Η ΕΥΘΕΙΑ ΣΤΟ ΕΠΙΠΕΔΟ

x+y+=0, 0 0

(-)

http://www.perikentro.blogspot.gr/ :

- 1 -

. . .. . . . , , , , , ... , . 17 , - . , . : i. ii.

ernoulli

K

x, y C, C, , . (x , y)=0

C, : A (x1,y1) C ,

(x1 , y1)=0 ( , )=0 , ( , )

C

x

y

x

y

x

y

C

M

N

x1 y1

(,)

( , )

C : (x,y)=0


- 2 -

=

xx

xx xx . , xx 00 .

00 1800 0 .

xx .

: , xx , . xx , xx xx 900, xx , .

900

=00

=900

, .

: . xx : .

//

y

y

y

y

y

y

y

y

y

y


- 3 -

y

y

2

1x x1 2

A

B

12

1

2

.

)(//

(x1,y1) B(x2,y2)

21 xx 12

12

xxyy

: (x1,y1) B(x2,y2) , AB .

)yy,xx( 1212 ,

12

12

xxyy

=. : 12

12

xxyy

x y.

1 2

1 2 , :

2121 // 12121 :

1 ,2 1 2

1 , 2 1 ,2 , :

212121 ////

1212121

=+

1 2

1 2


- 4 -

: y-y0=(x-x0) :

Oxy A(x0,y0 ) . M(x,y) A(x0,y0 ) , AM

, AM . )yy,xx(AM 00 ,

0

0AM xx

yy

. , M(x,y)

0

0

xxyy

y-y0=(x-x0).

A(x0,y0 ).

1: A(x0,y0 ) : x=x0

( )

2:

.

)xx(

xxyy

yy

1

12

12

1

21 xx x=x1 x1=x2

A(x ,y )00

M(x,y)


- 5 -

(0,)

yy (0,) y=x+

, y=x.

xOy xyO

y=x y=-x .

A(x0,y0 ) xx , , y=y0.

Ax+y+=0 0A 0B (1)

, (1) .

: . yy (0,) , y=x+,

x+(-1)y+=0 (x0,y0), x=x0

x+0y+(-x0)=0

, , Ax+y+=0 0A 0B

(0,)

=

=

(0,0)

y

y

450

y=x

y

y

0135

y=-x

y000(x ,y )

00(x ,y )


- 6 -

Ax

+By+

=0

=(,-)

, Ax+y+=0 0A 0B

0B ,

xBA

y ,

BA

yy

,0 .

=0, , , 0A

x ,

xx

0,P .

Ax+y+=0 0A 0B .

Ax+y+=0 ),( .

: Oxy Ax+y+=0 . : 0B ,

BA

),( . =0,

yy

),( .

Ax+y+=0 ),( .

:


- 7 -

00 (x ,y )0

d

x+By+=0

),(

),(

0ABAB)B,A()A,B( .

, Ax+y+=0 )y,x(M 000 . 0 :

22

000

Byx,Md

(x1,y1), B(x2,y2), (x3,y3), . :

A,ABdet21

1: y=x+1 2: y=x+2

2

2121

1),(d

Ax+B

y+=

0

=(,)

1 1

B(x2 y2),

(x ,y )

(x ,y )3 3

1

2

1

2


- 8 -

f(x,y)=0 (c) :

(x,y)

(x0,y0) (c)

. ,

(x,y) :

xx (x,-y)

yy (-x,y)

(-x,-y)

1 3 (y,x)

f(x,y)=f(x,-y) (x,y) f xx

f(x,y)=f(-x,y) (x,y) f yy

f(x,y)=f(-x,-y) (x,y) f

f(x,y)=f(y,x) (x,y) f 1 3 .


- 9 -

,

.

: . xx . . . : 0xx// yy//

0900

0900

21 1=0 2 :

0

()

1 ,2

1 ,2 : 2121 // 12121

x y. , , , ..

, ,

//

>0

=0


- 10 -

() : . . :

y-y0=(x-x0)

)xx(xxyy

yy 112

121

21 xx

x=x1 x1=x2 (,0)

(0,) : 1yx

P(x0,y0) , P(x0,y0) : : x=x0 : y-y0=(x-x0), IR i. : x=x0 ii. () IR

yy x=x0 xx y=y0 yy (0,) y=x+

Ax+y+=0 , , , , , , .

Ax+y+=0 x+y+k=0

Ax+y+=0

x-y+k=0

Ax+y+=0 0 0

A

=0


- 11 -

: 1+2=0, (1) IR : 1 : . , , (1). , (1) . 2 : ( ). , 1 , , .

(1) , .

1 , 2 1 , 2 .

21, :

11//

22//

21,

: |||| 21

21

.

),( 21

),( 12

21

12

21

1

),(

-

2 1


- 12 -

x2 +Bxy +y

2 +x +Ey+Z=0 , . 2 x y ( )

. , , .

(x,y) x y , . , , . , : (x,y) x y . . , .


- 13 -

, , , .

1,2 . (x,y) . : d(M,1)=d(,2)

. 1

2 yy . . ( ) ( )

21

.

