Mathimatika Kateuthinsis b Lykeiou

7/29/2019 Mathimatika Kateuthinsis b Lykeiou

1/136

-

:

[ - : , , -

, . , ,

, , , -

.

[ , - .

[ .[ ,

.

[ .

[ .

[ .[ -

.

[ , .

[ .

[ .

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10. -

: -

-

AB =JJJG JJJG

.

AB A = = = =JJJG JJJG JJJG JJJG JJJG JJJG JJJG JJJG

, , GG G

, R :

:

// =G GG G

R , 0G G

// GG

G

G

, G

, R = +GGG

. G

-

G

G

.

:

2

+=

JJJG JJJGJJJJG

( )

A B

1. + = +G GG G

2. ( ) ( ) + + = + +G GG GG G

3. 0 + =GG G 4. ( ) 0+ = GG G

5. + = + =G GG GG G

6. x x 0+ = =GG GG G

7. x 0 x + = = GG GG G

8. ( ) ( ) ( ) + = + G GG G

9. AB B A= JJJG JJJG JJJG

10. + +G G GG G G

: + = +G G GG G G

= +G G GG G G

11. 0 0 = GG , 0 0 =G G

12. ( ) + = +G GG G

13. ( ) + = +GG G

14. ( ) ( ) =G G

15. 1 =G G

16. 0 0= =GG 0=

GG

17. ( ) ( ) ( ) = = G G G G 18. ( ) =

G GG G

19. ( ) = G G G

20. =GG

0 =GG

21. =

G G

0

GG

=

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11. -

xy

G

, ( )1 1 x , y=

G

1 1x , y -

1 1 x i y j= +

G GG.

1. ( ) ( )1 1 2 2 x , y , x , y= =GG

1 2 x x= =

GG

1 2y y=

2. ( ) ( )1 1 2 2 x , y , x , y= =GG

( )1 2 1 2 x x , y yGG

+ = + +3. R ( )1 1 x , y=

G ( )1 1 x , y=

G

4. ( ) ( )1 1 2 2 x ,y , x ,y :

( )2 1 2 1 x x , y y= JJJG

( ) ( )2 2

2 1 2 1 x x y y= + JJJG

(x,y) :1 2x xx

2+= , 1 2y yy

2+=

5. ( ) ( )1 1 2 2 x , y , x , yGG

= = : 1 1

1 2 2 1

2 2

x y || 0 x y x y 0

x y = =

GG

11

y

x=G , 2

2

y

x=G 1x 0 , 2x 0 || = GG

GG, ,

GG

( )n

, , (, 0). A 0 0, 0 = = = =

GG G G G G GG G GG G G G G G

( ) ( )1 1 2 2 x , y , x , yGG

= = :( )

1 2 1 2

1 2 1 2

2 2 2 2

1 1 2 2

x x y y

x x y y ,

x y x y

= +

+ = = + +

GG

GGGGGG

1. = G GG G

2. ( ) ( ) ( ) = =G G GG G G

3. ( ) + = + G GG G GG G

4. 22 = =G G G G . .

5. = G G GG G G

6. = G G GG G G

7. 0 =G GG G

8.1 2 1 2 x x y y 0 + =

GG

9. 1 = GGGG

G , G

10. 1 = G G G G

( 1 = GG G

)

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12. 1

1 G

a G

E .

G

D

G

E .

.

o

OAG

o

AMG

.

cO

o

AOG

cc

o

MAG

cc .o o

AOOAG

cc o o

MAAMG

cc ,o o

AAOO cc

o o

MMAA cc . ,o o

MMOO cc , o o

MOOM cc .

ao

ao

Eo

Eo

ao

Oc

E

o Mc

oo

aoo

a

-

, o :

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13. 1

A

o

OAG

o

OBG

,

G

G

D E -

O0

O$ 2%.

2 -

.

(1) aG

GG

G

( )

(2) )()( G

G

GG

G

( )

(1)

:

o o o

OMAMOA G

G

o o o

OMBMOB G

G

.

G

G G

G

D E E D .

(2) :o o o o o o

*** OBOBBABOA )()(

G

G

G

o o o o o o

.)()( *** OAOABABOA G

G

G

( ) ( )G

G

G G

G

G

D E J E J a .

