= | -3 - 5 |
= 8
AB = 2
212
21 yyxx )()(
AB = 222253 )()(
= 64
= 8
A(-3, 2) B(5, 2) 8
X
A(-3, 2) B(5, 2)
Y
-3 O 5
O
|x1- x
2|
|y1- y2|
X
Y
P(x1, y1) R(x2, y1)
X
P(x1, y1)
O
Y
Q(x1, y2)
X
Y
P(x1, y1) Q(x2, y1)
O
ใบความรทูี ่1.1 (ความรเูบือ้งตนเกีย่วกบัเรขาคณติวเิคราะห)
⌫ ⌫ ⌦
⌫
44
2 P(-3, 9) Q(-3, 4)
P(-3, 9) Q(-3, 4) Y
PQ = |y1 - y2|
= | 9 - 4 |
= 5
PQ = 2
212
21 yyxx )()(
Q = 224933 )()(
= 25
= 5
P(-3, 9) Q(-3, 4) 5
3 A(0, 0) B(-4, 2)
AB = 2
212
21 yyxx )()(
AB = 222040 )()(
= 416
= 20
= 2 5
A(0, 0) B(-4, 2) 2 5
4 (-3, 6) (-9, -2)
P(-3, 6) Q(-9, -2)
PQ = 222693 )()(
= 2286
= 6436
= 100
= 10
(-3, 6) (-9, -2) 10
--
--
P(-3, 6)
X
8
-2R(-3, -2)
|6 + 2|
|-3 + 9|
-10 -5 O 1
Y
4
Q(-9, -2)
X
Y
-4 -2 O A(0, 0)
B(-4, 2) 2 --
9
X-3 O
Y
P(-3, 9)
Q(-3, 4) 4
⌦
55
⌦
a, b c a2 + b
2 = c
2 a
2 + c
2 = b
2
b2 + c
2 = a
2
5 (1, 1), (-1, -1) (-4, 2)
A(1, 1), B(-1, -1) C(-4, 2)
AB = 221111 )()( = 44 = 8
BC = 222141 )()( = 99 = 18
AC = 222141 )()( = 125 = 26
AB2 + BC
2 = ( 8 )
2 + ( 18 )
2 = 8 + 18 = 26 AC
2 = ( 26 )
2 = 26
AB2 + BC
2 = AC
2
(1, 1), (-1, -1) (-4, 2)
A, B C AB + BC = AC AB + AC = BC BC + AC = AB
AB + BC AC AB + AC BC BC + AC AB
6
(1) A(-2, 3), B(-6, 1) C(-10, -1)
AB = 221362 )()(
= 2224
= 416
= 20 = 52
BC = 2211106 )()(
= 2224
= 416
= 20 = 52
AC = 2213102 )()(
= 2248
= 1664
= 80 = 54
AB + BC = AC
A, B C
(2) A(5, 3), B(-2, 2) C(-1, 1)
AB = 222325 )()(
= 2217
= 149
= 50 = 25
BC = 221212 )()(
= 2211)(
= 11 = 2
AC = 221315 )()(
= 2226
= 436
= 40 = 102
AB + BC AC AB + AC BC
BC + AC AB
A, B C
B C B
c a b a a c
A b C A c B C b A
a2 + b
2 = c
2 a
2+ c
2= b
2 b
2+ c
2= a
2
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⌫
66
(1) A(0, 0) B(-3, 4)
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(2) A(1, -2) B(0, 0)
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(3) A(-5, 0) B(4, 0)
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(4) A(3, 2) B(3, 5)
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(5) A(-5, -2) B(2, -2)
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(6) P(-2, 4) Q(5, 7)
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(7) P(1, 2) Q(4, 6)
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(8) P(5, -4) Q(13, 2)
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(9) P(-4, 8) Q(7, 5)
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(10) P(-4, -5) Q(8, -3)
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2. A(2, 3), B(9, 2) C(5, 6)
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3. P(-1, -2), B(5, -2) C(2, 2)
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ใบงานที ่1.1
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77
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4. (2, 3) (5, 7)
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5. X P(5, 9) Q(-2, -4)
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6. A(-2, -2), B(4, -2), C(4, 4) D(-2, 4)
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7.
(1) A(1, 2), B(4, 3) C(7, 4)
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(2) P(-2, -8), Q(1, -3) R(5, 5)
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⌫
88 2
4
2
1.
1.
2.
2. ( )
3.
P( x , y ) P1(x1, y1) P2(x2, y2)
x = 2
xx 21 y = 2
yy 21
P( x , y ) = (2
xx 21 , 2
yy 21 )
4.
1.
2.
3. 4-5 1.2
4.
5. 1.2
⌦
99
⌦
5.
1. 1.2
2. 1.2
3.
4.
6.
1.
2.
3.
1.
2.
1. 1.2
2.
1. 95 %
2. 95 %
90 %
90 %
7.
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8.
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⌫ ⌫ ⌦
⌫
1010
2
P1(x1, y1) P2(x2, y2) P( x , y ) P1
P2
P2PQ P2P1R
12
2
PP
PP =
RP
PQ
1 =
RP
QP
2
2 = 2
1
RP
PQ
1 =
12
2
xx
xx = 2
1
2x2 - x = x2 - x1
x = 2
xx 21
RP
QP
2
2 = 12
2
yy
yy =
2
1
2y2 - y = y2 - y1
y = 2
yy 21
P( x , y ) P1(x1, y1) P2(x2, y2)
x = 2
xx 21 y = 2
yy 21
P( x , y ) = (2
xx 21 , 2
yy 21 )
1
(1) A(1, 2) B(3, 4)
P( x , y ) A B
x = 2
31 = 2
4 = 2
y = 2
42 = 2
6 = 3
P( x , y ) = (2, 3)
(2) A(2
1, 2) B(3, -1)
P( x , y ) A B
x = 2
32
1
= 22
7
= 4
7
y = 2
12 = 2
1
P( x , y ) = (4
7,2
1)
(3) A(-5, 7) B(1, -2)
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(4) A(3, 2
5) B(-3, -9)
……………………………………………………………….
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……………………………………………………………….
P2(x2, y2)
P1(x1, y1) R(x2, y1)
P( x , y ) Q(x2, y )
X
Y
O
ใบความรทูี ่1.2 (ความรเูบือ้งตนเกีย่วกบัเรขาคณติวเิคราะห)
⌦
1111
⌦
2 (-3, 8) (8, 3)
P( x , y ) (-3, 8) (8, 3)
x = 2
83 = 2
5
y = 2
38 =
2
11
(2
5, 2
11)
3 (3, 1) (5, -7)
P( x , y ) = P(3, 1) P1(x1, y1) = P1(5, -7)
P2(x2, y2)
P( x , y ) P1 P2
x = 2
xx 21
3 = 2
x5 2
6 = 5 + x2
x2 = 6 - 5 = 1
y = 2
yy 21
1 = 2
y7 2
2 = -7 + y2
y2 = 2 + 7 = 9
(1, 9)
4 A(-9, -6), B(3, -8) C(-1, 2)
D BC
D (2
13, 2
28) = (
2
2, 2
6) = (1, -3)
E CA
E (2
91, 2
62) = (
2
10, 2
4) = (-5, -2)
F AB
F (2
39, 2
86) = (
2
6, 2
14) = (-3, -7)
(1, -3), (-5, -2) (-3, -7)
XO
F(x3, y3)
C(-1, 2)
D(x1, y1)
E(x2, y2)
Y
⌫ ⌫ ⌦
⌫
1212
1.
(1) A(0, 0) B(6, 8)
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(2) A(1, 2) B(7, 6)
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(3) A(-3, 4) B(5, -6)
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(4) P(-3, -4) Q(6, -7)
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(5) P(-2, -3) Q(-2, 3)
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(6) P(2
1, 42
1) Q(-2
2
1, -72
1)
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2. C(2, 3) A(-2, 0) B(x, y) B(x, y)
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3. (-3, 2) (5, 8)
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4. (-1, 2) (-3, -2)
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ใบงานที ่1.2
⌦
1313
⌦
5. A(5, 6), B(-2, -2) C(4, 2) ABC E F AB
AC EF = 2
1BC
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6. D(1, 3), E(3, 6) F(-1, 5) AB, BC CA ABC
A, B C
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7. A(6, 5) B(2, 1) AB 4
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8. A(-3, 1), B(5, 7) C(1, 11) ABC
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⌫ ⌫ ⌦
⌫
1414 3
4
2
1.
1.
2.
2. ( )
Y
3.
L P1(x1, y1) P2(x2, y2) x1 x2 m
L
m = 21
21
xx
yy m =
12
12
xx
yy
(1) m = 0 X
(2) Y
(3) m > 0 X X
(4) m < 0 X X
4.
1.
2. 4 - 5 1.3
3.
4.
5. 1.3
⌦
1515
⌦
5.
1. 1.3
2. 1.3
3.
4.
6.
1.
2.
3.
1.
2.
1. 1.3
2.
1. 95 %
2. 95 %
90 %
90 %
7.
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8.
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⌫ ⌫ ⌦
⌫
1616
P(3, 2)
3 O 2 4
C(3, 2)
X
O 2
-5
2
L
Y
D(3, -5)
-2 O 2 4 6X
E(2, 3)
F(6, 5)L
Y
L P1(x1, y1) P2(x2, y2) x1 x2 m L
m = 21
21
xx
yy
L4 L2
1. m = 0 X ( L1 0)
2. Y ( L2 )
3. m > 0 X X ( L3 0)
4. m < 0 X X ( L4 0)
1
(1) A(2, 3) B(-4, 3)
AB mAB = 42
33
= 6
0 = 0
m = 0
(3) E(2, 3) F(6, 5)
EF mEF = 62
53
= 4
2 = 2
1
m > 0
(2) C(3, 2) D(3, -5)
CD mCD = 33
52
= 0
7
(4) P(3, 2) Q(-3, 6)
PQ mPQ = 33
62
= 6
4 =
3
2
m < 0
L3
X
m = 21
21
xx
yy =
12
12
xx
yy
Y
X
L B(-4, 3) A(2, 3)
-4 2 O 2
Y
O X
L1
6
3
L
Q(-3, 6)
ใบความรทูี ่1.3 (ความรเูบือ้งตนเกีย่วกบัเรขาคณติวเิคราะห)
⌦
1717
⌦
C ( -2, -6)
1
1. = 0
2. = 90
3. 0 < < 90
4. 90 < < 180
2 A(5, 6) B(9, k) 2
1 k
m = 21
21
xx
yy
2
1 =
95
k6
2
1 =
4
k6
2(6 - k) = -4
12 - 2k = -4
-2k = -16
k = 8
3 A(-6, -2), B(2, -2), C D AB
DC A 24 BC
A(-6, -2), B(2, -2) AB = | -6 - 2 | = 8
AB DC DC = 4
= 2
1( )
24 = 2
1 (AB + DC)
24 = 2
1 (8 + 4)
24 = 6
= 4
D(-6, 2) C(-2, 2)
D (-6, -6) C ( -2, -6)
mBC = 22
22 = 4
4 = -1
mBC = 22
62 = 4
4 = 1
O
B(2, -2)A(-6, -2)
D (-6, -6)
D(-6, 2) C(-2, 2)
Y
X
⌫ ⌫ ⌦
⌫
1818
1.
(1) A(0, 0) B(-3, 4)
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(2) A(0, 0) B(-6, -9)
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(3) A(-2, 3) B(-4, 8)
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(4) A(-2, -6) B(3, -1)
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(5) P(2, 5) Q(-7, 5)
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(6) P(-6, 4) Q(-6, -3)
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(7) P(3, -8) Q(-5, 7)
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(8) P(t + 1, s) Q(2t, s - 3)
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2. x P Q m
(1) P(6, -3) Q(9, x) ; m = 3
2
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(2) P(x, 12) Q(5, 12) ; m = 0
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3. C(3, -2) D(3x - 2, 4x) -3 x
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4. A(x, 5) B(-2, 8) 2
3 x
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ใบงานที ่1.3
⌦
1919
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5. P(-4, 3) Q(2, y) 2 y
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6. A(2, k) B(5, 6) C(-2, 1) D(1, 5) k
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7. A(2, 4), B(-5, 4) C(7, 8) P(a, b) A
BC
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8. A(-2, -2), B(4, -2), C(x, y) D(-2, 2) AB
CD BC
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⌫
2020 4
4
2
1.
2.
3.
1.
1.
2.
3.
4.
2. ( )
Y
Y
-1
3.
Y
Y
-1
⌦
2121
⌦
4.
1.
2. 4 - 5 1.4
3.
4. 1.4
5.
1. 1.4
2. 1.4
3.
4.
6.
1.
2.
3.
1.
2.
1. 1.4
2.
1. 95 %
2. 95 %
90 %
90 %
7.
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8.
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⌫ ⌫ ⌦
⌫
2222
Y
Y
L1 L2 m1 = m2
1 A(1, 2) B(4, 6) C(9, -6) D(6, -10)
mAB = 41
62 =
3
4 = 3
4
mCD = 69
106 = 3
4
mAB = mCD
AB CD
2 ABCD A(6, 8), B(5, 4), C(3, 6) D(4, 10)
mAB = 56
48 =
1
4 = 4
mBC = 35
64 =
2
2 = -1
mCD = 43
106 =
1
4 = 4
mDA = 64
810 =
2
2 = -1
mAB = mCD AB CD
mBC = mDA BC DA
ABCD
3 A(-2, 3), B(-6, 1) C(-10, -1)
mAB = 62
13 =
4
2 =
2
1
mBC = 106
11 =
4
2 =
2
1
mAB = mBC
A(-2, 3), B(-6, 1) C(-10, -1)
1
3 mAB mAC mAC mBC
4 A(5, 3), B(-2, 2) C(-1, 1)
mAB = 25
23 = 7
1
mAC = 15
13 =
6
2 = 3
1
mAB mAC
A(5, 3), B(-2, 2) C(-1, 1)
X
L1 L2
m1 m2
O
ใบความรทูี ่1.4 (ความรเูบือ้งตนเกีย่วกบัเรขาคณติวเิคราะห)
⌦
2323
⌦
Y
-1
L1 L2 m1 m2 = -1
5 A(1, 2) B(4, 6) C(-9, 6) D(-5, 3)
mAB = 41
62 =
3
4 = 3
4
mCD = 59
36 =
4
3
mAB mCD = (3
4)(4
3) = -1
AB CD
6 L1 2
3 L2 L2
L1 2
3 L1 L2
-1
L2 = m
m (2
3) = -1
m = (-1) (3
2) = 3
2
L2 3
2
7 A(k, 7) B(-3, -2) C(3, 2) D(1, -4) k
mAB = 3k
27 =
3k
9 mCD =
13
42 = 2
6 = 3
mAB mCD = -1
(3k
9)(3) = -1
k + 3 = -27
k = -30
8
(1) A(1, 1), B(-1, -1) C(-4, 2)
mAB = 11
11 =
2
2 = 1
mAC = 41
21 =
5
1
mBC = 41
21 = 3
3 = -1
mAB mBC = (1)(-1) = -1
ABC B̂ = 90
(2) A(2, 6), B(4, 1) C(-1, -2)
mAB = 42
16 =
2
5
mAC = 12
26 = 3
8
mBC = 14
21 = 5
3
-1
ABC
O
L2 L1
Y
m2 m1
X
⌫ ⌫ ⌦
⌫
2424
1. A(5, 3) B(2, -3) C(-5, -6) D(1, 6)
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2. A(4, 6) B(a - 2, -3) C(2, 4) D(5, -1) a
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3. P(1, 2) Q(6, 7) R(-3, 4)
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4. A(-2, 3), B(4, 5), C(2, 9) D(-1, 8)
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5. A(5, 6), B(-1, 2) C(7, 0) ABC D E AB
AC DE BC DE = 2
1BC
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ใบงานที ่1.4
⌦
2525
⌦
6. A(-3, -1), B(5, 3), C(3, 5) D(1, 7) ABCD P, Q, R S
AB, BC, CD DA PQRS
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7. 2
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8. ABCD AB CD BC AD D(x, y)
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OX
C(6, -3)
Y
B(3, 3)
D(x, y)
A (1, -1)
⌫ ⌫ ⌦
⌫
2626
9. A(-2, 3) B(4, -5)
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10. A(1, 3) B(6, 5) C(-1, -3) D(-3, 2)
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11. A(4, 1) B(1, 4) C(x, 5) D(-2, 6) x
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12. A(a, b) B(-b, a) a2 + b
2 0 OA OB
O
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13. A(-2, 1), B(5, 5) C(9, y) ABC CB̂A
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…………………………………………………………………………………………………………………………….sm.tm
⌦
2727
⌦
14. A(-1, 5), B(2, 1) C(6, 4)
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15. A(2, 1), B(6, 4), C(6, 8) D(-1, 5) ABCD
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16.
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17. A(5, 5) P(3, 5) Q(5, 3)
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18. L A(6, -3) B(4, 2)
L
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………………………………………………………………………………………………………………………….…sm.tm
⌫ ⌫ ⌦
⌫
2828 5
4
2
1.
2.
3.
1.
1. X
2. Y
3. X Y
4.
5. X Y
6.
2. ( )
X
Y
X Y
X Y
3.
X { (x, y) R R | y = b } y = b
Y Y (0, b)
Y { (x, y) R R | x = a } x = a
X X (a, 0)
X Y
m (x1, y1)
{(x, y) R R | y - y1 = m(x - x1) } y - y1 = m(x - x1)
2 (x1, y1) (x2, y2) 1
1
xx
yy =
12
12
xx
yy
m Y (Y - intercept) c ( Y (0, c)
y = mx + c
(General Form) Ax + By + C = 0 A, B,C A, B
(m) = B
A
⌦
2929
⌦
4.
1.
2. 4 - 5 1.5
3.