1 2

, .

2

2121

1),(d

1: y=x+1 2: y=x+2.

.


- 14 -

1.1. . (1,3) , (2,6), (-5,-15) . (5,-4), (-2,3), (t,1-t), IRt . (1,3), (1,-2), (1,1). 1.2. y=(22 ++1)x++4 y=(2-+4)x+-5 . i. ii. , . iii. // 1.3. , (-3,7) (3,1) 1: 3x-y+2=0 2: 2x+y-7=0 . : i. , xx . ii. iii. . iv. . 1.4. (-3,1) : 4x+y-6=0 i. ii. . 1.5. (-1,2). : x+2y-4=0 : x-y+1=0 1.6. (2,3)

21

.

:

1


- 15 -

2.1. IR : y=(2-10)x+2010 xx 1350. 2.2. (x1,y1) (x2,y2)

y=x+. : 221 1|xx|)( 2.3. (1,2) 1: x-3y+1=0 2: y=1. . 2.4. (1,2) . y=x-5 x=2 . . 2.5. (1,3)

xx yy : 2

2.6. : y=3x+10 , 2. 2.7. : x+y-4=0 (-1,1) . . . . . .

2.8. )3,2(

j4

. (-1,3)

21

3v .

2.9. yy (0,3)

xOy .

:

2


- 16 -

3.1. (2-1)x+(+1)y+(2-3+2)=0 , IR (1) . (1) . . xx yy ; . ; 3.2. (2+2)x-(+2)y+5=0 . . . , (1,2). 3.3. (3-1)x+(-1)y +4-8=0. : . IR . IR.

3.4. ( (x)1

01y)1

, .

3.5. x-y+1=0 (+1)x-(1-)y+4=0 . IR . 1 1: 4x-2y+3=0 . 2 2: x-3y+4=0 . 1 2 IR . 1 2 . 2 IR

3.6.

2 04y||x|| 22

.

: 3


- 17 -

4.1. x-2+(2x+3y-1)y=0 , IR (1) . : i. IR (1) ii. (1) .

. : y= 3x21

;

4.2. : . 6x2-xy-y2=0 . 450 4.3. 1: x-y=+1 2: x-y=2 , IR 4.4. . (3+1, 2), IR . (2+, 3+5), , IR +=1 . x+(-1)y=2 (+1)x+y=2+1, IR . 4.5. : . (x+y+2)2-4=0 . |x|-|y|=0 . x2-2xy+y2-5x+5y+6=0 4.6. 1 : 1x+1y+1=0 2 : 2 x+2y+2=0 :

. 0//22

1121

. 0212121 4.7. 1 2 1 : x+y+1=0 2 : x+y+2=0 . . . 4.8. y=x. : x+2y+1=0 y=7x+2

: 4


- 18 -

5.1. 1: 3x+ 4y -7 = 0 2: 3x + 4y -16=0. . 1 2. . 1 2. 5.2. (3-1, 2+3) (2, -) (,4) . . (= ) . ;

5.3. . (2,4) : y=3x+2.

.

. .

. : 5()2 8

. :

x+y+1=0 x+y+2=0

N :

22

221 )(E

5.4. )6,4(

)8,2(

.

.

. A

. (3,1)

5.5. (-2,-1) (2,2)

. (2,6)

.

()=() .

:

5


- 19 -

6.1. x2+y2+2xy-2x-2y-3=0 . 1 2 . 1 2 . 1 2

6.2. : 2x-y+3=0 : x+2y-1=0 (2,-1). .

.

,

. . . . . .

6.3. (,0) (0,) 0

4111

. . . , 4 .. . ,

. 54

.

6.4. (-2,1) , (3,5) (2,4) . , , . ()=3(). .

6.5. : x-y-1=0 : x+y-=0 , >0 . , yy >0 . () . : ()(1)

:

6


- 20 -

7.1. (8,0) (0,4) Oxy. . . . () . . ().

: 222

2 1999. 7.2. xy, ( - 1)x + ( + 1)y - - 3 = 0, , . . . . (2, 2), (-1, 5) (1, 3). , . . . . . 2000 7.3. (2,1) xy . . . xx . . . 2000 7.4. x2-y2+6x+9=0. . 2 1 2. . 2 1 2 . . (, ) >0 >0 ),3(

1 2

)4,16(

. 2001

7.5. (2-1,3+2), (1,2) (2,3) 2,IR . , IR . =1 2003

:

7


- 21 -

1. () () xx.

2.

(x1, y1) (x2, y2) = 12

12

x- xy - y

.

3. (x1, y1) (x1, y2) .

4. 1, 2 - 1, 2 1 = 2 1 . 2 = - 1.

5. y = 1

x y = - x 0.

6. 2x + y = 1 x - 2y = 1 .

7. y = 3x + 1 3x - y = 4 .

8. y = - 3

x + 1 y = - x + 2 . = 3.

9. y = 2x + 1 4x - 2y + 5 = 0 .

10. xx, yy y = x

y= - x .

11. y = 2 y = 2x .