3 . :

a

o a

o o

E

a

o

a

o

Eo

Eo

Eo

a

o a

o o

E

a

o

a

o

Eo

Eo

Eo

a

o o o

E J

E Jo o

a

o o

E

a

o

Jo

Eo

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14. 1

.

. -

Ao

2 ,

.

.

o

$%

o o o

OBABOA o o o

AB = OB- OA .

4 DG

EG

| | | | | | | | | |G

G

G

G

G

G

D E D E D E d d

,

-

:

)()()(|)()(| ABOAOBABOA dd

| | | | | | | | | |

G

G

G

G

G

G

D E D E D E d d

5 EDG

G

, , 0GG

zE ,

EDG

G

// EODG

G

, RO .

DG

EG

, 0GG

zE , EODG

G

,

. (

O

a

o

a

o o

Eao

Eo

Eo

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15. 1

). . , DG

EG

0

GG

zE , O :

EODG

G

.

ND

E

| |

| |

G

G , |||| ENDG

G

:

x EDG

G

nn , ENDG

G

.

x ED

G

G

np ,

G

G

.

x 0G

G

D , EDG

G

0 .

O , ,

EODG

G

.

6 o

OM

:

o

AB -

.

o

OM

:

o o o

AMOAOM (1) o o o

BMOBOM (2)

, (1)

(2) :

o o o o o o o

OBOABMOBAMOAOM 2 . o

o o

2

OBOAOM

.

OA OBOM

2

o o

o

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16. 1

7 , DG

, -

-

iG

jG

.

Oxy DG

. -

o

OAG

. 1A 2A

xxc cy y ,

:

o o o

21 OAOAOA (1)

y, A ,

:o

xOAG

1 o

jyOAG

2 . (1)

jyix

GG

G (2)

DG

iG

jG

.

x y .

DG

iG

jG

. , DG

::

jyixGG

G

ccD (3)

(2) (3) : jyixjyixGGGG

cc

jyyixxGG

)()( cc

xx cz , 0zc xx , G G

iy y

jc

c

, , G G

i j/ / , , iG

jG

. xx c , yy c .

ao

ao

io

jo

A1

y

x

A2

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17. 1

8 DG

EG

,

G

G

, G

, R -

G

G

DG

EG

.

),( 11 yxDG

),( 22 yxEG

, :

x

G

GG G G G G G

D E ( ) ( ) ( ) ( )x i y j x i y j x x i y y j1 1 2 2 1 2 1 2

x jyixjyixGGGG

G

)()()( 1111 OOODO

G

G

D E ( , )x x y y1 2 1 2 OD O O G

( , )x y1 1

),(),(),( 21212211 yyxxyxyx

),(),( 1111 yxyx OOO

, OD PE G

G

:

),(),(),( 21212211 yyxxyxyx POPOPPOOEPDO G

G

.

9 ),( 11 yx$ ),( 22 yx% -

),( yx -

.

2

21 xxx

2

21 yyy

.

),( 11 yx$

),( 22 yx%

),( yx

.

:

y

x

A(x1,y1)

B(x2,y2)

(x,y)

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18. 1

o o o

)(

2

1OBOAOM ,

o

),( yxOM ,o

),( 11 yxOA ,o

),( 22 yxOB ,

)],(),[(2

1),( 2211 yxyxyx

2,

2

2121 yyxx

2

21 xxx

2

21 yyy

.

10 ),( yx -

),( 11 yxA ),( 22 yx% -

: 12 xxx 12 yyy .

),( 11 yx$

),( 22 yx%

),( yx -

o

AB .

,o o o

OAOBAB ,o

),( yxAB ,

o

),( 22 yxOB , o

),( 11 yxOA ,

:

),(),(),(),( 12121122 yyxxyxyxyx

:

),( yx ),( 11 yxA

),( 22 yx%

12 xxx 12 yyy .

11 ),( yxDG

.