4. 1.5
5.
1. 1.5
2. 1.5
3.
4.
6.
1.
2.
3.
1.
2.
1. 1.5
2.
1. 95 %
2. 95 %
90 %
90 %
7.
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8.
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⌫ ⌫ ⌦
⌫
3030
1. X { (x, y) R R | y = b } y = b
Y Y (0, b)
1.1 b > 0 (b ) X X b
1.2 b < 0 (b ) X X | b |
1.3 b = 0 y = 0 X
1 y = 3 y = - 2
_
_
_
2. Y { (x, y) R R | x = a } x = a
X X (a, 0)
2.1 a > 0 (a ) Y Y a
2.2 a < 0 (a ) Y Y | a |
2.3 a = 0 x = 0 Y
2 x = 3 x = - 2
3. X Y
L X Y m (x1, y1)
(x, y) L m = 1
1
xx
yy
m (x1, y1) {(x, y) R R | y - y1 = m(x - x1) }
y - y1 = m(x - x1)
O
Y
X
(x1, y1)
L
(x, y)
O
(3, 0) X
Y
x = -2 x = 3
(-2,0)
OX
(0, -2)
y = 3
y = -2
(0, 3)
Y
ใบความรทูี ่1.5 (ความรเูบือ้งตนเกีย่วกบัเรขาคณติวเิคราะห)
⌦
3131
⌦
3 (5, -2) m = 3
2
x1 = 5 , y1 = -2 m = 3
2
y - y1 = m(x - x1)
y + 2 = 3
2(x - 5)
3y + 6 = 2x - 10
2x - 3y - 16 = 0
2x - 3y - 16 = 0
4. 2 (x1, y1) (x2, y2) (x, y)
1
1
xx
yy =
12
12
xx
yy
4 (3, 4) (-3, 5)
1
1
xx
yy =
12
12
xx
yy
3x
4y =
33
45
3x
4y =
6
1
6(y - 4) = -1(x - 3)
6y - 24 = -x + 3
x + 6y - 27 = 0
x + 6y - 27 = 0
5. m Y (Y - intercept) c ( Y (0, c)
y = mx + c
5 2x - 3y = 4 X Y
2x - 3y = 4 y = mx + c
3y = 2x - 4
y =3
2x - 3
4
X X y = 0
2x = 4
x = 2
3
2 X (2, 0) Y (0, -
3
4)
6. (General Form) Ax + By + C = 0 A, B,C A, B
(m) = B
A
6 2x - 3y = 4
2x - 3y = 4 2x - 3y - 4 = 0
A = 2 B = -3 m = B
A
m = 3
2 = 3
2
m = 21
21
xx
yy =
33
54 =
6
1
y - y1 = m(x - x1)
y - 4 =6
1(x - 3)
6y - 24 = -x + 3
x + 6y - 27 = 0
⌫ ⌫ ⌦
⌫
3232
7 4x + 3y + 5 = 0 3y = -4x + 7
1 y = mx + c
4x + 3y + 5 = 0
y = 3
4x - 3
5
4x + 3y + 5 = 0 3
4
3y = -4x + 7
y = 3
4x + 3
7
3y = -4x + 7 3
4
2 4x + 3y + 5 = 0 A = 4, B = 3
(m) = B
A
= 3
4
3x + 4y + 5 = 0 3
4
3y = -4x + 7
4x + 3y - 7 = 0 A = 4, B = 3
= 3
4
3y = -4x + 7 3
4
8 2x + y = 8 2y - x + 5 = 0
1 y = mx + c
2x + y = 8
y = -2x + 8
2x + y = 8 -2
2y - x + 5 = 0
y = 2
1x - 2
5
2y - x + 5 = 0 2
1
= (-2)( 2
1) = -1
2 2x + y = 8 2x + y - 8 = 0
A = 2, B = 1
(m) = B
A
= 1
2 = -2
2x + y = 8 -2
2y - x + 5 = 0 A = -1, B = 2
= 2
1 = 2
1
2y - x + 5 = 0 2
1
= (-2)( 2
1) = -1
9 (2, -3) (1, 2) (3, 5)
(1, 2) (3, 5) (m) = 13
25 = 2
3
(2, -3) (1, 2) (3, 5)
2
3
y - y1 = m(x - x1)
(x1, y1) = (2, -3) m = 2
3
y + 3 = 2
3 (x - 2)
2(y + 3) = 3(x - 2)
2y + 6 = 3x - 6
3x - 2y - 12 = 0
3x - 2y - 12 = 0
⌦
3333
⌦
10 (3, 4) (-4, 5) (1, 7)
(-4, 5) (1, 7) (m) = 41
57 = 5
2
(3, 4) (-4, 5) (1, 7)
2
5 ( (
5
2)(2
5) = -1)
y - y1 = m(x - x1)
(x1, y1) = (3, 4) m = 2
5
y - 4 = 2
5 (x - 3)
2(y - 4) = -5(x - 3)
2y - 8 = -5x + 15
5x + 2y - 23 = 0
5x + 2y - 23 = 0
11 ax + 5y = 4 5x + 3y + 9 = 0 a
5x + 3y + 9 = 0 y = 3
5x - 3
5x + 3y + 9 = 0 = 3
5
ax + 5y = 4 5x + 3y + 9 = 0
ax + 5y = 4 y = 5
ax + 4
5
a = 5
3 ( (
3
5)(5
3) = -1 )
a = (5
3)(-5)
= -3
a = -3
12 x + y - 5 = 0 x - y - 3 = 0
5x + 6y + 7 = 0
x + y - 5 = 0 x - y - 3 = 0
x = 4 y = 1
(4, 1)
(4, 1) 5x + 6y + 7 = 0 (m) = 6
5
y - y1 = m(x - x1)
(x1, y1) = (4, 1) m = 6
5
y - 1 = 6
5 (x - 4)
6(y - 1) = -5(x - 4)
6y - 6 = -5x + 20
5x + 6y - 26 = 0
5x + 6y - 26 = 0
⌫ ⌫ ⌦
⌫
3434
1.
(1) X X 5
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(2) Y Y 3
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(3) X (0, -1) 4
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(4) Y (2, 0) 5
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2. X Y
(1) x + 2y + 3 = 0
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=………………….. X ……………………… Y ………………………………..
(2) 4x - 3y - 9 = 0
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=………………….. X ……………………… Y ……………………………….
(3) 5y - x = 10
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=………………….. X ……………………… Y ……………………………….
(4) x = 5
4
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=………………….. X ……………………… Y ……………………………….
(5) 5
x + 3
y = 1
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=………………….. X ……………………… Y ……………………………….
ใบงานที ่1.5
⌦
3535
⌦
3. 2x - 4y + 5 = 0 6x - 8y - 5 = 0
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4. 5x + 2y - 3 = 0 4x - 10y + 5 = 0
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5. ax + 10y = 6 x + 2y = 8 a
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6. 3x + by = 5 5x + 3y - 10 = 0 b
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7. (1, 2) (4, 3)
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8. (-3, 4) (5, -6)
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⌫ ⌫ ⌦
⌫
3636
9. 3x - 2y - 4 = 0 6x - 4y = 8
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10. (2, -1) 4
3
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11. (-1, -2) 2x - 3y + 4 = 0
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12. (-1, -2) 2x - 3y + 4 = 0
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13. (1, 4) 5x + 3y - 4 = 0
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⌦
3737
⌦
14. (-4, -5) (1, 2) (6, 5)
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15. x + y = 5 x - y = 1 4x - 5y + 6 = 0
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16. x + 2y = 3 3y - x = 2 5x - 2y + 6 = 0
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17. A(-1, 9), B(-2, 2) C(5, 6) ABC
A
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⌫ ⌫ ⌦
⌫
3838 6
4
4
1.
2.
3.
1.
1.
2.
3.
4.
2. ( )
3.
Ax + By + C = 0 (x1, y1)
d = 22
11
BA
CByAx
Ax + By + C = 0 (0, 0)
d = 22 BA
CB(0)A(0)
= 22 BA
C
Ax + By + C1 = 0 Ax + By + C2 = 0
d = 22
21
BA
CC
Ax + By + C1 = 0 Ax + By + C2 = 0 C1 C2
C1 = C2
Ax + By + C1 = 0 Ax + By + C2 = 0
⌦
3939
⌦
4.
1.
2.
3.
4. 4 - 5 1.6
5.
6.
7.
8. 1.6
5. /
1. 1.6
2. 1.6
3.
4.
6.
1.
2.
3.
1.
2.
1. 1.6
2.
1. 95 %
2. 95 %
90 %
90 %
7.
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8.
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⌫ ⌫ ⌦
⌫
4040
Ax + By + C = 0 (x1, y1)
d = 2B2A
|C1By1Ax|
Ax + By + C = 0 (0, 0)
d = 2B2A
|CB(0)A(0)|
= 2B2A
|C|
Ax + By + C1 = 0 Ax + By + C2 = 0
d = 2B2A
|2C1C|
Ax + By + C1 = 0 Ax + By + C2 = 0
C1 C2 (C1 C2) C1 = C2
1 3x - 4y + 2 = 0 (2, -3)
3x - 4y + 2 = 0 (2, -3)
A = 3, B = -4, C = 2, x1 = 2 y1 = -3
d = 2B2A
|C1By1Ax|
d = 24)(23
|23)4)(((3)(2)|
= 169
|2126|
= 25
|20| = 5
20 = 4
3x - 4y + 2 = 0 (2, -3) 4
2 3x + 4y = 5 (0, 0)
3x + 4y = 5 3x + 4y - 5 = 0 (0, 0)
A = 3, B = 4, C = -5, x1 = 0 y1 = 0
d = 22 BA
|C|
d = 22 43
|5|
= 25
5
= 5
5 = 1
3x + 4y = 5 (0, 0) 1
ใบความรทูี ่1.6 (ความรเูบือ้งตนเกีย่วกบัเรขาคณติวเิคราะห)
⌦
4141
⌦
3 6x + 8y - 5 = 0 3x + 4y + 10 = 0
6x + 8y - 5 = 0 3x + 4y + 10 = 0 6x + 8y + 20 = 0
A = 6, B = 8, C1 = -5, C2 = 20
d = 22
21
BA
|CC|
d = 22 86
|205|
= 100
|25|
= 10
25 = 2
5
6x + 8y - 5 = 0 3x + 4y + 7 = 0 2
5
4 3x - 4y + 9 = 0 (2, -1) 3
3x - 4y + C = 0 C
3x - 4y + C = 0 (2, -1) 3
A = 3, B = -4, x1 = 2, y1 = -1
d = 22
11
BA
|CByAx|
3 = 22 4)(3
|C1)4)(((3)(2)|
3 = 5
|C10|
| 10 + C | = 15
10 + C = 15
C = 5, -25
3x - 4y + 5 = 0 3x - 4y - 25 = 0
5 4x + 3y - 5 = 0 (-2, 1) 1
4x + 3y - 5 = 0 3x - 4y + C = 0
3x - 4y + C = 0 (-2, 1) 1
A = 3, B = -4, x1 = -2, y1 = 1
d = 22
11
BA
|CByAx|
1 = 22 4)(3
|C4)(1)(2)(3)(|
1 = 5
|C10|
| -10 + C| = 5
-10 + C = 5
C = 15, 5
3x - 4y + 15 = 0 3x - 4y + 5 = 0
⌫ ⌫ ⌦
⌫
4242
1.
(1) 3x + 4y + 5 = 0 (3, 4)
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(2) 4x - 3y - 8 = 0 (1, 2)
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(3) 6x + 8y + 7 = 0 (1, -1)
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2.
(1) 3x + 4y - 5 = 0 3x + 4y - 10 = 0
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(2) 4x - 3y - 10 = 0 8x - 6y + 9 = 0
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(3) 12x + 5y = 17 y = x5
12- 5
9
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ใบงานที ่1.6
⌦
4343
⌦
3. 5x + 12y - 2 = 0 3
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4. 6x - 8y + 2 = 0 (-2, 5) 5
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5. 5x - 12y + 4 = 0 (3, 3) 1
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6. 6x + 8y - 2 = 0 10
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7. X (3, 0) Y (0, 5) 3
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⌫ ⌫ ⌦
⌫
4444 7
4
2
1.
2.
1.
1.
2.
3.
4.
2. ( )
(Translation of Axes)
( X
Y)
3.
1. (x, y) P (x , y ) P
h, k
x = x + h x = x - h
y = y + k y = y - k
2.
4.
1.
2. 1.7
3.
4.
5.
6. x = x - h y = y - k
(h, k)
7. 1.7
⌦
4545
⌦
5.
1. 1.7
2. 1.7
3.
4.
6.
1.
2.
3.
1.
2.
1. 1.7
2.
1. 95 %
2. 95 %
90 %
90 %
7.
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8.
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⌫ ⌫ ⌦
⌫
4646
(Translation of Axes)
( X Y)
X Y
P(x, y) Y x
X X y
P(x, y) P
X Y X Y
O (h, k) h k
(x, y) P
(x , y ) P h, k
x = x + h x = x - h
y = y + k y = y - k
1 (-2, 3) A(0, 2), B(-5, 4), C(4, -1) D(-3, -5)
(x, y)
(x , y )
(h, k) = (-2, 3) h = -2 k = 3
x = x - h y = y - k
x = x + 2 y = y - 3
(1) A(0, 2) x = 0 , y = 2
x = 0 + 2 = 2 y = 2 - 3 = -1
A(0, 2) (2, -1)
(2) B(-5, 4) x = -5 , y = 4
x = -5 + 2 = -3 y = 4 - 3 = 1
B(-5, 4) (-3, 1)
sm.tm
A
B
C
D
O
(-2, 3)
X
X
Y Y
x
x
P(x, y)
P (x , y )
X
y
X
O
Y
y
O (h, k)
h k
X
X
O
y
P(x, y)
Y
ใบความรทูี ่1.7 (ภาคตดักรวย)
⌦
4747
⌦
(3) C(4, -1) x = 4 y = -1
x = 4 + 2 = 6 y = -1 - 3 = -4
C(4, -1) (6, -4)
(4) D(-3, -5) x = -3 y = -5
x = -3 + 2 = -1 y = -5 - 3 = -8
D(-3, -5) (-1, -8)
2 (3, -4) P(-4, 3), Q(-5, -2) R(2, 7)
(x, y) (x , y )
(h, k) = (3, -4) h = 3 k = -4
x = x + h y = y + k x = x + 3 y = y - 4
(1) P(-4, 3) x = -4 y = 3
x = -4 + 3 = -1 y = 3 - 4 = -1
P(-4, 3) (-1, -1)
(2) Q(-5, -2) x = -5 y = -2
x = -5 + 3 = -2 y = -2 - 4 = -6
Q(-5, -2) (-2, -6)
(3) R(2, 7) x = 2 y = 7
x = 2 + 3 = 5 y = 7 - 4 = 3
R(2, 7) (5, 3)
" (h, k) " " (h, k)"
3 (-3, 4) y = | x + 3 | + 4
(x , y ) (x, y)
(-3, 4) (h, k) = (-3, 4) h = -3, k = 4
x = x + h y = y + k x = x - 3 y = y + 4
y = | x + 3 | + 4 x x - 3 y y + 4
y + 4 = | x - 3 + 3 | + 4
y + 4 - 4 = | x - 3 + 3 |
y = | x |
4 (3, -7) x2 - 6x + y
2 + 14y - 2 = 0
(x , y ) (x, y)
(3, -7) (h, k) = (3, -7) h = 3, k = -7
x = x + h y = y + k x = x + 3 y = y - 7
x2 - 6x + y
2 + 14y - 2 = 0 x x + 3 y y - 7
(x + 3)2 - 6(x + 3) + (y - 7)
2 + 14(y - 7) - 2 = 0
(x )2 + 6x + 9 - 6x - 18 + (y )
2 - 14 y + 49 + 14y - 98 - 2 = 0
(x )2 + (y )
2 = 60
(x )2 + (y )
2 = 60
sm.tm
Y Y
X
XO
(3, -4)
P
Q
R
⌫ ⌫ ⌦
⌫
4848
5
(1) 2x - 3y + 12 = 0 2x = 3y
2x - 3y + 12 = 0
2x = 3y - 12
2x = 3(y - 4) ..............................(1)
x = x y = y - 4
x x y - 4 y (1)
2x = 3y (0, 4)
2x - 3y + 12 = 0 2x + 6 = 3y - 6
2(x + 3) = 3(y - 2) ...............................(2)
x = x + 3 y = y - 2
x + 3 x y - 2 y (2)
2x = 3y (-3, 2)
2x - 3y + 12 = 0
(2) y(x - 5) = 3 y x = 3
y(x - 5) = 3 ...........................................(1)
x = x - 5 y = y
x - 5 x y y (1)
y x = 3 (5, 0)
(3) x2 + y
2 - 8x + 6y + 24 = 0 (x )
2 + (y )
2 = 1
x2 + y
2 - 8x + 6y + 24 = 0
(x2 - 8x + 16) + (y
2 + 6y + 9) = -24 + 16 + 9
(x - 4)2 + (y + 3)
2 = 1 ...............................(1)
x = x - 4 y = y + 3
x - 4 x y + 3 y (1)
(x )2 + (y )
2 = 1 (4, -3)
(4) 25x2 - 9y
2 + 50x + 36y = 236 1
25
y
9
x22
)()(
25x2 - 9y
2 + 50x + 36y = 236
(25x2 + 50x) - (9y
2 - 36y) = 236
25(x2 + 2x) - 9(y
2 - 4y ) = 236
25(x2 + 2x + 1) - 9(y
2 - 4y + 4) = 236 + 25 -36
25(x + 1)2 - 9(y - 2)
2 = 225
125
2y
9
1x22
)()(.............................................(1)
x = x + 1 y = y - 2
x + 1 x y - 2 y (1)
125
y
9
x22
)()( (-1, 2)
sm.tm
⌦
4949
⌦
X
(h, k)
(0, 0) P(x, y) P(x , y ) x = x - h
y = y - k
6
(1) y = | x + 3 | + 4
y = | x + 3 | + 4
y - 4 = | x + 3 |
(-3, 4)
y = | x |
(2) y = (x - 5)2
y = (x - 5)2
(5, 0)
y = (x )2
(3) y = (x - 5)2 + 3
y = (x - 5)2 + 3
y - 3 = (x - 5)2
(5, 3)
y = (x )2
(4) y = x3 + 3x
2 + 3x + 3
y = x3 + 3x
2 + 3x + 3
y = (x3 + 3x
2 + 3x + 1) + 2
y - 2 = x3 + 3x
2 + 3x + 1
y - 2 = (x + 1)3
(-1, 2)
y = (x )3
(5) y = x3 + 3x
2 + 3x
y = x3 + 3x
2 + 3x
y + 1 = x3 + 3x
2 + 3x + 1
y + 1 = (x + 1)3
(-1, -1)
y = (x )3
sm.tm
X
XO
(-3, 4)
O X
X
Y Y
(-1, -1)
O
X
X
(-1, 2)
Y Y
Y Y
O
X
X
(5, 3)
Y Y
O
Y Y
(5, 0) X
⌫ ⌫ ⌦
⌫
5050
1. (5, -4) A(5, -10) , B(-7, 3), C(0, 0), D(-3, -2) E(9, 0)
(x, y) (x , y )
(h, k) = ............ h = ........ k = ............
x = x - h y = y - k x = ................... y = ..................