12. 5x + y = 1 x - 5y - 1 = 0 .

13. (- 2, - 1), (1, 4) (- 4, 2) .

14. (, ), (, ), (, ) .

15. ( + , ), ( + , ), ( + , )

.

16. (x1, y1) (x2, y2)

: y - y2 = 21

21

x- x

y - y (x - x2) (x1 x2).

17. (x0, y0)

.

: 8


- 22 -

18. (1, 2)

y = - 3x + 4, y - 2 = - 3 (x - 1).

19. (1, - 4) (- 1, - 5)

y = 21

x + 3

20. (- 3, - 1), (2, 2), (- 3, 4) (3, - 6).

.

21. (1, 1)

xx 135 x + y = 0.

22. x

+ y

= 1 , 0 (, 0)

(0, ).

23. 2y - 3x + 4 = 0 xx (34 , 0).

24. ,

x = x0.

25. x + y = 0 xx 45.

26. 3x + 3 y + 1 = 0 xx

120.

27. x + By + = 0 0 .

28. , x + By + = 0 .

29. x + By + = 0

. = 0.

30. x + By = 0.

31. n =(- 2, 1) x + y + 2 = 0.

32. x + By + = 0 = (, - ).

33. Ax + By + = 0 n =(,- ).

34. 1 = (, ) 2 = (- , ) .

35. = (, ) 0


- 23 -

: x + By + = 0.

36. d (0, ) 0 (x0, y0) (): Ax + By

+ = 0 By Ax 0 0 = d (0, ) 22 .

37. det ( , ).

38.

.

39. y = 2x + 1 xx

0.

40. x + (x - y) - = 0 xOy

.

41. 1: y = 2x + 1, 2: y = 2x - 1, 3: x + 2y + 1 = 0 4: x + 2y + 2 =

0 .

42. 1: y = x + 1 2: y = x + 2

: d (1, 2) = 2

21

1

.

43. : x + 3 = 0

(3, 2), y = 3.

44. 2x - 3y = 11 4y + 3x + 9 = 0 (- 1, 3).

45. y = x + 3 xx

R.

46. ( + 1) x - y = 0 3x + y - 7 = 0 ,

=2

47. 1: 7x + 3y + 2 = 0 2: 2x + 5y - 3 = 0 .

48. xy = x .

49. 3x+5+2=0 )3,5(

50. 2x+3-1=0 )3,2(


- 24 -

-

1 1.1. . . x=1 1.2. i. =22++1 ,=2--4 ii. =5 iii. =-3 =1 1.3. i. =-1,=1350, :y=x-4 ii. A(1,5) iii. AM: y=x+4, iv. y=2x-5

1.4. i. 47

x41

y

ii. (5,3)

1.5. )35

,32

(B (2,1)

1.6. 1:y=x+1 2:23

x49

y

2 2.1. = 3 2.2.

221

221 )yy()xx()AB( =

. 2.3. (-1,0) , (6,1) 2.4. (2,-3) , (3,-2) 2.5. y=-6x+9 2.6. 6 2.7. . (1,3) . (3,5) . (3,1),(-1,5) (-1,5) (3,1) 2.8. 6x-7y+27=0 2.9. y=-x+3 3 3.1. . 1--R . =1, yy . =1 =2 3.2. . 0 0 . =1, =2 3.3. . (2,2)

3.4. 1 1 (1/2,-1/2) 3.5. . =2 . =2 . . =450 . (-2,2) 3.6.

02)y-x2(2y)x2)((22

(-1/2,1) 4

4.1. .ii )35

,2(

. 6x+3y=-7 4.2. . 2o y 4.3. 1 (1,-1) =1 4.4. . : 2x-3y-2=0 . : y=-2x+7 . (+1,), x+y=1 4.5. . x+y=0, x+y+4=0 . y=x ,y=-x . x-y=2, x-y=3 4.6. ),( 111 , ),( 222 .

4.7. . 22

1221

||),(d

. 0 , x+By+ 02

21

=0, x=

2

21

5 5.1. . 6x+8y-23=0 5.2. . 2x-3y+11=0

5.3. A. . 5102

),A(d

. () d ..


- 25 -

. d =d2. 5.4. . )7,1(AM . =22. . >0 . (-1,-5) (5,-7) x+3y+16=0 5.5. . =8 . . 3x-4y-14=0, 3x-4y+18=0 6 6.1. . x+y-3=0, x+y+1=0 . x+y-1=0 6.2. . .x-3y+4=0 ,3x+y+2=0 . (-1,1) . x-3y-5=0, 3x+y-5=0 6.3. . x+y-=0 . (4,4) . =2,=-4 =-4,=2 . ==8 . =1 =-4/3 6.4. . .x-y-1=0, x-y+5=0 . x-y+2=0

6.5. . . 1

2E

2

. (1)=1 ()(1)

7 7.2. . (-1,2) . :y=2. : x=-1 : x-2y+5=0 . . ()=3. 7.4. . x+y+3=0, x-y+3=0 . . (3,4) 7.5. . 3x-2y+7=0 (-5,-4) . ()=3/2 .. 8 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

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