:

22|| yx DG

.

y

x

A(x1,y1)

B(x2,y2)

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19. 1

),( yxDG

-

-

o

OAG

. 1$

2$ xxc

yyc .

x y , -

||)( 1 x2$ ||)( 2 y2$ . 1 :

222222

21

21

21

22 ||||)()()()()(|| yxyx 2$2$$$2$2$DG

.

:

22|| yx DG

12 $( , )x y1 1

),( 22 yx% 2

122

12 )()()( yyxx $% .

$( , )x y1 1

),( 22 yx% .

)($%

o

),( 1212 yyxxAB ,

:

2 2

2 1 2 1() AB ( x x ) ( y y ) JJJG

:

$( , )x y1 1 ),( 22 yx%

212

212 )()()( yyxx $% .

ao

A1

y

x

A(x,y)2

y

x

A(x1,y1)

B(x2,y2)

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20. 1

13 EDG

G

, ,

G

D ( , )x y1 1 G

E ( , )x y2 2 -

O1 O2 : 21// G

G

G

D ( , )x y1 1 G

E ( , )x y2 2 O1

O2 , :

21

2

2

1

11221

22

110//

x

y

x

yyxyx

yx

yx G

G

.

14

G

D ( , )x y1 1 G

E ( , )x y2 2

.

o

OAG

o

OBG

. -

2%$2%2$2%2$$% ))((2)()()(

222

,, .

( ) ( ) ( )$% 2 2 12

2 12

x x y y , ( )2$ 2 12

12

x y ( )2% 2 22

22

x y .

, :

aoE

o

y

x

(x1,y1)

(x2,y2)

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21. 1

%2$2%2$

))((2)()( 2222

21

21

212

212 yxyxyyxx

%2$2%2$

))((222 2222

21

2121

22

2121

22

21 yxyxyyyyxxxx

G

G

%2$2%2$ ))(( , :

G

G

D E x x y y1 2 1 2

15 :

xR),()(

G

GG

G

G

x GG

G

GG

G

G

)(

x A 1 2 = -1

G

G

, G1 G2 , ( yy c//,

G

G

)

G

D ( , )x y1 1 ,G

E ( , )x y2 2 G

J ( , )x y3 3 , :

( ) ( , )( , ) ( ) ( ) ( ) ( )OD E O O O O O O D E G

G

G

G

x y x y x x y y x x y y1 1 2 2 1 2 1 2 1 2 1 2

G

G

G

G

D OE O O O O O O D E ( ) ( , )( , ) ( ) ( ) ( ) ( )x y x y x x y y x x y y1 1 2 2 1 2 1 2 1 2 1 2 .

,

( ) ( ) ( )OD E D OE O D E G

G

G

G

G

G

G

G

G

D E J ( ) ( , )( , ) ( ) ( )x y x x y y x x x y y y1 1 2 3 2 3 1 2 3 1 2 3

( ) ( ) ( ) ( )x x x x y y y y x x y y x x y y1 2 1 3 1 2 1 3 1 2 1 2 1 3 1 3

G

G

G G

D E D J .

1100 212

2

1

121212121 A

x

y

x

yxxyyyyxx

G

G

G

G

16 G

D ( , )x y1 1 G

E ( , )x y2 2 -

. A

22

22

21

21

2121yxyx

yyxx

.

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22. 1

G

G

G

G

D E D E VXQT | | | |

| | | |

G

G

G

G

.

G

G

D E x x y y1 2 1 2 , | |G

D x y12

12 | |

G

E x y22

22 .

,

22

22

21

21

2121yxyx

yyxx

17 v

GG

, 0

G

G

z .

GGGG

G .

o

OAG

o

OMG

.

o

OA 1M

.

o

1OM G

G

G

G :

o

1OM G

G .

(H G

G

).

G

G

:

o o o o o

OMMMOMMMOMvGGGGGGG

11111 )( G

G

:

GGGG

G

ao

M1

M

Avo

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23. 2

.

2003.

1.3 : 6, 7, 9

: 5, 6, 8

1.4 : 3, 5, 6, 8

: 2, 3, 5

1.5 : 6, 7, 12, 13

: 2, 4, 5

: 1, 3

.

