Knowledge Mat13

8/6/2019 Knowledge Mat13

1/24

1

1. 2

2. 2

3.

m : n ( )

4. 1.

2.

3.

2 )y,(xP 111 2

)2

y-1

(y2

)2

x-1

(x=2

P1

P

)y,x( PP )y,(xP )y,(xP 21222111

2

2y

1y

y 2

2x

1x

x

+=

+=

PP )y',(x'P' 2 )y,(xP )y,(xP 21222111

nm

mxnx x' 21 +

+=

nm

myny y' 21 +

+=

1 20


2/24

2

5. -

P1 (x1 , y1) = m

6.

P1 (x1 , y1) Ax + By + C = 0 d

11

33

22

11

yx

yx

yx

yx

ABC =

)yxyxy(x-)yxyxy(x2

1122321133221 ++++=

ABC 4

1PQR =

++++

3

yyy

3

xxx 321321 ,

0 CByAx )x-m(x y-y 1 =++=

0 CByAx )x-m(x y-y 1 =++=

22

11

BA

CByAx d

+++=

2 20


3/24

3

7.

2 Ax + By + C 1 = 0 Ax + By + C 2 = 0 d

8.

Ax + By + C = 0

1. Bx - Ay + C 2 = 0

4. C = 0 ; (0, 0) Ax + By = 0

7.

8. = -1

9. Ax + By + C = 0 kAx + kBy + C 2 = 0

k k 0

22

21

BA

C-C d

+=

BC - yx ; 0 A 2. ==

A

C - xy ; 0 B 3. ==

=

B

C-0, y ; 0 x 5.

= 0 AC

- x ; 0 y 6. ,


4/24

4

c (h, k) (r)

x2

+ y2

= r 2

(0, 0) r

(x + h)2

+ (y k)2

= r 2

(h, k) r

D2

+ E2

4F = 0

D2

+ E2

4F > 0D

2+ E

2 4F < 0

Q P1 (x1 , y1) P1Q

1. x2

+ y2

= r 2

2

4F-ED

2

E

2

D- 0 FEyDxyx

2222 +

=++++ ,

22

1

2

11 r -yx QP +=3 20


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5

2. (x - h)2

+ (y k)2

= r 2

3. x2

+ y2

+ Dx + Ey + F = 0

2

111 r -k)-(yh)-(x QP +=

FEyDxy-(x QP11

2

1

2

11+++=

4 20

5 20


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6

(Focus)

(Latus rectum) Focus

Focus

2 Focus Focul Focus

(Latus rectum)

6 21


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7

(h, k)

(x h)2

= 4c(y k) (y - k)2

= 4c(x h)

v(h, k) v(h, k)

F(h, k + c) FOCUS F(h + c, k)

y = k -c x = h - c

4c 4c

( ) c >0 ( ) ( ) c


8/24

8

(ELLIPSE)

2

(0, 0)

(0, 0) (0, 0)

V(a, 0), V(-a, 0) V(0, a), V(0, -a)

F(c, 0), F(-c, 0) FOCUS F(0, c), F(0, -c)

2a 2a

2b 2b

B(0, b), B(0, -b) B(b, 0), B(-b, 0)

1 b

x

a

y 1

b

y

a

x2

2

2

2

2

2

2

2

=+=+

a

2b a

2b22

9 , 10 21


9/24

9

(h, k)

(h, k) (h, k)

V(h + a, k), V(h - a, k) V(h, k + a), V(h, k - a)

F(h + c, k), F(h - c, k) FOCUS F(h, k + c), F(h, k - c)

B(h, k + b, B(h, k -b) B(h + b, k), B(h - b, k)

2a 2a

2b 2b

b2

= a2

- c2

1 b

h)-(x

a

k)-(y 1

b

k)-(y

a

h)-(x2

2

2

2

2

2

2

2

=+=+

a

2b

a

2b22

11 , 12 22


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10

(0, 0) (0, 0)

V(a, 0), V(-a, 0) V(0, a), V(0, -a)

F(c, 0), F(-c, 0) FOCUS F(0, c), F(0, -c)

B(0, b), B(0, -b) B(b, 0), B(-b, 0)

2a 2a

2b 2b

1 b

x

a

y 1

b

y

a

x2

2

2

2

2

2

2

2

==

a

2b

a

2b22

b

ax Y e)(Asympttol

a

bx Y ==

13 , 14 22


11/24

11

1.

1 0

2. a = b

3. x y

xy = k, k

(h, k)

(h, k) (h, k)

V(h + a, k), V(h - a, k) V(h, k + a), V(h, k - a)

F(h + c, k), F(h - c, k) FOCUS F(h, k + c), F(h, k - c)

B(h, k + b, B(h, k -b) B(h + b, k), B(h - b, k)

2a 2a

2b 2b

a, b, c b2

= a2

- c2

1 b

h)-(x

a

k)-(y 1

b

k)-(y

a

h)-(x2

2

2

2

2

2

2

2

=+=+

h)-(x b

a k -y h)-(x

a

b k -y ==

15 , 16 22


12/24

12

1. I1 (-2, 0) (-1, 2) I2 I 1 x I1 I2

1. 0.5 2. 0.6

3. 0.8 4. 1

2. y2 = 5 + x x + y = 1

1. 25 2. 23

3. 4.

.

