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E-learning

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ii

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1

1.1

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2

1.2

P (x1, y1) Q Q(x2, y2)

P (x1, y1) Q(x2, y2)

P Q

1.1.

1. P (3, 2) Q(1, −1)

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1.2 3

2. P (3, 0) Q(6, 0)

3. P (1, −2) Q(1, 2)

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4

1.3

P (x1, y1) Q(x2, y2) x1 = x2

m P Q

m = y2 − y1

x2 − x1

= tan θ

θ X 0◦ ≤ θ < 180◦

1. X (θ = 0◦

)

2. X (0◦ < θ < 90◦)

3. Y (θ = 90◦)

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1.3 5

4. X

(90◦ < θ < 180◦

)

1.

2. ( Y)

3. ( Y) −1

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6

1.4 1. X Y (0, b)

y = b

2. Y X (a, 0)

x = a

3. X Y

• (x1, y1) m

y − y1 = m(x − x1)

• m Y (0, c)

y = mx + c

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1.4 7

4.

Ax + By + C = 0

A, B C A B

Ax + By + C = 0

By = −Ax − C

y = −A

Bx − C

B

Ax + By + C = 0 − A

B Y −C

B

X (a, 0) a X

(x-intercept) Y (0, b) b Y

(y-intercept)

1.2. Y

3x − 2y − 6 = 0

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8

1.5

l Ax + By + C = 0 P (x1, y1)

l d P

l

d = |Ax1 + By1 + C |

√ A2

+ B2

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1.5 9

1.3. ( A-NET )

l1 (5, 2) (1, −6) l2 (3, −1)

−1 (a, b) a+b

1. − 2 2. − 1 3. 1 4. 2

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10

1.6

1.4. (circle)

(center) (radius)

1.6.1

• (0, 0) r

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1.6 11

x2 + y2 = r2

• (h, k) r

(x

−h)2 + (y

−k)2 = r2

(3, −2) 5

(x − 3)2 + (y + 2)2 = 52

x2 + y2 − 6x + 4y − 12 = 0

•

x2 + y2 + Ax + By + C = 0

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12

x2 + y2 + Ax + By + C = 0

(x2 + Ax) + (y2 + By) = −C

(x2 + 2(A

2 )x +

A2

4 ) + (y2 + 2(

B

2 )y +

B2

4 ) = −C +

A2

4 +

B2

4

(x +

A

2 )2

+ (y +

B

2 )2

= −C +

A2

4 +

B2

4

x2 + y2 + Ax + By + C = 0

(−A

2, −B

2 )

– −C + A2

4 +

B2

4 > 0

r =

−C +

A2

4 +

B2

4

– −C + A2

4 +

B2

4 = 0

r = 0 ()

– −C +A2

4 +

B2

4 < 0

(imaginary circle)

1.5.

x2 + y2 − 12x + 10y + 45 = 0

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1.6 13

1.6.2

1. X

(h, k) X |k| (h, 0)

2. Y

(h, k) Y |h| (0, k)

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14

3. X Y

(h, k) Ax + By + C = 0 r = |Ah + Bk + C |√

A2 + B2

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1.6 15

1.6. ( A-NET )

a ax + 12y + 15 = 0

x2 + y2 − 14x + 4y + 49 = 0 a 1. (0, 4] 2. (4, 8] 3. (8, 12] 4. (12, 16]

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16

1.7. ( A-NET )

l x2 − 2x + y2 + 10y − 39 = 0 (2, 3) l

1. (4, −43

8 ) 2. (5, −9

2)

3. (2, −13) 4. (3, −11)

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1.7 17

1.7 1.8. (parabola)

(focus)

(directrix)

(parabola axis)

(axis of symmetry) (latus rectum)

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18

(0, 0)

(x)2 = 4c(y) (y)2 = 4c(x)

(0, 0)

c > 0 c < 0 c > 0 c < 0

(0, c) (c, 0)

y = −c x = −c

x = 0 y = 0

|4c|

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1.7 19

(h, k)

(x − h)2 = 4c(y − k) (y − k)2 = 4c(x − h)

(h, k)

c > 0 c < 0 c > 0 c < 0

(h, k + c) (h + c, k)

y = k − c x = h − c

x = h y = k

|4c|

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20

x2 + Ax + By + C = 0

y2 + Ax + By + C = 0

x2 + Ax + By + C = 0

x2 + 2(A

2 ) +

A2

4 = −By − C +

A2

4

(x + A

2 )2 =

−B(y

−C − A2

4

−B )

(x + A

2 )2 = 4(

−B

4 )(y +

C − A2

4B

)

x2+Ax+By+C = 0 (−A2

, −C − A2

4B

)

B

4 < 0

B

4 > 0

y2 + Ax + By + C = 0

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1.7 21

1.9. (3,

−8)

y = −2

1.10.

y2 + 2y + 8x + 25 = 0

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22

1.11. ( 2548 )

y2 − 4y − 16x − 12 = 0

l

3x − 2y + 5 = 0

y l

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1.7 23

1.12. ( A-NET )

y = 5 x2 + y2 − 6x + 2y − 6 = 0 1. x2 − 6x + 12y − 15 = 0 2. x2 − 6x − 12y + 33 = 0

3. x2 − 6x + 12y + 21 = 0 4. x2 − 6x − 12y − 3 = 0

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24

1.8

1.13. (Ellipse) 2

2 F F

P F F

(vertex) A

A

A A

(major axis) C

(center)

B B

(minor axis) a

b c

a2 = b2 + c2

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1.8 25

P (x, y) |P F | + |P F | = |2a| (0, 0)

x2

a2 +

y2

b2 = 1 a > b

x2

b2 +

y2

a2 = 1 a > b

(0, 0) (0, 0)

(±a, 0) (0, ±a)

(±c, 0) (0, ±c)

(0, ±b) (±b, 0)

|2a| |2a|

|2b| |2b|

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26

(h, k)

(x

−h)2

a2 +

(y

−k)2

b2 = 1

(x

−h)2

b2 +

(y

−k)2

a2 = 1

(h, k) (h, k)

(h ± a, k) (h, k ± a)

(h ± c, k) (h, k ± c)

(h, k ± b) (h ± b, k)

|2a| |2a|

|2b| |2b|

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1.8 27

Ax2 + By2 + Dx + Ey + F = 0

A B

. A > B > 0

. B > A > 0

1.14.