1:

: 3, 5

: 7, 11

2:

: 3

: 1, 5, 6

3:

: 1, 3, 5, 7, 8, 12

: 3, 4, 5

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24. 3

1. x :

1A AB

3=

JJJG JJJG,

1A A

5=

JJJG JJJG, =

JJJG JJJG

i. JJJG

JJJG

AB =JJJG G

A =JJJG G

, ,

.

ii. JJJG

JJJG

G

G

.

:

: 1 1

A AB 3 3

= =JJJG JJJG G

1 1

A A 5 5

= =JJJG JJJG G

AB AB A = = + = = JJJG JJJG JJJG JJJG JJJG JJJG G G

i. 2 5 B 2A 3 3

= + = = = JJJG JJJG JJJG JJJG JJJG G G G G G

( )11 1 1 5 1

A 5 3 5 3 5

= + = = = =

JJJG JJJG JJJG JJJG JJJG G G JJJGG G

JJJG JJJG

&

, , .

ii. 1

A 5

= + = JJJG JJJG JJJG G G

1

A A A 3

= + = = JJJG JJJG JJJG JJJG JJJG G G

1 1

5 3

=

JJJG JJJG G G G G

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25. 3

2 22 21 1 1 16 1 1

15 5 3 15 5 3

+ = G G G G G G G G G G

x :2

0 x 60

2 = =

G G G G.

( ) x= =G

G

2

2 216 x 1 1 x x 015 2 5 3

= =JJJG JJJG

. BE

2. (6,-1), (1,3), (1,2), (-1,-1) (1,-1).i.

= JJJJG JJJJG

, 1 .ii. , .

iii. , = JJJG JJJG

, = JJJG JJJG

= JJJJG JJJJG

, 1 = .:

i. (x,y).

( ) ( ) ( )( )

( )

x 1 6 x x 1, y 3 6 x, 1 y

y 3 1 y

= = =

=

JJJJG JJJJG

:1 6

x1

+=

+ ( )

3 y 1

1

=

+

ii. ( ) x 1, y 2= JJJG

( )M x 1, y 1= + +JJJJG

x 1 y 2 M 0

x 1 y 1

=

+ +

JJJG JJJJG& 3x 2y 1 0 + = (1) :

1 6 3 23 2 1 0 1 1 21

+ + = =+ +

iii. ( ) ( )k 2, 0 k 7, 0 = =JJJG JJJG

2

7 =

( ) ( )1

0,1 0, 33

= = = JJJG JJJG

105 84 10 8 21 , ,

23 23 23 23 2

= = =

JJJJG JJJJG

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26. 3

:2 1 21

1

7 3 2

= =

3.i. (3,2) (-1, 4).

ii.A (,5) , .

:

i. (x,y) .

:

( )

3 x1 3 x 2 x 5

2 A 5,6

2 y4 2 y 8 y 6

2

+ = + = = + = + = =

ii. , A, :

( )A det A ,A 0 =JJJJG JJJG JJJJG JJJG

& (1)

( ) 8,4 = JJJJG

( )3, 3 = JJJG

(1) :

8 40 24 4 12 0 4 12 3

3 3

= + = = =

4. , 0G G

. ( ) G G& ( ) G G G& : ( ) GG& .

:

( ) +GG G& R : (1)+ =G GG

( ) +G G G

& R : ( ) 2+ =GG G

(1) (2) : + = G GG GG G

( ) ( ) ( ) 1 1 3+ = + + = +G G GG G G

1 + 1 + , 1 0+

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27. 3

(3) :( ) 1

. 1

+=

+

G GG G& , .

: 1 0 1+ = = 1 0 1+ = = ,

(1): ( ) ( ) . + = = + +G G GG G GG G G

&

5. (x1,y

1) (x

2,y

2) (x

3,y

3) :

2 1 3 2x - x = x - x . 1, 2, 3 , , JJJG JJJG JJJG

, : 1 2 3 2 =:

( ) 2 12 1 2 1 12 1

y y x x , y y ,

x x

= =

JJJG, ( ) 3 23 2 3 2 2

3 2

y yx x , y y ,

x x

= =

JJJG,

( ) 3 13 1 3 1 33 1

y yA x x , y y ,

x x

= =

JJJG

( )( )3 2 2 1 3 22 1

1 2 2 1 3 22 1 3 2 2 1

y y y y y yy y

x x x x 1x x x x x x

+

+ = + = =

:1 3

2 1 3 2 2 1 3 2

x xx x x x 2x x x x

2

+ = = + = (2)

(1) (2) :

3 11 2

1 31

y y

x xx

2

+ =

+

3 1 3 13

3 1 3 1

y y y y2 2

x x x x

2

= = =

6. , ( ) 2,1 ( ) 3,1G xx , :

4+ = .