(1) A(5, -10) x = 5 y = -10
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(2) B(-7, 3) x = ………. y = …………
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(3) C(0, 0) x = ………. y = …………
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(4) D(-3, -2) x = ………. y = …………
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(5) E(9, 0) x = ………. y = …………
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2. (4, 3) P(-9, 3), Q(-7, -8), R(5, -1), S(10, 8) T(-4, -3)
(x, y) (x , y )
(h, k) = .............. h = ............. k = ...............
x = x + h y = y + k x = ............... y = ..............
(1) P(-9, 3) x = -9 y = 3
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(2) Q(-7, -8) x =………. y = ………..
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(3) R(5, -1) x =………. y = ………..
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(4) S(10, 8) x =………. y = ………..
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(5) T(-4, -3) x =………. y = ………..
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.............................................................................................................................................................................................sm.tm
ใบงานที ่1.7
⌦
5151
⌦
3. (2, 7) y = | x - 2 | + 7 (x , y )
(x, y)
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4. (-4, 0) x2 + 8x - y + 16 = 0 (x , y )
(x, y)
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5. (-3, 2) x2 + 6x + 4y
2 - 16y + 21 = 0
(x , y ) (x, y)
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6. (5, -4) 2x2 - 20x - 3y
2 - 24y - 16 = 0
(x , y ) (x, y)
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⌫ ⌫ ⌦
⌫
5252
7.
(1) 5x - 3y - 7 = 0 5x = 3y
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(2) y2 - 4y + 2x - 6 = 0 (y )
2 + 2x = 0
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(3) x2 + y
2 - 6x + 4y - 3 = 0 (x )
2 + (y )
2 = 16
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(4) 4x2 + 9y
2 - 8x + 18y = 23 1
4
)y(
9
)x( 22
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⌦
5353
⌦
8.
(1) | y - 4 | = | x + 5 |
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(2) | x - 3 | + | y + 1 | = 2
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(3) x (y + 3) = 5
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(4) (x - 4)(y - 7) = -6
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…………………………………………………………………………..sm.tm
⌫ ⌫ ⌦
⌫
5454
9.
(1) y = x2 + 6x + 12
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(2) x = y2 - 8y - 20
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(3) y = x3 - 3x
2 + 3x + 5
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(4) x = y3 + 3y
2 + 3y + 3
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⌦
5555
⌦
8
( ) 4
7
1.
2.
1.
1.
2.
3. (0,0)
4. (h, k)
2. ( )
(Conic Section) (circle) (parabola)
(ellipse) (hyperbola)
3.
1. (0, 0) r x2 + y
2 = r
2
2. (h, k) r (x - h)2 + (y - k)
2 = r
2
3. x2 + y
2 + Dx + Ey + F = 0
(2
E
2
D, ) F4ED
2
1 22
4.
1.
2. 1.8
3.
4. 1.8
5. (0, 0)
r
6. (0, 0)
⌫ ⌫ ⌦
⌫
5656
7. (h, k)
r
8. (h, k)
9. 1.8
5.
1. 1.8
2. 1.8
3.
4.
5.
6.
6.
1.
2.
3.
1.
2.
1. 1.8
2.
1. 95 %
2. 95 %
90 %
90 %
7.
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8.
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⌦
5757
⌦
(Conic Section) (circle) (parabola) (ellipse)
(hyperbola)
1.
2.
3.
4.
sm.tm
ใบความรทูี ่1.8 (ภาคตดักรวย(วงกลม))
⌫ ⌫ ⌦
⌫
5858
(Circle)
1 C(0,0) r x2 + y
2 = r
2
P(x, y)
CP = r
CP = 22 )0y()0x( = 22 yx
22 yx = r
x2 + y
2 = r
2
(0, 0) r
{ (x, y) R R | x2 + y
2 = r
2 }
1 (0, 0) 5
{ (x, y) R R | x2 + y
2 = 25 }
2 (0, 0) 7
{ (x, y) R R | x2 + y
2 = 7 }
2 C(h, k) r (x - h)2 + (y - k)
2 = r
2
1 P(x, y)
CP = r
CP = 22)ky()hx(
22)ky()hx( = r
(x - h)2 + (y - k)
2 = r
2
sm.tm
X
r
Y
P(x, y)
C(h, k)
O
X
Y
P(x, y)
r
C (0, 0)
⌦
5959
⌦
⌫ ⌫ ⌦
⌫
6060
6 (2, -1) (10, 5)
(2, -1) (10, 5)
)2
51,
2
102( = (6, 2)
(6, 2) (2, -1) r = 22)12()26( = 5
(x - 6)2 + (y - 2)
2 = 5
2
x2 - 12x + 36 + y
2 - 4y + 4 = 25
x2 + y
2 - 12x - 4y + 15 = 0
{ (x, y) | x2 + y
2- 12x - 4y + 15 = 0 }
7 3 Y (0, 2)
3 Y (0, 2)
2 (3, 2)
(-3, 2)
(3, 2) 3
(x - 3)2 + (y - 2)
2 = 3
2
x2 - 6x + 9 + y
2 - 4y + 4 = 9
x2 + y
2 - 6x - 4y + 4 = 0
(-3, 2) 3
(x + 3)2 + (y - 2)
2 = 3
2
x2 + 6x + 9 + y
2 - 4y + 4 = 9
x2 + y
2 + 6x - 4y + 4 = 0
{ (x, y) | x2 + y
2 - 6x - 4y + 4 = 0 x
2 + y
2 + 6x - 4y + 4 = 0 }
8 (7, -6) 3x - 4y - 20 = 0
(7, -6) 3x - 4y - 20 = 0
r = 22)4(3
|20)6)(4()7(3|
= 5
25 = 5
(x - 7)2 + (y + 6)
2 = 5
2
x2 - 14x + 49 + y
2 + 12y + 36 = 25
x2 + y
2 - 14x + 12y + 60 = 0
{ (x, y) | x2 + y
2 - 14x + 12y + 60 = 0 }
9 C(-5, -1) y = 3
C(-5, -1) y = 3
C(-5, -1) y = 3
r = | -1 - 3 | = 4
(x + 5)2 + (y + 1)
2 = 4
2
x2 + 10x + 25 + y
2 + 2y + 1 = 16
x2 + y
2 + 10x + 2y + 10 = 0
{ (x, y) | x2 + y
2+ 10x + 2y + 10 = 0 }
sm.tm
y = 3
C(-5, -1) X
O
Y
O
X
Y
3x - 4y - 20 = 0
(7, -6)
O X
Y
(-3, 2) (0, 2) (3, 2)
⌦
6161
⌦
10 A(5, 3) B(6, 2)
x + y = 5
C(h, k) x + y = 5
h + k = 5 .................................(1)
A(5, 3) B(6, 2)
AC = BC ( )
22)3k()5h( = 22
)2k()6h(
(h - 5)2 + (k - 3)
2 = (h - 6)
2 + (k - 2)
2
h2 - 10h + 25 + k
2 - 6k + 9 = h
2 - 12h + 36 + k
2 - 4k + 4
h - k = 3 ...............................(2)
(1) + (2) 2h = 8 h = 4
h 4 (1) 4 + k = 5 k = 1
C(4, 1) = AC = 22)31()54( = 5
22)1y()4x( = ( 5 )
2
x2 - 8x + 16 + y
2 - 2y + 1 = 5
x2 + y
2 - 8x - 2y + 12 = 0
{ (x, y) | x2 + y
2 - 8x - 2y + 12 = 0 }
11 (-1, -3) (-2, 4) (2, 1)
(-2, 4) (2, 1) y - 4 = 22
14(x + 2)
y - 4 = 4
3 (x + 2) 3x + 4y - 10 = 0
(-1, -3) 3x + 4y - 10 = 0
r = 2243
|10)3(4)1(3| = 5
25 = 5
(x + 1)2 + (y + 3)
2 = 5
2 x
2 + y
2 + 2x + 6y - 15 = 0
12 (5, 3), (6, 2) (3, -1)
x2 + y
2 + Dx + Ey + F = 0
(5, 3) 52 + 3
2 + 5D + 3E + F = 0
5D + 3E + F = -34 .........................(1)
(6, 2) 62 + 2
2 + 6D + 2E + F = 0
6D + 2E + F = -40 .........................(2)
(3, -1) 32 + (-1)
2 + 3D - E + F = 0
3D - E + F = -10 ........................(3)
(2) - (1) D - E = -6 ........................(4)
(2) - (3) 3D + 3E = -30 D + E = -10 ........................(5)
(4) + (5) 2D = -16 D = -8
D -8 (5) -8 + E = -10 E = -2
D -8 E -2 (3) 3(-8) - (-2) + F = -10 F = 12
x2 + y
2 - 8x - 2y + 12 = 0
sm.tm
B(6, 2)
A(5, 3)
x + y = 5
C(h, k)
⌫ ⌫ ⌦
⌫
6262
(h, k) r (x - h)2 + (y - k)
2 = r
2
(x - h)2 + (y - k)
2 = r
2
x2 + y
2 - 2hx - 2ky + h
2 + k
2 - r
2 = 0 h, k r
-2h = D, -2k = E h2 + k
2 - r
2 = F D, E F
x2 + y
2 + Dx + Ey + F = 0
-2h = D h = 2
D -2k = E k = 2
E
(h, k) = (2
D , 2
E )
h2 + k
2 - r
2 = F r
2 = h
2 + k
2 - F
r2 = (
2
D )2+ (
2
E )2 - F ( h k )
r2 =
4
D2
+ 4
E2
- F
r2 =
4
1 (D2 + E
2 - 4F)
r = 2
1 F4ED22
x2 + y
2 + Dx + Ey + F = 0 (
2
D , 2
E )
r = 2
1 F4ED22 r = F)
2
E()
2
D(
22
D2 + E
2 - 4F
(1) D2 + E
2 - 4F 0 r 0 (
2
D , 2
E )
2
1 F4ED 22
(2) D2 + E
2 - 4F = 0 r = 0 x
2 + y
2 - 2x + 4y + 5 = 0
D = -2, E = 4 F = 5 (-2)2 + 4
2 - 4(5) = 0 r = 0
(3) D2 + E
2 - 4F 0 r x
2 + y
2 + 2x + 4y + 10 = 0
D = 2, E = 4 F = 10 22 + 4
2 - 4(10) = -20
13 x2 + y
2 - 6x + 4y - 3 = 0
1 x2 + y
2 - 6x + 4y - 3 = 0 D = -6, E = 4 F = -3
(2
D , 2
E ) = (2
6 , 2
4 ) = (3, -2)
r = F4ED2
1 22 = F4ED2
1 22 = )3(44)6(2
1 22
= )1216362
1 = 642
1 = 2
1 (8) = 4
(3, -2) 4
2 x2 + y
2 - 6x + 4y - 3 = 0 (x - h)
2 + (y - k)
2 = r
2
(x2 - 6x) + (y
2 + 4y) = 3
(x2 - 6x + 9) + (y
2 + 4y + 4) = 3 + 9 + 4
(x - 3)2 + (y + 2)
2 = 16
(x - 3)2 + (y + 2)
2 = 4
2
(3, -2) 4
sm.tm
r = F)2
E()
2
D(
22
r = F)2
E()
2
D(
22
(2
D)2 = (2
D )2 (
2
E)2 = (
2
E)2
(-1, -2) C(3,-2) (7, -2)
X X
(3, -6)
(3,2)
Y Y
O
⌦
6363
⌦
14
(1) x2 + y
2 + 10x + 24y + 69 = 0
1 x2 + y
2 + 10x + 24y + 69 = 0 D = 10, E = 24 F = 69
(2
D , 2
E ) = (2
10 , 2
24 ) = (-5, -12)
r = F)2
E()
2
D( 22
= 69)12()5( 22
= 6914425
= 100
= 10
(-5, -12) 10
2 x2 + y
2 + 10x + 24y + 69 = 0 (x - h)
2 + (y - k)
2 = r
2
(x2 +10x) + (y
2 + 24y) = -69
(x2 +10x + 25) + (y
2 + 24y + 144) = -69 + 25 + 144
(x + 5)2 + (y + 12)
2 = 100
(x + 5)2 + (y + 12)
2 = 10
2
(-5, -12) 10
(2) 2x2 + 2y
2 - 3x + 4y + 3 = 0
1 2x2 + 2y
2 - 3x + 4y + 3 = 0 2
x2 + y
2 -
2
3 x + 2y + 2
3 = 0 D = -2
3 , E = 2 F = 2
3
(2
D , 2
E ) = 2
2
3
, 2
2 = (4
3 , -1)
r = F)2
E()
2
D( 22
= 2
3)1()
4
3( 22
= 2
31
16
9
= 16
24169
= 16
1
= 4
1
(4
3 , -1) 4
1
2 2x2 + 2y
2 - 3x + 4y + 3 = 0 2
x2 + y
2 -
2
3 x + 2y + 2
3 = 0 (x - h)2 + (y - k)
2 = r
2
(x2 -
2
3 x) + (y2 + 2y) = -
2
3
(x2 -
2
3x +
16
9 ) + (y2 + 2y + 1) = -
2
3+
16
9 + 1
(x -4
3 )2 + (y + 1)
2 =
16
1
(x -4
3 )2 + (y + 1)
2 = (
4
1 )2
(4
3 , -1) 4
1
sm.tm
X
X
O
C(4
3 , -1)
Y Y
O X
(-5, -22)
(-5, -2)
C(-5, -12)
Y Y
(-15, -12) (5, -12) X
⌫ ⌫ ⌦
⌫
6464
1
C(h, k) L ( Y) P(x1, y1)
( L CP P)
P(x1, y1) L
(1) CP = m = hx
ky
1
1
L (2) L = m
1 ( m (m
1 ) = -1)
(3) L y - y1 = m
1 (x - x1)
15 C(-2, 7) P(1, 3)
C(-2, 7) P(1, 3)
CP = 21
73 = 3
4 = 4
3
y - 3 = 4
3 (x - 1)
4y - 12 = 3x - 3 C(-2, 7)
3x - 4y + 9 = 0 P(1, 3)
3x - 4y + 9 = 0
2
(x - h)2 + (y - k)
2 = r
2 L P(x1, y1)
L (x1 - h)(x - h) + (y1 - k)(y - k) = r2
16 x2 + (y -2)
2 = 25 (-4, 5)
x2 + (y - 2)
2 = 25 (0, 2) (-4, 5)
(-4 - 0)(x - 0) + (5 - 2)(y - 2) = 25
-4x + 3y - 6 = 25
4x - 3y + 31 = 0
4x - 3y + 31 = 0
17 x2 + y
2 + 5x - 6y - 21 = 0 (2, -1)
x2 + y
2 + 5x - 6y - 21 = 0 (x - h)
2 + (y - k)
2 = r
2
(x2 + 5x +
4
25 )+ (y2- 6y + 9) = 21 +
4
25 + 9
(x + 2
5 )2 + (y - 3)
2 =
4
145 (-2
5 , 3) (2, -1)
(x1 + 2
5 )(x + 2
5 ) + (y1 - 3)(y - 3) = 4
145
(2 + 2
5 )(x + 2
5 ) + (-1 - 3)(y - 3) = 4
145
2
9 (x + 2
5 ) + (-4)(y - 3) = 4
145 2
9 x +4
45 - 4y + 12 = 4
145
18x + 45 - 16y + 48 = 145 18x - 16y - 52 = 0
9x - 8y - 26 = 0
9x - 8y - 26 = 0 sm.tm
C(h, k)
⌦
6565
⌦
1.
(1) (0, 0) 6
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(2) (0, 0) 5
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(3) (0, 3) 4
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(4) (-5, 0) 3
2
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. .....................................................................................................................................................................................................
(5) (2, 3) 5
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(6) (3
2 ,2
1 ) 3
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(7) (3
1 ,3
2 ) 2
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…….....................................................................................................................................................................................sm.tm
ใบงานที ่1.8
⌫ ⌫ ⌦
⌫
6666
2. (-1, 2) (4, 3)
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3. (3, -4) 12
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. .....................................................................................................................................................................................................
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.. ...................................................................................................................................................................................................
4. 2 y = 5 (3, 5)
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. .....................................................................................................................................................................................................
