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371.95

691

: 2551

151

ISBN 978-974-559-198-1

1. -

2. - 3.

. 30 / 2551

1 2551

1,000

99/20 10300

0-2668-7974 0-2668-7123 2530

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580 8 . 34 1

. 10230

0-2943-8373-4 0-2510-7753

. .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

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11

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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

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⌫

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)

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⌫

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


⌦

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

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⌫

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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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)


⌦

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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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


⌦

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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⌦

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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⌦

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


⌦

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 ……………………………….


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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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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

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3939

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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


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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

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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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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

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4545

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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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.………………………………………………………………………………………………………………….

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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)

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4747

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(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

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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)

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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

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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

.......................................................................................................................................................................................................

.......................................................................................................................................................................................................

(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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5151

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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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⌫

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

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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

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⌫

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

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2 ,2

1 ) 3

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(7) (3

1 ,3

2 ) 2

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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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⌦

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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⌦

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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⌫ ⌫ ⌦

⌫

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))

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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

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⌫

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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⌫ ⌫ ⌦

⌫

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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.............................................................................................................................................................................................sm.tm

⌦

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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⌫ ⌫ ⌦

⌫

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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⌦

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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⌫ ⌫ ⌦

⌫

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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⌦

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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⌫ ⌫ ⌦

⌫

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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⌦

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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⌫ ⌫ ⌦

⌫

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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⌦

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

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(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

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a b 0 b2 = a

2 - c

2

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(h, k - c) (h, k + a) (h, k - a) 1a

ky

b

hx2

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a b 0 b2 = a

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2

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1.

2. 1.10

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4.

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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

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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 -

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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

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a4 - 2a

2cx + c

2x

2 = a

2(x

2 - 2cx + c

2 + y

2)

a4 - 2a

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2x

2 = a

2x

2 - 2a

2cx + a

2c

2 + a

2y

2

a2x

2 - c

2x

2 + a

2y

2 = a

4 - a

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2

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2)

x

2 + a

2y

2 = a

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2 - c

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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)

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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

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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


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

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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

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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

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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

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a

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2

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x = x - h y = y - k

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1b

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a

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2

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2 a b 0 b

2 = a

2 - c

2

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(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

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b

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2

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x = x - h y = y - k

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b

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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

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2

2

2

1

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)2y(

4

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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

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b

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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

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a

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2

2

2

12

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3

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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

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b

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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)

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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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(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

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⌫

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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(10) (0, -4) (0, 4) e = 3

2

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⌦

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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⌫ ⌫ ⌦

⌫

108108

(4) 4x2 + 9y

2 = 36

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(5) x2 + 4y

2 - 100 = 0

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(6) 5x2 + 2y

2 - 4 = 0

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⌦

109109

⌦

3.

(1) (-3, 2) (3, 2) 10

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⌫

110110

(4) (6, 3) (-2, 3) 10

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111111

⌦

(7) (-5, 2) (1, 2) (-2, 6)

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⌫

112112

4.

(1) 116

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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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⌦

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.

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⌫ ⌫ ⌦

⌫

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


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


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


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


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


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


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


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


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


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


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


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


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


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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(2) (0, -4) (0, 4) 4

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(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


⌦

135135

⌦

(5) (0, - 10 ) (0, 10 ) (2, 5)

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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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(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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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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(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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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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(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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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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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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