:

1

1

2= =

2

1

3= =

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28. 3

( )

1 1 5 5

2 3 6 6

1 1 1 1 51 41 1

2 3 6 6

++

+ = = = = = =

4

+ = .

7. , , G G

: 1= =G G , G , m( ) , 6

=G m( )

, 3=G

, 3 2G G

G

.

:

( )22

2 2 23 2 3 2 3 4 2 3 4 3 4+ = + = + + + =G G G G GG G G G GG G G G G

( ) ( )2 2 2

3 4 4 3 , 4 , = + + G GG GG G G G

3 13 1 1 4 1 4 3 1 1 4 1 1 8 6 2 02 2

= + + = = .

3 2 0+ =GG G

3 1

3 2 0 2 2

+ = = +G GGG GG G

.

8. , ,x ,yG G G

, ,

G& .

. x y= G G

x y= G GG G

, ;

i. x yG& ii. x yG iii. x yG

. 1 , : ( )2 2 G G:

. ( ) x y x y 0 x y 0 = = = G G G G GG G G G G G

0=GG

x y=G G

( ) x y G G G

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29. 3

( ) x y x y 0 x y 0 = = = G G G G GG G G G G G

0=G G x y=G G ( ) x y G G G .

0=G

0=G

, GG

( ) x y G G G

( ) x y G G G

GG

&

.

x y=G G

.

. ( )2

2 =G GG

m( )

22 2 2

, = = G G G G GG G G

m( )2 12 22

, =

= G G GG G m( )

2

, 1=GG

m( ) m( )2 , 1 , 1 = = G GG G

GG

& . ( )2

2 G GG

9. G

. -

, :

+ =G G G (1)

+ = GG (2) :

G.

:

(1) :.

+ = GG GG G G G

( )22

+ = + =G G GG G G G G

(3)

(2) :.

+ = GG G G GG G

( )

22

+ = + =

G G G G GG G G

(4)

(3) (4) :22

= =G GG G

.

10. , G

m( ) ,3

=G + GG .. :

2 G .

. G G

.

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30. 3

:

. : 2

1 12. . , . 2 2

= = = G G GG G G GG G G

( )2 2 2

21 1 1

2 2 2 = + = + =

G G G G G G G GG G G

m( ) 2 21 , 2

+ = G G G GG G

1= =GG

: 1 1 1

1 1 1 1 13 2 2 2

+ = + = =

. 1

= :

=

GG G

m( ) ( ) ( )2

, 1

= = =

G GG G GGG GG GG G GG G G

:2

2 1 1 1 1 1

3 2 2 = = =

G G G GG G

( )2 222 22 2 1

2 2 2

= = = + = +G G G G GG G G G G G

1 1 1 1 1= + = =G

(1) :m( )

1 12 ,1 1 2

= =

G G, : m( ) 2,

3 3= =

G G

11. , G

:

( )

1,2G ( ) 2,1Gi. .

ii.

G.

:

i. ( )* R

|| 1, 2

= = G GG GG G G

( )*

R

|| 2,1

= =G GG GG G G

: ( )( ) 1, 2 2,1 = G G

0

1 2 2 1 0

= + = G GG G , 0

0

= GG G

G GG G

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31. 3

ii.1

= GJJG G

2 = GJJG G

G G

:

( ) ( ) ( ) ( )1 2 1,2 2,1 1 2,2 1 1,3= + = + = + + =JJG JJGG

12. , , , , -.

i. : ( )2 2 2 2 AB A B B A + = + + JJJG JJJG JJG JJJG JJJG JJGii.