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5. (3, 2) 3x - 4y + 10 = 0
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...........................................................................................................................................................................................sm.tm
⌦
6767
⌦
6. A(1, -2) B(4, 3) Y
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7. A(2, 3) B(3, 6) 2x + y - 2 = 0
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8. (4, 5), (1, -4) (3, -2)
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⌫ ⌫ ⌦
⌫
6868
9.
(1) x2 + y
2 + 6y - 7 = 0
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(2) x2 + y
2 - 8x + 10y + 5 = 0
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(3) 4x2 + 4y
2 + 12x - 16y - 11 = 0
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(4) 2x2 + 2y
2 - 5x + 3y + 2 = 0
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............................................................................................................................................................................................ sm.tm
⌦
6969
⌦
10. C(3, -2) P(7, 1)
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11. x2 + y
2 + 2y - 19 = 0 (-4, -3)
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12. x2 + y
2 + 4x + 8y - 21 = 0 (2, 1)
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13. x2 + y
2 - 4x - 1 = 0 (3, -2)
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14. x2 + y
2 - 2x + 6y -15 = 0 (4, 1)
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15. x2 + y
2 - 2x + 8y +16 = 0 (2, -4)
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.............................................................................................................................................................................................sm.tm
⌫ ⌫ ⌦
⌫
7070 9
( ) 4
7
1.
2.
1.
1.
2. (0, 0)
3. (0, 0)
4. (h, k)
5. (h, k)
2. ( )
1.
2. (Directrix)
3.
4. (Vertex)
5.
(Latus rectum)
3.
1.
2.
2.1 (0, 0) (c, 0) x = -c
X y2 = 4cx
2.2 (0, 0) (0, c) y = -c
Y x2 = 4cy
2.3 (h, k) (h + c, k) x = h -c
X y = k (y - k)2 = 4c(x - h)
2.4 (h, k) (h, k + c) y = k -c
Y x = h (x - h)2 = 4c(y - k)
⌦
7171
⌦
4.
1.
2. 1.9
3. 1.9
4.
(0, 0) (c, 0) X
5.
(0, 0) (0, c) Y
6.
(h, k) (h + c, k) X y = k
7.
(h, k) (h, k + c) Y x = h
8.
9. 1.9
5.
1. 1.9
2. 1.9
3.
4.
6.
1.
2.
3.
1.
2.
1. 1.9
2.
1. 95 %
2. 95 %
90 %
90 %
7.
…………………………………………………………………………………………………………………..
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8.
…………………………………………………………………………………………………………………..
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⌫ ⌫ ⌦
⌫
7272
(Parabola)
1.
2. (Directrix)
3.
4. (Vertex)
5.
(Latus rectum)
1. (0, 0) (c, 0) x = -c X
c 0 c 0
P(x, y) PD D(-c, y)
PF = PD
22 )0y()cx( = | x - (-c)|
(x - c)2 + y
2 = (x + c)
2
x
2 - 2cx + c
2 + y
2 = x
2 + 2cx + c
2
y2 = 4cx
y2 = 4cx (c, 0) x = -c
X
c 0 y2 = 4cx
c 0 y2 = 4cx sm.tm
D(-c, y) P(x, y)
Y
X O F(c, 0)
x = -c
P(x, y) D(-c, y)
F(c, 0) O
x = -c
X
Y
(Vertex)
(Directrix)
(Latus rectum)
ใบความรทูี ่1.9 (ภาคตดักรวย(พาราโบลา))
⌦
7373
⌦
2. (0, 0) (0, c) y = -c Y
c 0 c 0
P(x, y) PD D(x, -c)
PF = PD
22 )cy()0x( = | y - (-c)|
x 2 + (y - c)
2 = (y + c)
2
x
2 + y
2 - 2cy + c
2 = y
2 + 2cy + c
2
x2 = 4cy
x2 = 4cy (0, c) y = -c
Y
c 0 x2 = 4cy
c 0 x2 = 4cy
1. ( )
2. | c |
3. (Latus rectum) | 4c |
1
(1) (5, 0) (0, 0)
(5, 0) (0, 0)
X ( y = 0)
c = 5
x = -5
| 4(5) | = 20
y2 = 4cx
y2 = 4(5)x y
2 = 20x
(2) (-3, 0) x = 3
(-3, 0) x = 3
X ( y = 0)
c = -3
(0, 0) | 4(-3) | = 12
y2 = 4cx
y2 = 4(-3)x
y2 = -12x
sm.tm
(-3, -6)
X
Y
O
(-3, 6) x = 3
F(-3, 0)
X
Y
(5,10)
O F(5, 0)
(5, -10) x = -5
O
D(x,-c)
X
Y
y = -c
P(x, y) F(0, c)
X
D(x, -c)
P(x,-y)
y = -c
F(0, c)
Y
O
⌫ ⌫ ⌦
⌫
7474
2
(1) (0, 5) (0, 0)
(0, 5) (0, 0)
Y ( x = 0)
c = 5
y = -5
| 4(5) | = 20
x2 = 4cy
x2 = 4(5)y
x2 = 20y
(2) (0, -3) y = 3
(0, -3) y = 3
Y ( x = 0)
c = -3 (0, 0)
| 4(-3) | = 12
x2 = 4cy
x 2 = 4(-3)y
x2 = -12y
(3) y = 4
3 (0, 0)
y = 4
3 (0, 0)
Y ( x = 0)
c = 4
3 (0, 4
3 )
| 4(4
3 ) | = 3
x2 = 4cy
x 2 = 4(
4
3 )y
x2 = 3y
(4) (-2
1 , 0) x = 2
1
(-2
1 , 0) x = 2
1
X ( y = 0)
c = -2
1
(0, 0)
| 4(-2
1 ) | = 2
y2 = 4cx
y2 = 4(-
2
1 )x
y2 = -2x
sm.tm
O
O
F(-2
1 , 0)
(-2
1 , 1)
(-2
1 , -1)
x = 2
1
X
Y
O
F(0, 4
3 )
(-2
3 , 4
3 ) (2
3 , 4
3 )
y = 4
3
Y
X
O
F(0, -3)
(-6, -3)
X
(6, -3)
Y
y = 3
(-10, 5)
X
(10, 5)
F(0, 5)
y = -5
Y
⌦
7575
⌦
y = -2
3
3
(1) y2 = 20x
y2 = 20x y
2 = 4(5)x
(0, 0)
X
c = 5
(5, 0)
x = - 5
| 4(5) | = 20
(2) y2 = - 6x
y2 = - 6x y
2 = 4(-
2
3 )x
(0, 0)
X
c = -2
3
(-2
3 , 0)
x = 2
3
| 4(-2
3 ) | = 6
(3) x2 = 6y
x2 = 6y x
2 = 4(
2
3 )y
(0, 0)
Y
c = 2
3
(0, 2
3 )
y = -2
3
| 4(2
3 ) | = 6
(4) x2 = - 10y
x2 = - 10y x
2 = 4(-
2
5 )y
(0, 0)
Y
c = - 2
5
(0, - 2
5 )
y = 2
5
| 4(-2
5 ) | = 10
sm.tm
X
F(0, -2
5 )
O
F(0, 2
3 )
(-3, 2
3 ) (3, 2
3 )
Y
y =2
5
O X
Y
(-5, -2
5 ) (5, -2
5 )
O X
(-2
3 , 3)
(-2
3 , -3)
Y
F(-2
3 , 0)
x = 2
3
(5, -10)
O
(5, 10)
F(5, 0)
x = - 5
X
Y
⌫ ⌫ ⌦
⌫
7676
(h, k) X Y
1. X
c 0 c 0
V(h, k) F(h + c, k) (h, k)
(0, 0) (c, 0)
P(x , y )
(y )2 = 4cx y = y - k x = x - h
(y - k)2 = 4c(x - h)
(y - k)2 = 4c(x - h) (h, k) (h +c, k)
x = h - c X y = k
(Latus rectum) | 4c |
c 0 c 0
2. Y
c 0 c 0
V(h, k) F(h , k + c) (h, k)
(0, 0) (0, c)
P(x , y )
(x )2 = 4cy x = x - h y = y - k
(x - h)2 = 4c(y - k)
(x - h)2 = 4c(y - k) (h, k) (h , k + c)
y = k - c Y x = h
(Latus rectum) | 4c |
c 0 c 0 sm.tm
D
X
X
F(h, k + c)
O
Y Y
y = k -c
P(x, y)
V(h, k)
D
X
X
F(h, k + c)
O
Y Y
y = k - c
V(h, k)
P(x, y)
O
X
x = h - c
Y Y
D
X
P(x, y)
V (h, k) F(h + c, k)
P(x, y)
X
O
F(h + c, k) V(h, k)
x = h - c
X
Y Y
D
⌦
7777
⌦
4
(1) (-2, 3) (1, 3)
V(-2, 3) = V(h, k) h = - 2 , k = 3
F(1, 3) = F(h + c, k) h + c = 1 - 2 + c = 1 c = 3
X y = 3 ( y = k)
x = -5 ( x = h - c = - 2 - 3 = - 5)
( c 0)
| 4(3) | = 12
(y - k)2 = 4c(x - h)
(y - 3)2 = 4(3)(x + 2)
y 2- 6y + 9 = 12x + 24
y 2- 6y - 12x - 15 = 0
(2) (-3, -4) x = 2
V(-3, -4) = V(h, k) h = - 3, k = -4
x = 2 h - c = 2 - 3 - c = 2 c = -5
X y = - 4 ( y = k )
( c 0)
F(h + c, k) = F(-3 - 5, - 4) = F(-8, -4)
| 4(-5) | = 20
(y - k)2 = 4c(x - h)
(y + 4)2 = 4(-5)(x + 3)
y2+ 8y + 16 = -20x - 60
y2+ 8y + 20x + 76 = 0
(3) x = 5 (2, 4)
x = 5 (2, 4)
F(2, 4) = F(h + c, k) h + c = 2 k = 4
x = h - c = 5 h - c = 5
h + c = 2 h - c = 5 h = 2
7 c = 2
3
X y = 4 ( y = k )
(2
7 , 4)
| 4(2
3 ) | = 6
(y - k)2 = 4c(x - h)
(y - 4)2 = 4 (
2
3)(x -
2
7)
y 2- 8y + 16 = - 6x + 21
y 2- 8y + 6x - 5 = 0
O
Y Y
(-8, -14)
(-8, 6)
F(-8, -4) V( 3,-4)
x = 2
X
V(-2, 3) F(1, 3)
(1, 9)
X
X
x = - 5
(1, -3)
Y Y
O
X
X
O
Y Y
(2, 7)
F(2, 4) V (2
7 , 4)
(2, 1)
x = 5
⌫ ⌫ ⌦
⌫
7878
5
(1) (-2, 3) (-2, 7)
V(-2, 3) = V(h, k) h = - 2 , k = 3
F(-2, 7) = F(h, k + c) k + c = 7 - 3 + c = 7 c = 4
Y x = - 2 ( x = h)
y = - 1 ( y = k - c = 3 - 4 = - 1) Y Y
( c 0)
| 4(4) | = 16
(x - h)2 = 4c(y - k)
(x + 2)2 = 4(4)(y - 3)
x 2+ 4x + 4 = 16y - 48 y = - 1
x 2+ 4x - 16y + 52 = 0
(2) (2, 3) y = 8
V(2, 3) = V(h, k) h = 2, k = 3
y = 8 k - c = 8 3 - 8 = 2 c = -5
Y x = 2 ( x = h )
( c 0) Y Y
F(h , k + c) = F(2, 3 - 5) = F(2, -2)
| 4(-5) | = 20 y = 8
(x - h)2 = 4c(y - k) V(2, 3)
(x - 2)2 = 4(-5)(y - 3) (-8, -2) (12, -2)
x2- 4x + 4 = -20y + 60
x2- 4x + 20y - 56 = 0
(3) y = - 5 (2, 4)
y = -5 (2, 4)
F(2, 4) = F(h , k + c) h = 2 k + c = 4
y = k - c = -5 k - c = -5
k + c = 4 k - c = -5 k = -2
1 c = 2
9
Y x = 2 ( x = h ) Y Y
(2, -2
1 )
| 4(2
9 ) | = 18
(x - h)2 = 4c(y - k)
(x - 2)2 = 4(
2
9)(y +
2
1 )
x2- 4x + 4 = 18y + 9
x2- 4x - 18y - 5 = 0
sm.tm
F(2, -2)
F(-2, 7)
V(-2, 3)
O
O
X X
X
X
V(2, 2
1 )
X
X (-10, 7) (6, 7)
⌦
7979
⌦
(4) (4, -3) x = 4 8
V(4, -3) = V(h, k) h = 4 k = - 3
8 | 4c | = 8 c = 2
Y x = 4
(x - h)2 = 4c(y - k)
(x - 4)2 = 4(2)(y + 3)
x2- 8x + 16 = 8y + 24
x2- 8x - 8y - 9 = 0
x - 4)2 = 4(-2)(y + 3)
x2- 8x + 16 = -8y - 24
x2- 8x + 8y + 40 = 0
x2- 8x - 8y - 9 = 0 x
2- 8x + 8y + 40 = 0
6
(1) y2 - 6y - 20x + 109 = 0
y2 - 6y - 20x + 109 = 0
y2 - 6y = 20x - 109
y2 - 6y + 9 = 20x - 109 + 9
y2 - 6y + 9 = 20x - 100
(y - 3 )2 = 20(x - 5)
(y - 3 )2 = 4(5)(x - 5)
V(h, k) = V(5, 3) h = 5 k = 3
X y = 3 ( y = k )
c = 5 F(h + c, k) = F (5 + 5, 3) = F(10, 3)
x = 0 Y ( x = h - c = 5 -5 = 0)
| 4(5) | = 20
(2) y2 - 2y + 6x + 19 = 0
y2 - 2y + 6x + 19 = 0 y
2 - 2y = - 6x - 19 Y
y2 - 2y + 1 = - 6x - 19 + 1
y2 - 2y + 1 = - 6x - 18
(y - 1)2 = - 6(x + 3)
(y - 1)2 = 4(-
2
3 )(x + 3)
V(h, k) = V(-3, 1) h = -3 k = 1
X y = 1 ( y = k ) x = 2
3
c = -2
3 F(h + c, k) = F (-3 -2
3 , 1) = F(-2
9 , 1)
x = -2
3 ( x = h - c = - 3 +
2
3 = -2
3)
| 4(-2
3) | = 6
sm.tm
V( - 3, 1)
(-2
9 , 4)
F(-2
9 , 1)
X O
(-2
9 , -2)
X
Y
V(5, 3)
(10, -7)
O
(10, 13)
F(10, 3)X
x = 0
X
Y Y
O
x2- 8x + 8y + 40 = 0
X
X
Y Y
x2- 8x - 8y - 9 = 0
⌫ ⌫ ⌦
⌫
8080
7
(1) x 2+ 4x - 16y + 52 = 0
x 2+ 4x - 16y + 52 = 0
x 2+ 4x = 16y - 52
x 2+ 4x + 4 = 16y - 52 + 4
x 2+ 4x + 4 = 16y - 48
(x + 2)2 = 4(4)(y - 3)
V(h, k) = V(-2, 3) h = - 2 , k = 3
Y x = - 2 ( x = h)
c = 4 F(h, k + c) = F(-2, 3 + 4) = F(-2, 7)
y = - 1 ( y = k - c = 3 - 4 = - 1)
| 4(4) | = 16
(2) x2- 4x + 20y - 56 = 0
x2- 4x + 20y - 56 = 0
x2- 4x = -20y + 56
x2- 4x + 4 = -20y + 56 + 4
x2- 4x + 4 = -20y + 60
(x - 2)2 = 4(-5)(y - 3)
V(h, k) = V(2, 3) h = 2, k = 3
Y x = 2 ( x = h )
c = -5 F(h , k + c) = F(2, 3 - 5) = F(2, -2)
y = 8 ( y = k - c = 3 - (- 5) = 8
| 4(-5) | = 20
(3) 3x2- 9x - 5y - 2 = 0
3x2- 9x - 5y - 2 = 0 3x
2- 9x = 5y + 2
x2- 3x =
3
5 y + 3
2
x2- 3x +
4
9 = 3
5 y + 3
2 +4
9
x2- 3x +
4
9 = 3
5 y + 12
35
( x -
2
3 )2 =
3
5 (y + 4
7 )
( x -
2
3 )2 = 4(
12
5 )(y + 4
7 )
V(h, k) = V(2
3 , -4
7 ) h = 2
3 , k = -4
7
Y x = 2
3 ( x = h )
c = 12
5 F(h , k + c) = F(
2
3, -4
7+12
5) = F(
2
3, -3
4)
y = -6
13 ( y = k - c = -
4
7 - 12
5 = -12
26 = -
6
13)
| 4(12
5) | =
3
5
sm.tm
O
X V(
2
3 , -4
7 ) y = -
6
13
F(2
3 , -3
4 )
X
Y Y
(3
2 ,-3
4 ) (3
7 ,-3
4 )
O
y = 8
V (2, 3)
F(2,-2)
O
X
X
Y Y
(-8, -2) (12, -2)
y = - 1
F(-2, 7)
V(-2, 3)
X
X
(-8, 7) (6, 7)
Y Y
⌦
8181
⌦
1.
(1) (0, 0) (6, 0)
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(2) (0, 0) (0, -3)
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(3) x = 3 (0, 0)
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(4) y = -4 (0, 0)
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.............................................................................................................................................................................................sm.tm
ใบงานที ่1.9
⌫ ⌫ ⌦
⌫
8282
(5) x = 3 (-3, 0)
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(6) y = 4 ( 0, -4)
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(7) x = -2
3 (2
3 , 0)
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(8) y = 4
3 (0, -4
3 )
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⌦
8383
⌦
(9) (0, 0) X (-1, -2)
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(10) (0, 0) Y (1, 5)
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(11) (0, 0) X 12
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(12) y = 0 8
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.............................................................................................................................................................................................sm.tm
⌫ ⌫ ⌦
⌫
8484
2.