16x2 + 25y2 + 128x − 50y − 119 = 0

1.15. (5, 2) (5, −2) 12

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28

1.16. ( A-NET )

x = y (x − 1)2

9 + (y − 1)2

4 = 1 A

B F 1 F 2 AF 1 + AF 2 + BF 1 + BF 2

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1.8 29

1.17. ( A-NET )

C x2 + 2y2 + 4x − 4y + 2 = 0 C y = −x A B AB

1.√

3

2.

√ 5

3.

√ 6 4.

√ 8

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30

1.18. ( A-NET )

X 4x2 + 3y2 − 16x + 4 = 0

1. y2 = x 2. y2 = 4x

3. 2y2 = x 4. 8y2 = x

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1.9 31

1.9 1.19. (Hyperbola) 2

P P F P

F A A

A

A

C C

AA AA

BB

BB

(conjugate axis) |2c| |2a| |2b|

c2 = b2 + a2

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32

(0, 0)

(x)

2

a2 − (y)

2

b2 = 1 y

2

a2 − x

2

b2 = 1

(0, 0) (0, 0)

(±a, 0) (0, ±a)

(±c, 0) (0, ±c)

(0, ±b) (±b, 0)

|2a

| |2a

||2b| |2b|

asymtotex

a ± y

b = 0

y

a ± x

b = 0

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1.9 33

(h, k)

(x − h)2

a2 − (y − k)

2

b2 = 1 (y − k)

2

a2 − (x − h)

2

b2 = 1

(h, k) (h, k)

(h ± a, k) (h, k ± a)

(h ± c, k) (h, k ± c)

(h, k ± b) (h ± b, k)

|2a| |2a|

|2b| |2b| asymtote

x − h

a ± y − k

b = 0

y − k

a ± x − h

b = 0

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34

Ax2 + By2 + Cx + Dy + E = 0

A B

. A > 0 B < 0

. A < 0 B > 0

1.20.

3x2 − y2 + 18x + 4y + 20 = 0

1.21. (2, −1) (2, 3) 8

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1.9 35

1.22. ( A-NET )

9x2 − 16y2 − 18x − 64y − 199 = 0

3x + 4y − 8 = 0

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36

1.23. F 1 F 2 16y2−9x2+36x−32y−164 = 0 V x2+y2+2x−4y+1 = 0 F 1F 2V

1. 13 + 2√

5

2. 13 + 3√

5

3. 15 + 2√ 54. 15 + 3

√ 5

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1.9 37

1.24. ( A-NET )

F 6x2−10y2−12x−40y−94 = 0

F

1. y2 + 4y−

4x−

9 = 0

2. y2 + 4y − 4x + 24 = 0

3. y2 + 4y − 16x − 44 = 0

4. y2 + 4y − 16x + 84 = 0

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38

1.25. ( A-NET )

(−4, 0) (−5, 0) (1, 0)

1. (x + 2)2

4 +

y2

5 = 1

2. (x + 2)2

5 +

y2

9 = 1

3. (x − 2)2

4 +

y2

9 = 1

4. (x

−2)2

4 +

y2

5 = 1

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1.9 39

( A-NET)

1. (5,-1) y =

x

2 X

1. y = 7 2. y = 7

2 3. y = 5

2 4. y = 6

2. y = 5 x2 + y2 − 6x + 2y − 6 = 0

1. x

2

−6x+12y−15 = 0 2. x

2

−6x−12y+33 = 03. x2−6x+12y +21 = 0 4. x2−6x−12y−3 = 0

3. 4x2 + 9y2 − 48x + 72y + 144 = 0 3x + 4y = 5

1. 4x − 3y + 12 = 0 2. 4x − 3y − 12 = 03. 4x − 3y − 36 = 0 4. 4x − 3y + 36 = 0

4. E 6x2 + 5y2 + 12x − 20y − 4 = 0 H E E

E H

1. x2 − 5y2 − 2x − 20y + 14 = 0

2. x2 − 5y2 + 2x + 20y − 14 = 0

3. x2 − 5y2 + 2x + 20y − 18 = 0

4. 5x2

−y2

−2x + 20y + 18 = 0

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40

5. (h, k) 2x+3y = 6 h, k 2x − y = 1 2x + 7 = −3 1. [2, 4] 2. [4, 5] 3. [5, 6] 4. [6, 7]

6. x2 + y2 + 2ax + 2by + c = 0 y

2 x 1. a2 > c b2 > c 2. a2 > c b2 < c

3. a2 < c b2 > c 4. a2 < c b2 < c

7. (10, 7)

5x2 + 5y2 − 20x − 10y − 100 = 0

8. A 5x − 4y − 15 = 0 x +

y − 12 = 0 B C 4y2 − 9x2 − 16y + 18x − 29 = 0 ABC

9. P x2 + 8y + 2x + a = 0 a < 0 y = 4 P x A A P

10.

186 162 ( )

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[1] , A-NET ,www.neutron.rmutphysics.com

[2] , . , www.atom.rmutphysics.com

[3] , .4-5-6 , ,.

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