JJG JJG

,

:

( ) ( )2 2 2 2 1 1 + + :

i. ( )22 2 2 2

2 2 A AB A 2A AB AB= = = = = + =

JJG JJG JJG JJJG JJG JJG JJJG JJJG

2 22 2 2 B B 2 B + = +

JJG JJG JJJG JJJG JJG JJJG

: 2 2 2 2 B = +JJG JJJG

, 2 2 22 = + JJJG JJG

(1)

:

2 2 22A = + JJJG JJG

(2) 2 2 22 B = + JJJG JJG

(3) .

(1), (2) (3) :

2 2B BA 2A B + + =JJJG JJG JJG JJJG JJJG JJG

2 2 2 2 2 2 2 2 2 + + + + +

( ) 2 2 22 + + = + +JJJG JJG JJG JJJG JJJG JJG

.

ii. ( ) ( )2 22

2 B B B = = + = + =

JJJG JJJG JJJG JJJG JJG

2 22 22 2

2 2 + + = + =JJJG JJJG JJG JJG JJJG JJJG JJG JJG

2 2 2 2 +JJJG JJG

2 2 2

2A = +

JJJG JJG

( (2) i.) - :

( )2 2 2 2 2 2 2 2 2 2 2 2 2A = + + = + + =

= ( ) ( )2 2 2 1 1 + + .

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32. 4

1. / / G

R GG 2 3+G G .

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

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

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

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

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

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

2. , , G G

. ( ) // G G ( ) // G G G + GG .

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

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

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

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

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

3. , JJJG G

JJJG G

2JJJG JJJG

. ,

JJJG

, JJJG

JJJJG

-

G

G

. 2JJJG JJJJG

.

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

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33. 4

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

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

4. =JJJG JJJG

, = JJJG G

= JJJG G

:

i. =JJJG

;

. G

. ( ) + G . G . Gii. = ;

JJJG

. ( )1

2+ G . 1

2+ G . 1

2+ G . ( )1

2 + G

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

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

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

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

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

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

5. 1

( ) 3,5 x+ 2

-

( ) 12,x 4G , x : 1 2 2 3= .............................................................................................................................

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

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

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

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

6. (2,1), (6,4) (5,5).

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34. 4

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

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

7. (2,5) ( ) 6,4JJJG :. (6,4) . (8,9) . (4,9) . (8,1)

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

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

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

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

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

8. 2+JJJG G 2 JJJG G . JJJG JJJG , ( )2 22

3 = G G G . 4

5G

(m( ) ,G ).

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

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

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

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

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

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

9. (,) (,) . 0

, n ),JJJG JJJG .............................................................................................................................

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35. 4

10. ( ) 1,3 , G G 2=G 1=G . 2 3 +G G G .

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

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

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

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

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

11. , G

i 2 j+G G 1G m( ) ,

6=G . u, u 2 3+ G

2 GG .............................................................................................................................

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

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

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

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

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

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

12. , G

: 2 2=G , 2 8=G

n ) 2, 2 3+ =G GG . G .............................................................................................................................

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

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

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36. 4

13. R ( ) 3,4 1 ( ) 3 9, 3 + G .

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

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

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

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

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

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

14. , B = = =JJJG G JJJG G JJJG G

,

: 3= =G G

, 7=G

, 2 3 0+ =G G G G

. , , .

. , , G G GG G G

m( ),GG

.

. xG

( )x G G G

& ( ) ( )x + +G G G G

:

i. N : ( )21x 4= G G G ii. xG

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

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

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

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

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

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

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

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

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

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

15. AB 2 +JJJG GG , 3JJJG G G ,

G

1GG = = m( ) 2,

3

GG = . :

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37. 4

. :

i. G

G ii. G

G 4 2G

G

. AMJJJJG

, BJJJG

G

,

G

AM BJJJJG JJJG

AMJJJJG

BJJJG

.

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

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

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

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

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

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

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

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

16. (0,3), (2,1) ( ) 2 3,1 .:

i.

JJJG JJJG

ii. iii.

JJJJG

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

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

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

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

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

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

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

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

17. . G , G

2

= GGGG G

G

. ( )v 1, 2G ( )u 3, 4G .