(1) y2 = 4x
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(2) y2 = - 8x
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(3) x2 = 18y
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(4) x2 = -12y
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.............................................................................................................................................................................................sm.tm
⌦
8585
⌦
(5) y2 + 5x = 0
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..................................... .................................................................................................................................................................
(6) 2y2 - 5x = 0
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(7) 5x2 + 4y = 0
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(8) 10y - 3x2 = 0
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................................................................... .........................................................................................................................sm.tm
⌫ ⌫ ⌦
⌫
8686
3.
(1) (1, 2) (5, 2)
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(2) (-2, -1) (-2, 5)
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(3) x = 1 (-2, 3)
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(4) y = -4 (3, -9)
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.............................................................................................................................................................................................sm.tm
⌦
8787
⌦
(5) y = -3 (1, 3)
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(6) x = 1 (9, 3)
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(7) x = 2 (-5, -4)
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(8) y = 6 (2, -2)
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.............................................................................................................................................................................................sm.tm
⌫ ⌫ ⌦
⌫
8888
4.
(1) (-3, 4) (2, -1)
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(2) (-2, 2) (-2, 4)
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(3) (3, 3), (6, 5) (6, -3) X
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(4) x2 + y
2 + 4x - 6y - 3 = 0 12
Y
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.............................................................................................................................................................................................sm.tm
⌦
8989
⌦
5.
(1) y2 - 2y - 16x + 33 = 0
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(2) y2 + 4y + 12x -32 = 0
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(3) x2 + 6x + 8y + 1 = 0
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(4) x2 + 6x - 20y + 49 = 0
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.............................................................................................................................................................................................sm.tm
⌫ ⌫ ⌦
⌫
9090
(5) y2 + 3y - 6x + 33 = 0
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(6) x2 + 10y + 5y + 30 = 0
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(7) 3x2 + 6x + 8y + 4 = 0
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(8) 4y2 - 4x - 32y + 17 = 0
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.............................................................................................................................................................................................sm.tm
⌦
9191
⌦
10
( ) 1.10 4
7
1.
2.
1.
1.
2.
3. (0, 0) X (c, 0)
(-c, 0)
4. (0, 0) Y (0, c)
(0, -c)
5. (h, k) X
6. (h, k) Y
7.
2. ( )
1.
2.
3.
4. (major axis)
5.
(minor axis)
6.
⌫ ⌫ ⌦
⌫
9292
3.
1.
2.
2.1 (0, 0) X (c, 0) (-c, 0)
(a, 0) (-a, 0) 1b
y
a
x2
2
2
2
a b 0 b2 = a
2 - c
2
2.2 (0, 0) Y (0, c) (0, -c)
(0, a) (0, -a) 1a
y
b
x2
2
2
2
a b 0 b2 = a
2 - c
2
2.3 (h, k) X (h + c, k)
(h - c, ) (h + a, k) (h - a, k) 1b
ky
a
hx2
2
2
2)()(
a b 0 b2 = a
2 - c
2
2.4 (h, k) Y (h, k + c)
(h, k - c) (h, k + a) (h, k - a) 1a
ky
b
hx2
2
2
2)()(
a b 0 b2 = a
2 - c
2
4.
1.
2. 1.10
3. 1.10
4.
(0, 0) X (c, 0) (-c, 0) (a, 0) (-a, 0)
5.
(0, 0) Y (0, c) (0, -c) (0, a) (0, -a)
6.
(h, k) X (h + c, k) (h - c, k) (h + a, k) (h - a, k)
7.
(h, k) Y (h, k + c) (h, k - c) (h, k + a) (h, k - a)
8.
9. 1.10
⌦
9393
⌦
5.
1. 1.10
2. 1.10
3.
4.
6.
1.
2.
3.
1.
2.
1. 1.10
2.
1. 95 %
2. 95 %
90 %
90 %
7.
…………………………………………………………………………………………………………………..
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8.
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⌫ ⌫ ⌦
⌫
9494
(Ellipse)
1. F F
2. C
3. A A
4. AA (major axis)
5. BB
(minor axis)
6.
(0, 0)
1. X F(c, 0) F (- c, 0) c 0
P(x, y) = FF = 2c
PF + PF 2c PF + PF = 2a a 0 2a 2c
220ycx )()( +
220ycx )()( = 2a
22ycx )( = 2a -
22ycx )(
x2+ 2cx + c
2 + y
2 = 4a
2 - 4a
22ycx )( + x
2 - 2cx + c
2 + y
2
4cx - 4a
2 = - 4a
22ycx )(
-4 a2 - cx = a
22ycx )(
a4 - 2a
2cx + c
2x
2 = a
2(x
2 - 2cx + c
2 + y
2)
a4 - 2a
2cx + c
2x
2 = a
2x
2 - 2a
2cx + a
2c
2 + a
2y
2
a2x
2 - c
2x
2 + a
2y
2 = a
4 - a
2c
2
(a2 - c
2)
x
2 + a
2y
2 = a
2(a
2 - c
2)
sm.tm
B(0, b)
X
A (- a, 0) F (- c, 0) O F(c, 0) A(a, 0)
P(x, y)
Y
B (0, -b)
B
A F C F A
B
ใบความรทูี ่1.10 (ภาคตดักรวย(วงร)ี)
⌦
9595
⌦
P(x, y) B(0, b) POF PF = a b2 = a
2 - c
2
a2 - c
2 b
2 b
2x
2 + a
2y
2 = a
2b
2
a
2b
2 1
b
y
a
x2
2
2
2
a b 0 b2 = a
2 - c
2
1. X
2. (0, 0)
3. A(a, 0) A (- a, 0) AA AA 2a (a 0)
4. B(0, b) B (0, -b) BB BB 2b (b 0)
5. F(c, 0) F (- c, 0) FF 2c (c 0)
6. a
b22
7. (eccentricity) e = a
c
8. x = e
a x =
c
a2
2. Y F(0, c) F (0, - c) c 0
Y
F (0, c)
P(x, y)
A (0, -a)
P(x, y) = FF = 2c
PF + PF 2c
PF + PF = 2a a 0 2a 2c
1b
x
a
y2
2
2
2
1a
y
b
x2
2
2
2
a b 0 b2 = a
2 - c
2
1. Y
2. (0, 0)
3. A(0, a) A (0, -a) AA AA 2a (a 0)
4. B(b, 0) B (-b, 0) BB BB 2b (b 0)
5. F(0, c) F (0, -c) FF 2c (c 0)
6. a
b2 2
7. (eccentricity) e = a
c
8. y = e
a y = c
a 2
sm.tm
A(0, a)
F (0, -c)
X
B (-b, 0) O B(b, 0)
⌫ ⌫ ⌦
⌫
9696
1
(1) (3, 0) (-3, 0) ( ) 8
(3, 0) (-3, 0) (0, 0) c = 3
8 2a = 8 a = 4
(4, 0) (-4, 0)
b2 = a
2 - c
2 b
2 = 4
2 - 3
2 = 16 - 9 = 7 b = 7
(0, 7 ) (0, - 7 )
a
b2 2
= 4
)7(2 = 2
7
X
1b
y
a
x2
2
2
2
1)7(
y
4
x2
2
2
2
17
y
16
x 22
7x2 + 16y
2 = 112
(2) (0, 4) (0, -4) ( ) 10
(0, 4) (0, -4) (0, 0) c = 4
10 2a = 10 a = 5
(0, 5) (0, -5)
b2 = a
2 - c
2 b
2 = 5
2 - 4
2 = 25 - 16 = 9 b = 3
(3, 0) (- 3, 0)
a
b2 2
= 5
92 )( = 5
18
Y
1b
x
a
y2
2
2
2
13
x
5
y2
2
2
2
19
x
25
y 22
25x2 + 9y
2 = 225
(3) (5, 0) (-5, 0) (2, 0)
(5, 0) (-5, 0) (0, 0) a = 5
(2, 0) c = 2 (-2, 0)
b2 = a
2 - c
2 b
2 = 5
2 - 2
2 = 25 - 4 = 21 b = 21
(0, 21 ) (0, - 21 )
a
b2 2
= 5
)21(2 = 5
42
X
1b
y
a
x2
2
2
2
1)21(
y
5
x2
2
2
2
121
y
25
x 22
21x2 + 25y
2 = 525
sm.tm
F (0, 4)
A (-5, 0) F (-2, 0) O F(2,0) A(5,0)
B(0, 21 )
B (0, - 21 )
X
Y
O B (-3, 0) B(3, 0)
A (0, -5)
F (0, -4)
A(0, 5)
X
Y
B (0, - 7 )
B (0, 7 )
XA (-4, 0) A(4, 0)
F (-3, 0) O F(3, 0)
Y
⌦
9797
⌦
(4) (3, 0) (-3, 0) 8
(3, 0) (-3, 0) (0, 0) c = 3
8 2a = 8 a = 4 (4, 0) (-4, 0)
b2 = a
2 - c
2 b
2 = 4
2 - 3
2 = 16 - 9 = 7 b = 7
(0, 7 ) (0, - 7 )
a
b22
= 4
72 )( = 2
7
X
1b
y
a
x2
2
2
2
17
y
4
x2
2
2
2
)(
17
y
16
x22
7x2 + 16y
2 = 112
(5) (0, 6) (0, -6) (2, 14 )
(0, 6) (0, -6) (0, 0) a = 6 Y
1b
x
a
y2
2
2
2
a = 6 (2, 14 ) 1b
2
6
142
2
2
2)(
1b
4
36
142
36
141
b
42
36
22
b2 = 4(
2236
) b2 = 11
72
172
x11
36
y22
11x2 + 2y
2 = 72
(6) X (4, 3) (6, 2)
X 1b
y
a
x2
2
2
2
(4, 3) 2
2
a
4 + 1b
32
2
1b
9
a
1622 .................................................(1)
(6, 2) 1b
2
a
62
2
2
2
1b
4
a
3622 ..................................................(2)
(1) 4 4b
36
a
6422
.................................................(3)
(2) 9 9b
36
a
32422
.................................................(4)
(4) - (3) 2a
260 = 5 a
2 = 52
a2 52 (1) 1
b
9
52
162
13
9
52
36
52
161
b
92
b2 = 9
9
13 = 13
113
y
52
x22
x2 + 4y
2 = 52 sm.tm
B (0, - 7 )
B (0, 7 )
X A (-4, 0) A(4, 0)
Y
F (-3, 0) O F(3, 0)
⌫ ⌫ ⌦
⌫
9898
2
(1) 19
y
16
x 22
19
y
16
x 22
a2 = 16 a = 4 b
2 = 9 b = 3
b2 = a
2 - c
2 c
2 = a
2 - b
2 c
2 = 16 - 9 = 7 c = 7
X
1. (0, 0)
2. (4, 0) (-4, 0)
3. ( 7 , 0) (- 7 , 0)
4. (0, 3) (0, -3)
5. a
b2 2
= 4
)9(2 = 2
9
(2) 19
y
4
x 22
19
y
4
x 22
a2 = 9 a = 3 b
2 = 4 b = 2
b2 = a
2 - c
2 c
2 = a
2 - b
2 c
2 = 9 - 4 = 5 c = 5
Y
1. (0, 0)
2. (0, 3) (0, -3)
3. (0, 5 ) (0, - 5 )
4. (2, 0) (-2, 0)
5. a
b2 2
= 3
)4(2 = 3
8
(3) 3x2 + y
2 = 3
3x2 + y
2 = 3 1
3
y
1
x 22
a2 = 3 a = 3 b
2 = 1 b = 1
b2 = a
2 - c
2 c
2 = a
2 - b
2 c
2 = 3 - 1 = 2 c = 2
Y
1. (0, 0)
2. (0, 3 ) (0, - 3 )
3. (0, 2 ) (0, - 2 )
4. (1, 0) (-1, 0)
5. a
b22
= 3
122
)(= 3
2 = 3
32
sm.tm
A (0, 3 )
F (0, 2 )
A (0, - 3 )
F (0, - 2 )
Y
XB (-1, 0) O B(1, 0)
B (-2, 0) O B(2, 0)
Y
A (0, -3)
A(0, 3)
F (0, - 5 )
X
F (0, 5 )
A (-4,0) F (- 7 ,0) O F( 7 , 0) A(4,0)
B (0, -3)
B(0, 3)
X
Y
⌦
9999
⌦
B (0, -2 5 )
(4) 4x2 + 5y
2 = 100
4x2 + 5y
2 = 100 1
20
y
25
x 22
a2 = 25 a = 5 b
2 = 20 b = 20 = 52
b2 = a
2 - c
2 c
2 = a
2 - b
2 c
2 = 25 - 20 = 5 c = 5
X
1. (0, 0)
2. (5, 0) (-5, 0)
3. ( 5 , 0) (- 5 , 0)
4. (0, 2 5 ) (0, -2 5 )
5. a
b2 2
= 4
)20(2 = 10
(5) 9x2 + 2y
2 = 18
9x2 + 2y
2 = 18 1
9
y
2
x 22
a2 = 9 a = 3 b
2 = 2 b = 2
b2 = a
2 - c
2 c
2 = a
2 - b
2 c
2 = 9 - 2 = 6 c = 6
Y
1. (0, 0)
2. (0, 3) (0, -3)
3. (0, 6 ) (0, - 6 )
4. ( 2 , 0) (- 2 , 0)
5. a
b2 2
= 3
)2(2 = 3
4
(6) x2 +4y
2 = 16
x2 + 4y = 16 1
4
y
16
x 22
a2 = 16 a = 4 b
2 = 4 b = 2
b2 = a
2 - c
2 c
2 = a
2 - b
2 c
2 = 16 - 4 = 12 c = 12 = 2 3
X
1. (0, 0)
2. (4, 0) (-4, 0)
3. (2 3 , 0 ) (-2 3 , 0)
4. (0, 2) (0, -2)
5. a
b22
= 4
222
)(= 2
sm.tm
X
B (0, -2)
F (-2 3 , 0) O F(2 3 , 0)
A (-4, 0)
B(0, 2)
A(4, 0)
Y
X
F (0, - 6 )
F(0, 6 )
A(0, 3)
A (0,-3)
B (- 2 , 0) O B( 2 , 0)
Y
X
B(0, 2 5 )
A (-5, 0) F ( 5 ,0) O F ( 5 , 0) A(5, 0)
Y
⌫ ⌫ ⌦
⌫
100100
(h, k)
1. (h, k) X
(h, k) X (h, k)
(h, k) 1b
)y(
a
)x(2
2
2
2
x = x - h y = y - k
(h, k)
1b
)ky(
a
)hx(2
2
2
2 a b 0 b
2 = a
2 - c
2
(1) X y = k
(2) (h, k)
(3) F(h + c, k) F (h - c, k) FF 2c (c 0)
(4) A(h + a, k) A (h - a, k) AA AA 2a (a 0)
(5) B(h, k + b) B (h, k - b) BB BB 2b (b 0)
(6) a
b2 2
2. (h, k) Y
(h, k) Y
(h, k)
1a
)y(
b
)x(2
2
2
2
x = x - h y = y - k
(h, k)
1a
)ky(
b
)hx(2
2
2
2 a b 0 b
2 = a
2 - c
2
(1) Y x = h
(2) C(h, k)
(3) F(h , k + c) F (h , k - c) FF 2c (c 0)
(4) A(h , k + a) A (h, k - a) AA AA 2a (a 0)
(5) B(h + b, k) B (h - b, k) BB BB 2b (b 0)
(6) a
b2 2
1.
2.