3. I1 3x + 4y 12 = 0 I 2 4x 3y + 12 = 0

R P (1, -3) PQ I 1 S PQRS

1. 19 2. 20

3. 21 4. 22

4. C(2, 3) 3x 4y -9 = 0 1. x

2+ y

2+ 4x + 6y + 4 = 0

2. x2

+ y2- 4x + 6y + 4 = 0

3. x2

- y2+ 4x - 6y + 4 = 0

4. x2

+ y2- 4x - 6y + 4 = 0

35 7417 23

18 23


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13

5. 3x + 2y 18 = 0 (2, 6) (-3, 7)

1. x2

+ y2- 2x - 8y - 8 = 0

2. x2

+ y2+ 2x + 8y -92 = 0

3. x2

+ y2

+ 2x - 8y + 4 = 04. x

2+ y

2- 2x + 8y - 14 = 0

6. x2 + y2- 2x - 2y - 2 = 0x

2+ y

2- 6x - 2y + 6 = 0

4.

7. 40 60

( )

1. 10 2. 15

3. 20 4. 30

323

8 1.

33

4 2.

3

4 3.

19 23

20 23

25 24


14/24

14

8. ABCD A B 200 45 A B

5 25

1. 22.5 2. 27.5 3. 35 4. 37.5

9. A B 4y = 12 x2 4x = 12 y

2 |AB |

10. . x2 10x 32y 39 = 0 F a 1 F A(x 1, y1) B(x 2, y2) a 2 C(x 3, y3) D(x 4, y4)

a 1 a 2

1. y 1 + y 2 + y 3 + y 4 + 40

2. y 1 + y 2 + y 3 + y 4 + 8

26 3. 28 4.

40)x-(x)x-(x)y-(y)y-(y 3.2

43

2

21

2

43

2

21++++

8)x-(x)x-(x)y-(y)y-(y 4.2

43

2

21

2

43

2

21++++

23 1. 24 2.

26 24


15/24

15

11. (0, -1) 3x2

+ 4y2

16y + 4 = 0

12. 12x2

+ 16y2

192 = 0

F1 F2 A

B B

3. (-2, 3)

4. (2, 3)

13. (3, 1) (5, 1) y (0, 1)

(-2, 1) 1. x

2+ y

2+ 4x - 2y = 0 2. x

2+ y

2+ 4x - 2y - 5 = 0

3. x2

+ y2+ 4x - 2y - 4 = 0 4. x

2+ y

2+ 4x - 2y - 1 = 0

1 3

2 1.

,

1 2

3 2.

,

1 2

1 3.

,

1 2

1

3.

,

7

9- ,

7

26 1.

7

26- ,

7

9 2.

27 24

28 24


16/24

16

14. 45O x x2 4x + 3y 2- 2 = 0

3. 2

4. 4

15. ABC A x2

y2

= a2 B C

ABC

16. E 6x2

+ 5y2+ 12x - 20y - 4 = 0 H

E E E

H

1. x2

- 5y2- 2x - 20y + 14 = 0

2. x2

- 5y2+ 2x + 20y + 18 = 0

3. x2

- 5y2+ 2x + 20y - 14 = 0

4. 5x2

- y2- 2x + 20y + 18 = 0

17. ( )

9x2

- 16y2- 18x - 64y - 199 = 0

22 1.

242.

a3 1.2

a33 2. 2

a 2

3 3.

2

a 2

33 4.

2

21 23

22 23

23 23


17/24

17

18. 3 3

x, x 2 x

19. (3, 1) (5, 1) y (0, 1)

(-2, 1)

1. x2

+ y2

+ 4x 2y = 0

2. x2

+ y2

+ 4y 2y - 4 = 0

3. x2

+ y2

+ 4x 2y - 1 = 0

4. x2

+ y2

- 4x 2y - 15 = 0

20. H 16x2 64x - 9y

2- 80 = 0 H

H

1. 16x2 64x + 25y

2- 464 = 0

2. 25x2 100x + 16y

2- 464 = 0

3. 16x2 64x + 25y

2 336 = 0

4. 25x2 100x + 16y

2- 336 = 0

21. H 12y2 4x2 + 72y + 16x + 44 = 0 F1 F2 E

H F1 F2 y E X A

B AB

x4

2x

2

1-x

2

1 1.

222 +

x2

2x

2

1-x

2

1 2.

222 +

x4

2x

3

1-x

2

1 3.

222 +

x2

2x

3

1-x

2

1 4.

222 +

8 1. 7 2.

6 3. 5 4.

24 23


18/24

18

22. C y = 1 8(x 2) 2 3x 4y + 5 = 0C C

23. 25x2

+ 16y2

- 100x + 96y - 156 = 0

20

1. 320

2. 640

3. 960

4. 1,280

24. a (10, 7) x2

+ y2

- 4x - 2y - 20 = 0 b

(10, 7) x2

+ y2

- 4x - 2y - 20 = 0 a+b

1. 10

2. 15

3. 20

4. 40

25. 0 3x + 4y 12 = 0 x A y B C1 C2

ABO

C1, C2

)51(0, 1. +

0),22-1( 2.

1)-1,(- 3.

2)-(2, 4.

21 1.

1 2.

3

2 3.

2

3 4.

29 24

30 24

31 24


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19

Entrance

1. 3 2. 3 3. 3 4. 4 5. 3

6. 1 7. 1 8. 2 9. 4 10. 1

11. 1 12. 1 13. 1 14. 1 15. 2

16. 3 17. (96) 18. 1 19. 1 20. 321. 2 22. 23. 1 24. 3 25. 1


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1 2

3 4

5


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21

6

( h,k )

7 8

( 0,0 )

9 10


22/24

22

( h,k )

11 12

13 14

( h,k )

15 16


23/24

23

17

21

18

22

19 23

20 24


24/24

24

25 29

26 30

27 31

28

Knowledge Mat13

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