. ()=, ()=.

: ( ) ( ) 0=JJJG JJJG G

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38. 4

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

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

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

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

18. , 0G GG 2G GG ,

3

8=G G G .

. : 2 23

2 8

= =G GG G

. : 3

4=

G

G

. m( ) ,GG .

. v +G GG w GGG R.

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

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

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

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39. 5

1

.1. i , jG G

xx yy

( ) x, y=G xi yj= +G GG .

G

iG

jG

.

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

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

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

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

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

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

( 5)

2. ( )2 , 3 2= +G

i , jG G

.

( ) i j + :. 2= . 1= 2=. 3= .

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

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

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

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

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

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

( 8)

.

:

1. ( ) 1,2 , ( ) 2,3 ( ) 6, :

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40. 5

. 1= . 0= 2=

. 0= 2 3=. 7=

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

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

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

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

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

( 7)

2. () () -

.

, GG

:

. // =G GG G

. = =G GG G G G

. = G G GG G G

. G GG G

= =

. = G GG G

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

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

( 5)

20

, GG

( )2 4 3,2+ =GG

( ) 4 9,4 =GG

.

. ( ) 4,2=G

5 1

,4 2=

G

.

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

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

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

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

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

( 7)

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41. 5

. 8+GG

4+GG

-

.

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

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

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

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

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

( 8)

.

G

G.

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

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

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

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

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

( 10)

30

. , GG

=GG G

. ( ) 2 G G G

.

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

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

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

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

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

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

( 10)

. , ( ) 3=

9 =JJJG JJJG

. JJJG JJJG

.

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

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42. 5

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

............................................................................................................................( 15)

40

( ) 3, 5= G

, ( ) 2,1= G

( ) 3, 1= G

. G

G

G

.

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

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

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

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

( 25)

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35/136

-

:

[ .[ -

, .

[ (x

0, y

0) x=x

0

( ) = 0 0y y x x , R . ( )1 1A x , y ( )2 2B x , y ,

1 2x x , ( )

=

2 10 0

2 1

y yy y x x

x x,

.

[

-

.

[ - .

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36/136

44. -

E: -

. () ( )0 0x , y

:

( )0 0: y y x x =

. () ( )1 1x , y ( )2 2x , y

1 2x x : ( )

2 11 1

2 1

y y : y y x x

x x

=

, 2 1

2 1

y y

x x

= =

JJJJG

1x x 2 1x x

. () yy (0, )

: y x = + , ()

. () (0, 0)

yy : : y x=

. () ( ) y,x

xx : 0: y y =

. () ( ) y,x

yy : 0: x x =

+ + = Ax By 0 , 0 0

: Ax By 0 , 0 0+ + = (1)

(1) .

: 0 :

. (1) :

= .

. (1)

( )1 ,

= ( )2 ,

= .

. (1) ( )k A, B

= .

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37/136

45. -

1. d 0 0 0(x , y ) :

Ax By 0+ + = , 0 +

:

0 00

2 2

x Byd d(M , )

+ + = =

+

2. ,

:1

(AB) det(,)2

=

det( , )

. : 1(AB ) det( , )

2

=

1det( , )

2

=

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38/136

46. 1

1 :

,

.

G

.

G

G xx c ,

. . :

, .

G

o

=

x

y

G

o

=+

x

y

2 :

),( 11 yxA ),( 22 yxB , 21 xx z 12

12

xx

yy

.

),( 11 yxA ),( 22 yxB .

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39/136

47. 1

-

)-,-( 1212 yyxxAB o

, 12

12

--

xxyy .

3 Oxy

),( 00 yxA .

-

.

Oxy

),( 00 yxA .

-

. ),( yxM

),( 00 yxA , ,

o

AM , , -

o

AM .

),( 00 yyxxAM o

, 0

0

xx

yy

AM

o

.

, ),( yxM , , :

xx

yy

0

0 )( 00 xxyy .

),( 00 yxA . -

:

)( 00 xxyy

x

y

M(x,y)

(x0,y0)

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40/136

48. 1

4 Oxy

),( 11 yxA ),( 22 yxB .

),( 11 yxA ),( 22 yxB .