3. a
4. c
5. b sm.tm
F(h, k+ c)
X
X
A (h, k - a)
A(h, k + a)
B (h - b, k) C(h, k) B(h + b, k)
O
F (h, k- c)
Y Y
B(h, k - b)
B (h, k - b)
X
O
C(h, k)
X
A (h - a, k) F (h - c, k) F(h + c, k) A(h + a, k)
Y Y
⌦
101101
⌦
3
(1) (4, 2) (-2, 2) 8
(4, 2) (-2, 2) (1, 2)
(1, 2) h = 1, k = 2 8 2a = 8 a = 4
(1, 2) (4, 2) 3 c = 3
b2 = a
2 - c
2 b
2 = 4
2 - 3
2 = 16 - 9 = 7 b = 7 X y = 2
1b
)ky(
a
)hx(2
2
2
2
1
)7(
)2y(
4
)1x(2
2
2
2
17
)2y(
16
)1x( 22
7(x - 1)2 + 16(y - 2)
2 = 112
7(x2 - 2x + 1) + 16(y
2 - 4y + 4) = 112
7x2 - 14x + 7 + 16y
2 - 64y + 64 = 112
7x2+ 16y
2 - 14x - 64y - 41 = 0
(2) (-2, -1) (-2, 3) 10
(-2, -1) (-2, 3) (-2, 1)
(-2, 1) h = -2 , k = 1 10 2a = 10 a = 5
(-2, 1) (-2, 3) 2 c = 2
b2 = a
2 - c
2 b
2 = 5
2 - 2
2 = 25 - 4 = 21 b = 21
Y x = -2
1a
)ky(
b
)hx(2
2
2
2
125
)1y(
21
)2x( 22
25(x + 2)2 + 21(y - 1)
2 = 525
25(x2 + 4x + 4) + 21(y
2 - 2y + 1) = 525
25x2 + 100x + 100 + 21y
2 - 42y + 21 = 525
25x2 + 21y
2 + 100x - 42y - 404 = 0
(3) (2, -3) (2, 2) (2, 1)
(2, -3) h = 2, k = -3 (2, 2)
Y x = 2
(2, -3) (2, 2) 5 a = 5
(2, -3) (2, 1) 4 c = 4
b2 = a
2 - c
2 b
2 = 5
2 - 4
2 = 25 - 16 = 9 b = 3
1a
ky
b
hx2
2
2
2)()(
125
3y
9
2x22
)()(
25(x2 - 4x + 4) + 9(y
2 + 6y + 9) = 225
25x2 - 100x + 100 + 9y
2 + 54y + 81 = 225
25x2 + 9y
2 -100x + 54y - 44 = 0
sm.tm
O
C(2,-3)
X
X
A (2, -8)
F(2,1)
F (2,-7)
B(5, -3) B (-1, -3)
A(2, 2)
Y Y
O X
A (-2, -4)
A(-2, 6)
F(-2, 3)
B (-2 - 21 , 1) C(-2, 1) B(-2 + 21 , 1) X
F (-2, -1)
Y Y
A (-3, 2) F (-2, 2) F(4, 2) A(5, 2)
C (1, 2)
O
B(1, 2 + 7 )
B (1, 2 - 7 )
X
X
Y Y
⌫ ⌫ ⌦
⌫
102102
(4) (-1, 3) (5, 3) 4
(-1, 3) (5, 3) (2, 3)
(2, 3) h = 2, k = 3 4 2b = 4 b = 2
(2, 3) (5, 3) 3 a = 3
b2 = a
2 - c
2 c
2 = a
2 - b
2 = 9 - 4 = 5 c = 5 X y = 3
1b
)ky(
a
)hx(2
2
2
2
12
)3y(
3
)2x(2
2
2
2
14
)3y(
9
)2x( 22
4(x - 2)2 + 9(y - 3)
2 = 36
4(x2 - 4x + 4) + 9(y
2 - 6y + 9) = 36
4x2 - 16x + 16 + 9y
2 - 54y + 81 = 36
4x2+ 9y
2 - 16x - 54y + 61 = 0
(5) (3, 4) (3, -4) (0, 0)
(3, 4) (3, -4) (3, 0)
(3, 0) h = 3 , k = 0 (3, 0) (3, 4) 4 a = 4
Y x = 3
1a
)ky(
b
)hx(2
2
2
2 (0, 0)
14
)0(
b
)30(2
2
2
2 1
b
92
b2 = 9 b = 3
c2 = a
2 - b
2 c
2 = 16 - 9 = 7 c = 7
116
y
9
)3x( 22
16(x - 3)2 + 9y
2 = 144
16(x2 - 6x + 9) + 9y
2 = 144
16x2 - 96x + 144 + 9y
2 = 144
16x2+ 9y
2 - 96x = 0
(6) (-1, 5) (-1, -1) (4, 2)
(-1, 5) (-1, -1) (-1, 2)
(-1, 2) h = -1 , k = 2
X y = 2
(-1, 2) (4, 2) 5 a = 5
(-1, 2) (-1, 5) 4 b = 3
c2 = a
2 - b
2 c
2 = 5
2 - 3
2 = 25 - 9 = 16 c = 4
1b
ky
a
hx2
2
2
2)()(
19
2y
25
1x22
)()(
9(x2 + 2x + 1) + 25(y
2 - 4y + 4) = 225
9x2 + 18x + 9 + 25y
2 - 100y + 100 = 225
9x2 + 25y
2 + 18x - 100y – 116 = 0
sm.tm
O
B (-1, -1)
O
C(-1, 2)
X
X A (-6, 2)
F (-5, 2) F(3, 2)
B (-1,5)
A(4, 2)
Y Y
F(3, 7 )
F (3 , - 7 )
X
A (3, -4)
A(3, 4)
C(3, 0)
B(6, 0) B (0, 0)
Y Y
F (2 - 5 , 3) F(2 + 5 , 3)
A(5, 3)C (2, 3)
O
B(2, 5 )
B (2, 1)
X
X
A (-1, 3)
Y Y
⌦
103103
⌦
4
1. 7x2+ 16y
2 - 14x - 64y - 41 = 0
7x2+ 16y
2 - 14x - 64y - 41 = 0
(7x2 - 14x) + (16y
2 - 64y) = 41
7(x2 - 2x ) + 16(y
2 - 4y) = 41
7(x2 - 2x + 1) + 16(y
2 - 4y + 4) = 41 + 7 + 64
7(x - 1)2 + 16(y - 2)
2 = 112
17
)2y(
16
)1x( 22
a2 = 16 a = 4 b
2 = 7 b = 7
c2 = a
2 - b
2 c
2 = 16 - 7 = 9 c = 3
X y = 2
(1) C(h, k) = C(1, 2) h = 1 k = 2
(2) F(h + c, k) = F(1 + 3, 2) = F(4, 2)
F (h - c, k) = F (1 - 3, 2) = F (- 2, 2)
(3) A(h + a, k) = A(1 + 4 , 2) = A(5, 2)
A (h - a, k) = A (1 - 4, 2) = A (-3, 2)
(4) B(h, k + b) = B(1, 2 + 7 )
B (h, k - b) = B (1, 2 - 7 )
(5) a
b2 2
= 4
)7(2 =2
7
2. 25x2 + 21y
2 + 100x - 42y - 404 = 0
25x2 + 21y
2 + 100x - 42y - 404 = 0
(25x2+ 100x) + (21y
2 - 42y) = 404
25(x2+ 4x ) + 21(y
2 - 2y ) = 404
25(x2+ 4x + 4) + 21(y
2 - 2y + 1) = 404 + 100 + 21
25(x + 2)2 + 21(y - 1)
2 = 525
125
)1y(
21
)2x( 22
a2 = 25 a = 5 b
2 = 21 b = 21
c2 = a
2 - b
2 c
2 = 25 - 21 = 4 c = 2
Y x = -2
(1) C(h, k) = C(-2, 1) h = -2 , k = 1
(2) F(h , k + c) = F(-2 , 1 + 2) = F(-2 , 3)
F (h , k - c) = F (-2 , 1 - 2) = F (-2 , -1)
(3) A(h , k + a) = A(-2 , 1 + 5) = A(-2 , 6)
A (h, k - a) = A (-2, 1 - 5) = A (-2, - 4)
(4) B(h + b, k) = B(-2 + 21 , 1)
B (h - b, k) = B (-2 - 21 , 1)
(5) a
b22
= 5
212 )( = 542
sm.tm
X
A (-2, -4)
A(-2, 6)
C(-2, 1) X
F (-2 , -1)
B(-2 + 21 , 1)B (-2 - 21 ,1)
F(-2, 3)
Y Y
O
A (-3, 2) F (-2, 2) F(4, 2) A(5, 2)
C (1, 2)
B(1, 2 + 7 )
B (1, 2 - 7 )
X
X
Y Y
⌫ ⌫ ⌦
⌫
104104
3. 4x2+ 9y
2 - 16x - 54y + 61 = 0
4x2+ 9y
2 - 16x - 54y + 61 = 0
(4x2- 16x) + (9y
2 - 54y) = - 61
4(x2- 4x) + 9(y
2 - 6y) = - 61
4(x2- 4x + 4) + 9(y
2 - 6y + 9) = - 61 + 16 + 81
4(x - 2)2 + 9(y - 3)
2 = 36
14
)3y(
9
)2x( 22
a2 = 9 a = 3 b
2 = 4 b = 2
c2 = a
2 - b
2 c
2 = 9 - 4 = 5 c = 5
X y = 3
(1) C(h, k) = C(2, 3) h = 2 k = 3
(2) F(h + c, k) = F(2 + 5 , 3)
F (h - c, k) = F (2 - 5 , 3)
(3) A(h + a, k) = A(2 + 3 , 3) = A(5, 3)
A (h - a, k) = A (2 - 3, 3) = A (-1, 3)
(4) B(h, k + b) = B(2, 3 + 2) = B(2, 3 + 2)
B (h, k - b) = B (2, 3 - 2) = B (2, -1)
(5) a
b2 2
= 4
)4(2 = 2
4. 16x2+ 9y
2 - 96x = 0
16x2+ 9y
2 - 96x = 0
(16x2 - 96x) + 9y
2 = 0
16(x2 - 6x) + 9y
2 = 0
16(x2 - 6x + 9) + 9y
2 = 144
16(x - 3)2 + 9y
2 = 144
116
y
9
)3x( 22
a2 = 16 a = 4 b
2 = 9 b = 3
c2 = a
2 - b
2 c
2 = 16 - 9 = 7 c = 7
Y x = 3
(1) C(h, k) = C(3, 0) h = 3 , k = 0
(2) F(h , k + c) = F(3 , 0 + 7 ) = F(3 , 7 )
F (h , k - c) = F (3 , 0 - 7 ) = F (3 , - 7 )
(3) A(h , k + a) = A(3 , 0 + 4) = A(3 , 4)
A (h, k - a) = A (3, 0 - 4) = A (3, - 4)
(4) B(h + b, k) = B(3 + 3, 0) = B(6, 0)
B (h - b, k) = B (3 - 3 , 0) = B (0 , 0)
(5) a
b2 2
= 4
)9(2 = 2
9
sm.tm
F (3 , - 7 )
O
B (0, 0)
F(3, 7 )
X
A (3, -4)
A(3, 4)
C(3, 0) B(6, 0)
Y Y
F (2 - 5 , 3) F(2 + 5 , 3) A(5, 3)
C (2, 3)
O
B(2, 5 )
B (2, 1)
X
X
A (-1, 3)
Y Y
⌦
105105
⌦
1.
(1) (-3, 0) (3, 0) 8
.......................................................................................................................................................................................................
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(2) (0, -4) (0, 4) 10
.......................................................................................................................................................................................................
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.......................................................................................................................................................................................................
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(3) (-10, 0) (10, 0) (-8, 0) (8, 0)
.......................................................................................................................................................................................................
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(4) (0, -6) (0, 6) 16
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(5) (0, 0) Y 8 2
9
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.............................................................................................................................................................................................sm.tm
ใบงานที ่1.10
⌫ ⌫ ⌦
⌫
106106
(6) (0, -6) (0, 6) (3, 2)
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(7) (-8, 0) (8, 0) (8, 5
18 )
.......................................................................................................................................................................................................
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(8) (0, 0) (0, -5) (0, -4)
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(9) (0, 0) 10 (-4, 0)
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(10) (0, -4) (0, 4) e = 3
2
.......................................................................................................................................................................................................
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.............................................................................................................................................................................................sm.tm
⌦
107107
⌦
2.
(1) 116
y
25
x 22
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(2) 125
y
9
x22
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(3) 112
y
8
x 22
.......................................................................................................................................................................................................
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.......................................................................................................................................................................................................
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............................................................................................................................................................................................sm.tm
⌫ ⌫ ⌦
⌫
108108
(4) 4x2 + 9y
2 = 36
.......................................................................................................................................................................................................
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..................................................................................................................................................................................................... .
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(5) x2 + 4y
2 - 100 = 0
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(6) 5x2 + 2y
2 - 4 = 0
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………………………………………………..…………………………………………………………………………. sm.tm
⌦
109109
⌦
3.
(1) (-3, 2) (3, 2) 10
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(2) (-1, -1) (7, -1) 10
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(3) (-2, -1) (-2, -5) 8
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⌫ ⌫ ⌦
⌫
110110
(4) (6, 3) (-2, 3) 10
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(5) (6, 3) (4, 3) (-4, 3)
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(6) (-4, 1) (-4, -5) 14
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⌦
111111
⌦
(7) (-5, 2) (1, 2) (-2, 6)
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(8) (-2, -7) (-2, 1) (-5, -3)
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(9) (-2, 2), (2, 5), (6, 2) (2, -1)
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.............................................................................................................................................................................................sm.tm
⌫ ⌫ ⌦
⌫
112112
4.
(1) 116
)1y(
25
)2x( 22
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(2) 25(x + 3)2 + 9(y + 1)
2 - 225 = 0
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(3) 9x2 + 16y
2 - 36x - 96y + 36 = 0
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................................................................................…………………………………………………….…………………sm.tm
⌦
113113
⌦
(4) 25x2 + 16y
2 + 50x + 64y - 311 = 0
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(5) 16x2 + 20y
2 - 64x - 40y - 236 = 0
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(6) 25x2 + 9y
2 + 100x + 18y - 116 = 0
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........................................................................................................................................................................................... sm.tm
⌫ ⌫ ⌦
⌫
114114
(7) x2 + 4y
2 - 4x - 8y = 92
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(8) 9x2 + 64y
2 + 54x + 128y = 431
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(9) 9x2 + 5y
2 - 36x - 20y = 124
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............................................................................................................................................................................................sm.tm
⌦
115115
⌦
11
( ) 1.11 4
7
1.
2.
1.
1.
2.
3. (0, 0)
X (c, 0) (-c, 0)
4. (0, 0) Y
(0, c) (0, -c)
5. (h, k)
X
5. (h, k)
Y
6.
2. ( )
1.
2.
3.
4.
5.
6. (asymptotes)
3.
3.1 (0, 0) X (c, 0)
(-c, 0) (a, 0) (-a, 0) 1b
y
a
x2
2
2
2
c > a > 0 b2 = c
2 - a
2
⌫ ⌫ ⌦
⌫
116116 3.2 (0, 0) Y (0, c) (0, -c)
(0, a) (0, -a) 1b
x
a
y2
2
2
2
c > a > 0 b2 = c
2 - a
2
3.3 (h, k) X (h + c, k)
(h - c, k) (h + a, k) (h - a, k) 1b
ky
a
hx2
2
2
2)()(
c > a > 0 b2 = c
2 - a
2
3.4 (h, k) Y (h, k + c)
(h, k - c) (h, k + a) (h, k - a) 1b
hx
a
ky2
2
2
2)()(
c > a > 0 b2 = c
2 - a
2
4.
1.
2. 1.11
3. 1.11
4.
(0, 0) X (c, 0) (-c, 0)
(a, 0) (-a, 0)
5.
(0, 0) Y (0, c) (0, -c)
(0, a) (0, -a)
6.
(h, k) X (h + c, k) (h - c, k)
(h + a, k) (h - a, k)
7.
(h, k) Y (h, k + c) (h, k - c)
(h, k + a) (h, k - a)
8.
9. 1.11
5.
1. 1.11
2. 1.11
3.
4.
⌦
117117
⌦
6.
1.
2.
3.
1.
2.
1. 1.11
2.
1. 95 %
2. 95 %
90 %
90 %
7.
…………………………………………………………………………………………………………………..
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…………………………………………………………………………………………………………………..
8.
…………………………………………………………………………………………………………………..
…………………………………………………………………………………………………………………..
.………………………………………………………………………………………………………………….
.………………………………………………………………………………………………………………….
…………………………………………………………………………………………………………………..
…………………………………………………………………………………………………………………..
…………………………………………………………………………………………………………………..
.………………………………………………………………………………………………………………….
⌫ ⌫ ⌦
⌫
118118
(Hyperbola)
1. F F
2. C
3. A A
4. AA (transverse axis)
5. BB (conjugate axis)
6. L1 L2 (asymptotes)
(0, 0)
1. X
(0, 0) F(c, 0) F (-c, 0) c 0
P(x, y) P 2a a 0
| PF - PF | = 2a
220ycx )()( -
220ycx )()( = 2a
22ycx )( = 2a +
22ycx )(
(x + c)2 + y
2 = 4a
2 + 4a
22ycx )( + (x - c)
2 + y
2
x2 + 2cx +c
2 + y
2 = 4a
2 + 4a
22ycx )( + x
2 - 2cx +c
2 + y
2
4cx - 4a2 = 4a
22ycx )(
4 cx - a2 = a
22ycx )( sm.tm
B
B
A (-a, 0) O A(a, 0)
B (0, -b)
X
y = -a
bx y =
a
bx
Y
P(x, y) B(0, b)
F(c, 0) F (-c, 0)
F A
L1 L2
C A F
ใบความรทูี ่1.11 (ภาคตดักรวย(ไฮเพอรโบลา))
⌦
119119
⌦
c2x
2 - 2a
2cx + a
4 = a
2(x
2 - 2cx +c
2 + y
2)
c2x
2 - 2a
2cx + a
4 = a
2x
2 - 2 a
2cx + a
2c
2 + a
2y
2
(c2x
2 - a
2x
2) - a
2y
2 = a
2c
2 - a
4
(c2 - a
2)
x
2 - a
2y
2 = (c
2 - a
2)a
2
(c2 - a
2)a
2 1
ac
y
a
x22
2
2
2
0 a c c2 - a
2 0 c
2 - a
2 = b
2 b 0
1b
y
a
x2
2
2
2
b2 = c
2 - a
2
1. C(0, 0)
2. X
3. F(c, 0) F (-c, 0)
4. A(a, 0) A (-a, 0) AA = 2a
5. B(0, b) B (0, -b) BB = 2b
6. (asymptotes) y = a
bx
7. a
b22
8. (eccentricity) e = a
c
2. Y
(0, 0) F(0, c) F (0, -c) c 0
P(x, y) P 2a a 0
| PF - PF | = 2a
1b
x
a
y2
2
2
2
b2 = c
2 - a
2
1. C(0, 0)
2. Y
3. F(0, c) F (0, -c)
4. A(0, a) A (0, -a) AA = 2a0
5. B(b, 0) B (-b, 0) BB = 2b
6. (asymptotes) y = b
ax
7. a
b22
8. (eccentricity) e = a
c sm.tm
F (0, -c)
X
A (0, -a)
F (0, c)y = b
ax
y = -b
ax
B (-b, 0) O B(b, 0)
A (0, a)
P(x, y)
Y
⌫ ⌫ ⌦
⌫
120120
1. 1b
y
a
x2
2
2
2 1
b
x
a
y2
2
2
2 a = b
(rectangular hyperbola) 1
16
y
16
x 22 x2 - y
2 = 16
1
10
x
10
y 22 y2 - x
2 = 10
2. 1
b
y
a
x2
2
2
2 0
b
y
a
x2
2
2
2
y2 =
2
2
a
b x2 y =
a
b x
y = a
b x y = -a
b x 1b
y
a
x2
2
2
2
3. 1b
x
a
y2
2
2
2
0b
x
a
y2
2
2
2
y2 =
2
2
b
a x2 y =
b
a x
y = b
a x y = -b
a x 1b
x
a
y2
2
2
2
1
(1) (5, 0) (-5, 0) 8
(5, 0) (-5, 0) (0, 0) c = 5
X 8 2a = 8 a = 4
b2 = c
2 - a
2 b
2 = 5
2 - 4
2 = 25 - 16 = 9 b = 3
1b
y
a
x2
2
2
2
19
y
16
x22
9x2 - 16y
2 = 144
A(4, 0) A (-4, 0)
B(0, 3) B (0, -3)
y = 4
3 x
3x + 4y = 0 3x - 4y = 0
4
)3(2 2 =
2
9
eccentricity e = 45
(2) (0, 5) (0, -5) 6
F(0, 5) F (0, -5) (0, 0) c = 5
Y 6 2a = 6 a = 3
b2 = c
2 - a
2 b
2 = 5
2 - 3
2 = 25 - 9 = 16 b = 4 1
b
x
a
y2
2
2
2
116
x
9
y22
16y2 - 9x
2 = 144
A(0, 3) A (0, -3)
B(4, 0) B (-4, 0)
y = 4
3 x
3x + 4y = 0 3x - 4y = 0
3
422
)( = 3
32
eccentricity e = 3
5
B (0, -3)
F (0, -5)
A(0, 3) X
B (-4, 0) A (0, -3) B(4, 0)
F(0, 5)
Y
3x + 4y = 0 3x - 4y = 0
O
A(4, 0)
F(5, 0)F (-5, 0)
A (-4, 0) X
3x - 4y = 0 3x + 4y = 0
B(0, 3)
Y
⌦
121121
⌦
2
(3) (13, 0) (-13, 0) 24
(13, 0) (-13, 0) c = 13 (0, 0) X
24 2b = 24 b = 12
b2 = c
2 - a
2 a
2 = c
2 - b
2 = 13
2 - 12
2 = 169 - 144 = 25 a = 5
1b
y
a
x2
2
2
2 1
144
y
25
x 22 144x
2 - 25y
2 = 3600
A(5, 0) A (-5, 0)
B(0, 12) B (0, -12)
y = 5
12 x
12x + 5y = 0 12x - 5y = 0
5
)12(2 2 =
5
288
eccentricity e = 5
13
(2) (0, 1) (0, -1) (0, -4)
(0, 1) (0, -1) a = 1 (0, 0) Y
(0, -4) c = 4
b2 = c
2 - a
2 b
2 = 4
2 - 1
2 = 16 - 1 = 15 b = 15
1b
x
a
y2
2
2
2
115
x
1
y 22 15y
2 - x
2 = 15
F(0, 4) F 0, -4)
B( 15 , 0) B (- 15 , 0)
y = 15
1 x = 15
15 x
1
152 )( = 30
eccentricity e = 4
(3) (0, 0) (-2, 0) (2 2 , 3)
(0, 0) (-2, 0) X a = 2
1b
y
a
x2
2
2
2 (2 2 , 3) 1
b
3
2
)22(2
2
2
2 b
2 = 9 b = 3
c2 = a
2 + b
2 c
2 = 2
2 + 3
2 = 13 c = 13
19
y
4
x 22 9x
2 - 4y
2 = 36
A(2, 0) A (-2, 0)
F( 13 , 0) F (- 13 , 0)
B(0, 3) B (0, -3)
y = 2
3x 3x + 2y = 0 3x - 2y = 0
2
92 )( = 9
eccentricity e = 2
13
sm.tm
B (0, -12)
F( 13 , 0)F (- 13 , 0)
B(0, 3)
B (0, -3)
A (-2, 0) O A(2, 0)
Y
X
3x - 2y = 0 3x + 2y = 0
B (- 15 , 0) B( 15 , 0)
y = -15
15 x 15 x+15y = 0 F(0, 4)
A (0, -1)
A(0, 1)
y = 15
15 x 15 x - 15y = 0
X
Y
F (0, -4)
F (-13, 0) O
A(5, 0)
F(13, 0)
A (-5, 0) X
B(0, 12)
12x - 5y = 0 Y 12x + 5y = 0
⌫ ⌫ ⌦
⌫
122122
3
eccentricity e
(1) 4x2 - 9y
2 = 36
4x2 - 9y
2 = 36 36 1
4
y
9
x 22
X a2 = 9 a = 3 b
2 = 4 b = 2
c2 = a
2 + b
2 c
2 = 3
2 + 2
2 = 13 c = 13
1. C(0, 0)
2. A(3, 0) A (-3, 0)
3. F( 13 , 0) F (- 13 , 0)
4. B(0, 2) B (0, -2)
5. y = 3
2 x 2x +3y = 0 2x -3y = 0
6. 3
)4(2 = 3
8
7. eccentricity e = 3
13
(2) 25y2 - 16x
2 = 400
25y2 - 16x
2 = 400 400 1
25
x
16
y 22
Y a2 = 16 a = 4 b
2 = 25 b = 5
c2 = a
2 + b
2 c
2 = 16 + 25 = 41 c = 41
1. C(0, 0)
2. A(0, 4) A (0, -4)
3. F(0, 41 ) F (0, - 41 )
4. B(5, 0) B (-5, 0)
5. y = 5
4 x 4x + 5y = 0 4x - 5y = 0
6. 4
)25(2 = 2
25
7. eccentricity e = 4
41
(3) x2 - y
2 = 9
x2 - y
2 = 9 9 1
9
y
9
x 22
X a2 = 9 a = 3 b
2 = 9 b = 3
c2 = a
2 + b
2 c
2 = 9 + 9 = 18 c = 18 = 3 2
1. C(0, 0)
2. A(3, 0) A (-3, 0)
3. F(3 2 , 0) F (-3 2 , 0)
4. B(0, 3) B (0, -3)
5. y = x x + y = 0 x - y = 0
6. 3
92 )( = 6
7. eccentricity e = 3
23 = 2
B(0, 2)
F (-3 2 , 0) F(3 2 , 0)
B (0, -3)
X
Y
x - y = 0 x + y = 0
B(0, 3)
A (-3, 0) O A(3, 0)
A(0, 4)
B (-5, 0) O B(5, 0)
A (0, -4)
F (0, - 41 )
F(0, 41 )
X
Y
4x + 5y = 0
4x - 5y = 0
F (- 13 , 0) O F( 13 , 0)
B (0, -2)
X
2x -3y = 0
Y
2x +3y = 0
A(3, 0) A (-3, 0)
⌦
123123
⌦
4
eccentricity e
(1) 4x2 - y
2 = -16
4x2 - y
2 = -16 -16 1
4
x
16
y 22
Y a2 = 16 a = 4 b
2 = 4 b = 2
c2 = a
2 + b
2 c
2 = 16 + 4 = 20 c = 2 5
1. C(0, 0)
2. A(0, 4) A (0, -4)
3. F(0, 2 5 ) F (0, -2 5 )
4. B(2, 0) B (-2, 0)
5. y = 2x 2x + y = 0 2x - y = 0
6. 4
)4(2 = 2
7. eccentricity e = 4
52 = 2
5
(2) 5y2 - 2x
2 = 15
5y2 - 2x
2 = 15 15 1
15
x2
3
y 22
1
2
15
x
3
y 22
Y a2 = 3 a = 3 b
2 =
2
15 b = 2
30
c2 = a
2 + b
2 c
2 = 3 +
2
15 = 2
21 c = 2
42
1. C(0, 0)
2. A(0, 3 ) A (0, - 3 )
3. F(0, 2
42 ) F (0, -2
42 )
4. B(2
30 , 0) B (-2
30 , 0)
5. y = 30
32 x y = x5
10
6. 3
)2
15(2
= 5 3
7. eccentricity e = 32
42 = 6
126 = 2
14
(3) x2 - y
2 = 1
x2 - y
2 = 1 X a
2 = 1 a = 1 b
2 = 1 b = 1
c2 = a
2 + b
2 c
2 = 1 + 1 = 2 c = 2
1. C(0, 0)
2. A(1, 0) A (-1, 0)
3. F( 2 ,0) F (- 2 , 0)
4. B(0, 1) B (0, -1)
5. y = x x + y = 0 x - y = 0
6. 2
7. eccentricity e = 2 sm.tm.
A (-1, 0) A(1, 0) B (0, -1)
X
Y
B(0, 1)
F (- 2 , 0) O F( 2 ,0)
x - y = 0
x + y = 0
y = - x5
10 10 x +5y = 0
F(0, 2
42 )
A(0, 3 )
y =5
10 x 10 x - 5y = 0
B (-2
30 , 0) O B(2
30 , 0)
F (0, - 2
42 )
X
Y
B (-2, 0) O B(2, 0)
Y
2x - y = 0
2x + y = 0
F (0, -2 5 )
F(0, 2 5 )
A (0, -4)
X
A(0, 4)
⌫ ⌫ ⌦
⌫
124124
(h, k)
1. X
(h, k) X y = k
1b
y
a
x2
2
2
2)()(
x = x - h y = y - k
1b
ky
a
hx2
2
2
2)()( b
2 = c
2 - a
2 0 a c
1. X y = k
2. C(h, k)
3. F( h + c, k) F (h - c, k)
4. A( h + a, k) A (h - a, k)
5. B( h, k + b) B (h, k - b)
6. (asymptotes) y - k = a
b (x - h)
7. a
b2 2
8. (eccentricity) e = a
c
2. Y
(h, k) Y x = h
1b
x
a
y2
2
2
2)()(
A(h, k+a)
C(h, k)
A (h, k-a)
X
XO
Y Y
F (h, k-c)
y - k = -b
a(x - h)
y - k = b
a(x - h)
F(h, k+c)
B (h-b, k) B(h+b, k)
A (h-a, k) C(h, k) A(h+a, k)
B (h, k-b)
X
X O
Y Y
B(h, k+b)
F(h+c, k) F (h-c, k)
y - k = -a
b (x - h) y - k = a
b (x - h)
⌦
125125
⌦
x = x - h y = y - k
1b
hx
a
ky2
2
2
2)()( b
2 = c
2 - a
2 0 a c
1. Y x = h
2. C(h, k)
3. F( h, k + c) F (h , k - c)
4. A( h, k + a) A (h, k - a)
5. B( h + b, k) B (h - b, k)
6. (asymptotes) y - k = b
a (x - h)
7. a
b2 2
8. (eccentricity) e = a
c
(h, k)
(h, k)
X Y
1b
ky
a
hx2
2
2
2)()( 1
b
hx
a
ky2
2
2
2)()(
(1) 1b
ky
a
hx2
2
2
2)()( 1
b
hx
a
ky2
2
2
2)()(
(h, k)
(2) 1b
ky
a
hx2
2
2
2)()(
0b
ky
a
hx2
2
2
2)()( y - k =
a
b (x - h)
1b
hx
a
ky2
2
2
2)()(
0b
hx
a
ky2
2
2
2)()( y - k =
b
a (x - h)
(3) a = b
(4)
Ax2 + By
2 + Cx + Dy + E = 0 A B
x2
x2 + Ay
2 + Bx + Cy + D = 0
(5) Ax2 + By
2 + Cx + Dy + E = 0 A 0, B 0 A 0, B 0
sm.tm
⌫ ⌫ ⌦
⌫
126126
5
(1) (4, 2)0 (-2, 2) 4
(4, 2) (-2, 2) (1, 2)
(1, 2) h = 1, k = 2 4 2a = 4 a = 2
(1, 2) (4, 2) 3 c = 3
b2 = c
2 - a
2 b
2 = 3
2 - 2
2 = 9 - 4 = 5 b = 5
X y = 2
1b
ky
a
hx2
2
2
2)()(
15
2y
2
1x2
2
2
2
)(
)()(
15
2y
4
1x22
)()(
5(x - 1)2 - 4(y - 2)
2 = 20
5(x2 - 2x + 1) - 4(y
2 - 4y + 4) = 20
5x2 - 10x + 5 - 4y
2 + 16y - 16 = 20
5x2- 4y
2 - 10x + 16y - 31 = 0
(2) (-2, -6) (-2, 4) 6
(-2, -6) (-2, 4) (-2, -1)
(-2, -1) h = -2 , k = -1 6 2a = 6 a = 3
(-2, -1) (-2, 4) 5 c = 5
b2 = c
2 - a
2 b
2 = 5
2 - 3
2 = 25 - 9 = 16 b = 4
Y x = -2
1b
hx
a
ky2
2
2
2)()(
14
2x
3
1y2
2
2
2)()(
16(y + 1)2 - 9(x + 2)
2 = 144
16(y2 + 2y + 1) - 9(x
2 + 4x + 4) = 144
16y2 + 32y + 16 - 9x
2 - 36x - 36 = 144
16y2- 9x
2 + 32y - 36x - 164 = 0
(3) (2, -3) (2, -1) (2, 2)
(2, -3) h = 2, k = -3 (2, -1)
Y x = 2
(2, -3) (2, -1) 2 a = 2
(2, -3) (2, 2) 5 c = 5
b2 = c
2 - a
2 b
2 = 5
2 - 2
2 = 25 - 4 = 21 b = 21
1b
hx
a
ky2
2
2
2)()(
121
2x
4
3y22
)()(
21(y + 3)2 - 4(x - 2)
2 = 84
21y2 + 126y + 189 - 4x
2 + 16x - 16 = 84
21y2 - 4x
2 + 126y + 16x + 89 = 0 sm.tm
O
C (1, 2)
A(2, -1)
A (2, -5)
C(2, -3)
X
X
F (2, -8)
F(2, 2)
Y Y
(-1, 2) A A(3, 2) O
B(1, 2 + 5 )
B (1, 2 - 5 )
X
X
(-2, 2) F F(4, 2)
Y Y
A (-2, -4)
X
F (-2, -6)
F(-2, 4)
X
A(-2, 2)
B (-6, -1) C(-2, -1) B(2, -1)
Y Y
⌦
127127
⌦
(4) (-1, 3) (1, 3) 4
(-1, 3) (1, 3) (0, 3)
(0, 3) h = 0, k = 3 4 2b = 4 b = 2
(0, 3) (1, 3) 1 a = 1
b2 = c
2 - a
2 c
2 = a
2 + b
2 = 1
2 + 2
2 = 5 c = 5
X y = 3
1b
ky
a
hx2
2
2
2)()(
12
3y
1
0x2
2
2
2)()(
x 2 - 1
4
3y2
)(
4x2 - (y - 3)
2 = 4
4x2 - (y
2 - 6y + 9) = 4
4x2 - y
2 + 6y - 9 = 4
4x2 - y
2 + 6y - 13 = 0
(5) (-4, 2) (-4, 6) X
(-4, 2) h = -4 , k = 2 (-4, 6)
(-4, 2) (-4, 6) 4 c = 4
X (-4, 0) (-4, 2) (-4, 0) 2 a = 2
b2 = c
2 - a
2 b
2 = 4
2 - 2
2 = 16 - 4 = 12 b = 12 = 2 3
Y x = -4
1b
hx
a
ky2
2
2
2)()(
112
4x
4
2y22
)()(
3(y - 2)2 - (x + 4)
2 = 12
3(y2 - 4y + 4) - (x
2 + 8x + 16) = 12
3y2 - 12y + 12 - x
2 - 8x - 16 = 12
3y2- x
2 - 12y - 8x - 16 = 0
(6) (-1, 5) (-1, -1) (3, 2)
(-1, 5) (-1, -1) (-1, 2)
(-1, 2) h = -1 , k = 2
X y = 2
(-1, 2) (3, 2) 4 a = 4
(-1, 2) (-1, 5) 3 b = 3
c2 = a
2 + b
2 c
2 = 4
2 + 3
2 = 16 + 9 = 25 c = 5
1b
ky
a
hx2
2
2
2)()(
19
2y
16
1x22
)()(
9(x2 + 2x + 1) - 16(y
2 - 4y + 4) = 144
9x2 + 18x + 9 - 16y
2 + 64y - 64 = 144
9x2 - 16y
2 + 18x + 64y - 199 = 0
sm.tm
A(-4,4)
A(3, 2) B (-1, -1) O
C(-1, 2)
X X
A (-5, 2)
F (-6, 2) F(4, 2)
B(-1, 5)
Y Y
B (-4-2 3 ,2) B(-4+2 3 ,2)
F(-4, 6 )
F (-4 , -2)
O
X
A (-4, 0)
C(-4,2)
X
Y Y
O
F (- 5 , 3) C(0,3) F( 5 , 3)
B(0, 5 )
B (0,1)
X
XA (-1, 3) A(1, 3)
Y
⌫ ⌫ ⌦
⌫
128128
6
eccentricity e
(1) 4x2 - y
2 + 6y - 13 = 0
4x2 - y
2 + 6y - 13 = 0
4x2 - (y
2 - 6y) = 13
4x2 - (y
2 - 6y + 9) = 13 - 9
4x2 - (y - 3)
2 = 4
1
x2
- 14
3y2
)(
h = 0 , k = 3 a2 = 1 a = 1, b
2 = 4 b = 2
b2 = c
2 - a
2 c
2 = a
2 + b
2 = 1 + 4 = 5 c = 5
1. X y = 3
2. C(h, k) = C(0, 3)
3. F( h + c, k) = F(0 + 5 , 3) = F( 5 , 3) F (h - c, k) = F (0 - 5 , 3) = F (- 5 , 3)
4. A( h + a, k) = A(0 + 1, 3) = A(1, 3) A (h - a, k) = A (0 - 1, 3) = A (-1, 3)
5. B( h, k + b) = B(0, 3 + 2) = B(0, 5) B (h, k - b) = B (0, 3 - 2) = B (0, 1)
6. (asymptotes) y - 3 = 2x 2x + y - 3 = 0 2x - y + 3 = 0
7. a
b22
= 1
42 )( = 8
8. (eccentricity) e = a
c =1
5 = 5
(2) 3y2- x
2 - 12y - 8x - 16 = 0
3y2- x
2 - 12y - 8x - 16 = 0
(3y2 - 12y) - (x
2 + 8x) = 16
3(y2 - 4y + 4) - (x
2 + 8x + 16) = 16 + 12 - 16
3(y2 - 4y + 4) - (x
2 + 8x + 16) = 12
3(y - 2)2 - (x + 4)
2 = 12
112
4x
4
2y22
)()(
h = -4 , k = 2 a2 = 4 a = 2 , b
2 = 12 b = 12 = 2 3
b2 = c
2 - a
2 c
2 = a
2 + b
2 = 4 + 12 = 16 c = 4
1. Y x = -4
2. C(h, k)
3. F( h, k + c) = F(-4, 2 + 4) = F(-4, 6) F (h , k - c) = F (-4, 2 - 4) = F (-4, -2)
4. A( h, k + a) = A(-4, 2 + 2) = A(-4, 4) A (h, k - a) =A (-4, 2 - 2) =A (-4, 0)
5. B( h + b, k) = B(-4 +2 3 , 2) B (h - b, k) = B (-4-2 3 , 2)
6. (asymptotes) y - 2 = 32
2 (x + 4) y - 2 = 3
1 (x + 4)
x + 3 y + 4 - 2 3 = 0 x - 3 y + 4 + 2 3 = 0
3 x + 3y + 4 3 - 6 = 0 3 x - 3y + 4 3 + 6 = 0
7. a
b22
= 2
122 )( = 12
8. eccentricity e = a
c = 24
= 2
sm.tm
X X
O
B (-4-2 3 ,2) B(-4+2 3 ,2)
F(-4, 6 )
F (-4 , -2) A (-4, 0)
A(-4, 4)
C(-4,2)
x + 3 y + 4 - 2 3 = 0
x - 3 y + 4 + 2 3 = 0
Y Y
O
F (- 5 , 3) C (0,3) F( 5 , 3)
B(0, 5 )
B (0,1)
X
X
2x + y - 3 = 0 2x - y + 3 = 0
Y
⌦
129129
⌦
(3) 5x
2- 4y
2 - 10x + 16y - 31 = 0
5x2- 4y
2 - 10x + 16y - 31 = 0
(5x2 - 10x) - (4y
2 - 16y) = 31
5(x2 - 2x) - 4(y
2 - 4y) = 31
5(x2 - 2x + 1) - 4(y
2 - 4y + 4) = 31+ 5 - 16
5(x2 - 2x + 1) - 4(y
2 - 4y + 4) = 20
5(x - 1)2 - 4(y - 2)
2 = 20
15
2y
4
1x22
)()(
h = 1, k = 2 a2 = 4 a = 2 , b
2 = 5 b = 5
b2 = c
2 - a
2 c
2 = a
2 + b
2 = 4 + 5 = 9 c = 3
1. X y = 2
2. C(h, k) = C(1, 2)
3. F( h + c, k) = F(1 +3, 2) = F(4, 2) F (h - c, k) = F (1 -3, 2) = F (-2, 2)
4. A( h + a, k) = A(1 + 2, 2) = A(3, 2) A (h - a, k) = A (1 - 2, 2) = A (-1, 2)
5. B( h, k + b) = B(1, 2 + 5 ) B (h, k - b) = B (1, 2 - 5 )
6. (asymptotes) y - 2 = 2
5 (x - 1)
5 x + 2y - 4 - 5 = 0 5 x - 2y + 4 - 5 = 0
7. a
b22
= 2
52 )( = 5
8. eccentricity e = a
c = 2
3
(4) 16y2- 9x
2 + 32y - 36x - 164 = 0
16y2- 9x
2 + 32y - 36x - 164 = 0
(16y2 + 32y) - (9x
2 + 36x) = 164
16(y2 + 2y ) - 9(x
2 + 4x ) = 164
16(y2 + 2y + 1) - 9(x
2 + 4x + 4) = 164 + 16 - 36
16(y + 1)2 - 9(x + 2)
2 = 144
116
2x
9
1y22
)()(
h = -2 , k = -1 a2 = 9 a = 3, b
2 = 16 b = 4
b2 = c
2 - a
2 c
2 = a
2 + b
2 = 9 + 16 = 25 c = 5
1. Y x = -2
2. C(h, k) = C(-2, -1)
3. F( h, k + c) = F(-2, -1 + 5) = F(-2, 4) F (h , k - c) = F (-2, -1 - 5) = F (-2, -6)
4. A( h, k + a) = A(-2, -1 + 3) = A(-2, 2) A (h, k - a) =A (-2, -1 - 3) =A (-2, -4)
5. B( h + b, k) = B(-2 + 4, -1) = B(2, -1) B (h - b, k) = B (-2 -4, -1) = B (-6, -1)
6. (asymptotes) y + 1 = 4
3 (x + 2)
3x + 4y + 10 = 0 3x - 4y + 2 = 0
7. a
b22
= 3
162 )( = 3
32
8. (eccentricity) e = a
c = 3
5
sm.tm
(-1, 2) A A(3, 2)
C (1, 2)
A(-2, 2)
A (-2, -4)
X
F (-2, -6)
F(-2, 4)
X
3x + 4y + 10 = 0
3x - 4y + 2 = 0
B (-6, -1) C(-2, -1) B(2, -1)
Y Y
O
B(1, 2 + 5 )
B (1, 2 - 5 )
X
X
(-2, 2) F F(4, 2)
5 x + 2y - 4 - 5 = 0 5 x - 2y + 4 - 5 = 0 Y Y
⌫ ⌫ ⌦
⌫
130130
(5) 4x2- 4y
2 + 8x + 16y - 21 = 0
4x2- 4y
2 + 8x + 16y - 21 = 0
(4x2+ 8x) - (4y
2 - 16y) = 21
4(x2+ 2x) - 4(y
2 - 4y) = 21
4(x2+ 2x + 1) - 4(y
2 - 4y + 4) = 21 + 4 - 16 C(-1, 2)
4(x + 1)2 - 4(y - 2)
2 = 9
19
2y4
9
1x422
)()(
1
4
9
2y
4
9
1x22
)()(
h = -1, k = 2 a2 = 49 a =
2
3 , b2 = 49 b =
2
3 c2 = a
2 + b
2 = 49 + 49 =
418 =
2
9 c = 2
23
1. X y = 2
2. C(h, k) = C(-1, 2)
3. F( h + c, k) = F(-1 +2
23 , 2) F (h - c, k) = F (-1 -2
23 , 2)
4. A( h + a, k) = A(-1 +2
3 , 2) = A(2
1 , 2) A (h - a, k) = A (-1 - 2
3 , 2) = A (-2
5 , 2)
5. B( h, k + b) = B(-1, 2 +2
3 ) = B(-1, 2
7 ) B (h, k - b) = B (-1, 2 - 2
3 ) = B (-1, 2
1 )
6. (asymptotes) y - 2 = (x + 1) x + y - 1 = 0 x - y + 3 = 0
7. a
b22
= 24
92 )(
= 4
9
8. eccentricity e = a
c = (
2
23)(3
2) = 2
(6 5x2- 4y
2 - 30x - 16y + 49 = 0
5x2- 4y
2 - 30x - 16y + 49 = 0
(5x2- 30x) - (4y
2 + 16y) = -49
5(x2- 6x) - 4(y
2 + 4y) = -49
5(x2- 6x + 9) - 4(y
2 + 4y + 4) = -49 + 45 - 16
5(x - 3)2 - 4(y + 2)
2 = -20
4(y + 2)2 - 5(x - 3)
2 = 20
14
3x
5
2y22
)()(
h = 3 , k = -2 a2 = 5 a = 5 , b
2 = 4 b = 2
b2 = c
2 - a
2 c
2 = a
2 + b
2 = 5 + 4 = 9 c = 3
1. Y x = 3
2. C(h, k) = C(3, -2)
3. F( h, k + c) = F(3, -2 + 3) = F(3, 1) F (h , k - c) = F (3, -2 -3) = F (3, -5)
4. A( h, k + a) = A(3, -2 + 5 ) A (h, k - a) =A (3, -2 - 5 )
5. B( h + b, k) = B(3 + 2, -2) = B(5, -2) B (h - b, k) = B (3 - 2, -2) = B (1, -2)
6. (asymptotes) y + 2 = 2
5(x - 3) 5 x + 2y + 4 -3 5 = 0 5 x - 2y - 4 -3 5 = 0
7. a
b22
= 5
42 )( = 5
58
8. eccentricity e = a
c =
5
3 =
5
53
A (3, -2 - 5 )
A(3, -2 + 5 )
F (3, -5)
F(3, 1)
X
Y Y
B (1, -2) C(3, -2) B(5, -2)
X
5 x + 2y + 4 -3 5 = 0 5 x - 2y - 4 -3 5 = 0
X
X
F(-1+2
23 , 2) F (-1 -2
23 , 2)
A (-2
5 , 2) A(
2
1 , 2)
B (-1, 2
1 )
B(-1, 2
7 )
x + y - 1 = 0 Y Y
x - y + 3 = 0
⌦
131131
⌦
(Rectangular Hyperbola)
x2- y
2 = 4 ,
15
y
5
x22
, 19
3y
9
2x22
)()(, xy = 7, xy = -10 , xy – y = 5, xy – x = -3 , (x- 2)(y + 3) = 5
xy = a a 0
1. ( 1 3)
2. (0,0)
3. y = x
4. ( a , a ) (- a ,- a )
5. ( a2 , a2 ) (- a2 ,- a2 )
6. X Y
xy = 9
1.
2. (0,0)
3. y = x
4. ( a , a ) = ( 9 , 9 ) = (3,3)
(- a ,- a ) = (- 9 ,- 9 ) = (-3,-3)
5. ( a2 , a2 ) = ( )9(2 , )9(2 )= (3 )23,2
(- a2 , - a2 ) = (- )9(2 ,- )9(2 ) = (- ,23 - 23 )
6. X Y
xy = a a 0
1. ( 2 4)
2. (0,0)
3. y = -x
4. ( a ,- a ) (- a , a )
( || a ,- || a ) (- || a , || a )
5. ( a2 ,- a2 (- a2 , a2 )
( a2 ,- a2 ) (- a2 , a2 )
6. X Y
xy = -4
1.
2. (0,0)
3. y = -x
4. ( a ,- a ) = ( 4 ,- 4 ) = (2,-2)
(- a , a ) = (- 4 , 4 ) = (-2,2)
5. ( a2 ,- a2 ) = ( 8,- 8) = (2 2 , -2 2 )
(- a2 , a2 ) = (- 8, 8) = (-2 2 , 2 2 )
6. X Y
sm.tm.
(-2,2) ( 2,-2) O (2 2 , -2 2 )
(-2 2 , 2 2 )
Y
X
y = -x
O X
y = -x
Y
O
Y
y = x
X
O
(3, 3)
(-3, -3)
(- ,23 - 23 )
Y
(3 )23,2
y = x
X
⌫ ⌫ ⌦
⌫
132132
(x - h)(y - k) = a a 0
(h, k)
X ( y = k) Y ( x = h)
x y = a a 0
(h, k)
(1) (x - 2)(y + 3) = 7
(x - 2)(y + 3) = 7
(2, -3)
X ( y = -3) Y ( x = 2)
x y = 7
(2, -3)
(2) xy - 6x = 1
xy - 6x = 1 x(y - 6) = 1
(0, 6)
X ( y = 6) Y( x = 0)
x y = 1
(0, 6)
(3) xy - 2y - 4x = 0
xy - 2y - 4x = 0
y(x - 2) - 4x = 0
y(x - 2) - 4(x - 2) = 8
(x - 2)(y - 4) = 8
C(2, 4)
X ( y = 4) Y ( x = 2)
x y = 8
(2, 4)
sm.tm.
O X
X C(2, 4)
Y Y
O X
X C(0, 6)
Y
Y Y
O X
X
C(2, -3)
O
X
X
C(h, k)
Y Y
⌦
133133
⌦
(x - h)(y - k) = a a 0
(h, k)
X ( y = k) Y ( x = h)
x y = a a 0
(h, k)
(1) (x - 2)(y + 3) = -4
(x - 2)(y + 3) = -4
(2, -3)
X ( y = -3) Y ( x = 2)
x y = -4
(2, -3)
(2) xy - 6x = -3
xy - 6x = -3 x(y - 6) = -3
(0, 6)
X ( y = 6) Y( x = 0)
x y = -3
(0, 6)
(3) xy + 3y - 4x - 11 = 0
xy + 3y - 4x - 11 = 0
y(x + 3) - 4x = 11
y(x + 3) - 4(x + 3) = 11 - 12
y(x + 3) - 4(x + 3) = -1
(x + 3)(y - 4) = -1
C(-3, 4)
X ( y = 4) Y ( x = -3)
x y = -1
(-3, 4)
sm.tm.
O X
X C(-3, 4)
Y Y
O
C(0, 6)
X
X
Y
O X
X
C(2, -3)
Y Y
O
X
X
C(h, k)
Y Y
⌫ ⌫ ⌦
⌫
134134
1.
(1) { (x, y) | 2x = 3y }
........................................................................................
(2) { (x, y) | x2 + y
2 = 1 }
........................................................................................
(3) { (x, y) | x2 - y
2 = 1 }
........................................................................................
(4) { (x, y) | 2y2 = 3x + 5 }
........................................................................................
(5) { (x, y) | 2x = y
3}
........................................................................................
(6) {(x, y) | 2x2 = 3y
2 - 12 }
........................................................................................
(7) { (x, y) | 2x = 3y + xy }
........................................................................................
(8) { (x, y) | 9y2 - x
2 - 45 = 0 }
........................................................................................
(9) { (x, y) | 5x2 + 10x + y
2 = 100 }
........................................................................................
(10) { (x, y) | 5
x2
= y }
........................................................................................
2.
(1) (-10, 0) (10, 0) 16
.......................................................................................................................................................................................................
.......................................................................................................................................................................................................
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.......................................................................................................................................................................................................
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(2) (0, -4) (0, 4) 4
.......................................................................................................................................................................................................
.......................................................................................................................................................................................................
.......................................................................................................................................................................................................
.......................................................................................................................................................................................................
.......................................................................................................................................................................................................
.......................................................................................................................................................................................................
(3) (-3, 0) (3, 0) (-5, 0) (5, 0)
.......................................................................................................................................................................................................
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(4) (0, -6) (0, 6) 10
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............................................................................................................................................................................................ sm.tm
ใบงานที ่1.11
⌦
135135
⌦
(5) (0, - 10 ) (0, 10 ) (2, 5)
.......................................................................................................................................................................................................
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.......................................................................................................................................................................................................
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(6) (-3, 0) (3, 0) e = 3
.......................................................................................................................................................................................................
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......................................................................................................................................................................................................
.......................................................................................................................................................................................................
.......................................................................................................................................................................................................
(7) (0, 0) (0, -4) (0, -5)
.......................................................................................................................................................................................................
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.................................................................................................................................................................................................... .
......................................................................................................................................................................................................
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(8) (0, 0) X 8 6
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.......................................................................................................................................................................................................
(9) (0, -4) (0, 4) e = 2
3
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.............................................................................................................................................................................................sm.tm
⌫ ⌫ ⌦
⌫
136136
3.
(1) 116
y
25
x22
.......................................................................................................................................................................................................
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(2) 125
x
9
y22
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(3) 116
x
9
y22
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.............................................................................................................................................................................................sm.tm
⌦
137137
⌦
(4) 4x2 - 9y
2 = 36
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(5) y2 - 4x
2 - 100 = 0
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(6) 5x2 - 2y
2 + 4 = 0
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...........................................................................................................................................................................................sm.tm
⌫ ⌫ ⌦
⌫
138138
3.
(1) (-3, 2) (3, 2) 4
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(2) (-1, -1) (7, -1) 6
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(3) (2, 3) (6, 3) (7, 3)
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............................................................................................................................................................................................sm.tm
⌦
139139
⌦
(4) (4, -5) (4, -2) 4
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(5) (4, 3) (6, 3) (-6, 3)
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(6) (-1, -3) (-1, 5) (-3, -5)
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..........................................................................................................................................................................................sm.tm
⌫ ⌫ ⌦
⌫
140140
4.
(1) 116
1y
25
2x22
)()(
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(2) 25(y + 3)2 - 9(x - 1)
2 - 225 = 0
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(3) 9x2 - 16y
2 - 36x - 96y + 36 = 0
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............................................................................................................................................................................................sm.tm
⌦
141141
⌦
(4) 4y2 - 100x
2 + 1000x - 24y = 2864
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(5) x2 - 4y
2 + 2x + 16y + 1 = 0
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(6) 9x2 - 4y
2 - 54x - 16y + 29 = 0
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.............................................................................................................................................................................................sm.tm
⌫ ⌫ ⌦
⌫
142142
(7) 9y2 - 16x
2 + 32x - 36y - 124 = 0
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(8) 7x2 - 9y
2 + 56x - 18y + 166 = 0
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(9) 7x2 - 9y
2 + 28x - 72y - 179 = 0
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............................................................................................................................................................................................sm.tm.
⌦
143143
⌦
5.
(1) xy = 16
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(2) x(y + 3) = -8
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(3) (x + 2)(y - 5) = -4
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(4) xy + 5y + 3x = 0
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(5) xy - 3x + 2y + 1 = 0
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⌫ ⌫ ⌦
⌫
144144 144
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:
⌦
145145
⌦
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