-

),( 11 yxA ),( 22 yxB .

21 xx z ,

12

12

xx

yy

)( 00 xxyy :

)( 112

121 xx

xx

yyyy

1 2 0x x x , -

.

),( 00 yxA ,

0x . : 0xx .

5 :

Ax By * 0 A z 0 B z 0 (1)

, (1)

.

.

yyc 6( , )0 E

, y x O E ,

O Ex y ( )1 0

P x y( , )0 0 ,

x x

0 , :

x

y

(x1,y1)

B(x2,y2)

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49. 1

x y x 0 00( ) .

, ,

Ax By * 0 A z 0 B z 0.

x ,

Ax By *

0 Az

0 Bz

0.

B z 0 , yA

Bx

B

*, -

O A

B yyc

0,

*

B.

B 0 , , , A z 0

xA

*

, cx x -

PA

*,0 .

Ax By * 0 A z 0 B z 0 -

.

x

y

P(x0,y0)

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50. 2

.

2003.

2.1: : 3, 7

: 1, 3, 5

2.2 : 4, 5, 6

: 1, 2, 4, 6

2.3 : 6, 8, 9, 10

: 1, 3, 4, 8, 10

: 1, 2, 6

.

4:

: 6, 7, 8

: 5, 8, 9

5:

: 1, 4, 5, 6

: 5, 7, 9

6:

: 2, 3, 5, 7, 9

: 1, 10, 12, 19, 20

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43/136

51. 3

1. . (2,1), (3,4) (2,2),

.

:

AB

4 1 5

3 2

+= =

, : ( )y 1 5 x 2 5x y 11 0+ = =

A Hx x 2= = , : x 2= .

B (3,4), : y 4=

4 2 1

2, 1 3 2 2

= = = =

, :

( )1

y 1 x 2 2y 2 x 22

+ = + = + x 2y 0+ =

:

y 4 y 4 y 4

x 8x 2y 0 x 2y

= = = = + = =

( ) 8,4 .

2. : ( ) ( ) ( )2 1 x 3 y 3 6 0 1+ + + = , R.i. (1) -

.

ii. (1)

.

:

i. H (1) x By 0+ + = =21 =3.

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44/136

52. 3

:

12 1 0

2 3 0 3

= = = =

R = 0 = 0. (1) R .

ii. 0= (1) : ( )1x 3y 6 0 : + =

1= (1) : ( )2x 2y 9 0 : + =

1

2:

( )y 3x 3y 6 0 5y 15 0 y 3, x 3

x 2y 9 0 x 2y 9 0 x 2 3 9 0= + = + = = = + = + = + =

(1) (

2) (3,3)

(3,3) (1) :

( ) ( )3 2 1 3 3 3 6 0 + + + = 6 3 3 9 3 6 0 + + + + = 0 0= . (1) (3,3).

3. / : x y 3 0+ = x y 1 0 = (4,2).

.

:

(4,2) -

.

( ) : x y 3 0 1+ + = ( )B : x y 1 0 2 = .

(1) (2).

: : x y 3 0 x 1

: x y 1 0 y 2

+ + = = = =

( ) 1, 2 .

( ) 1 // = = .

: ( )y 2 1 x 4 x y 6 0 = + = .

:x y 6 0

x y 1 0

+ = =

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45/136

53. 3

:

7x

x y 6 0 2x y 1 0 5

y2

= + = = =

7 5B ,2 2

.

( ) 1 A // = = .

: ( )y 2 1 x 4 x y 2 0 = =

():x y 2 0

x y 3 0

= + + =

:

1x

x y 2 0 2

x y 3 0 5y

2

= = + + = =

1 5

,2 2

.

. :

7 1 5 5

2 2 2 2 ,2 2

3

,02

4. : 2 2x y 8x 16 0+ + = .i. () ().

ii. () () .

iii. (,), > 0 > 0

( )1

4,G

() ()

( )2 8,2G .iv.

(0,0).

:

i. ( ) ( )22 2 2 2 2x y 8x 16 0 x 8x 16 y 0 x 4 y 0 + + = + + = + =

( )( )x 4 y x 4 y 0 x 4 y 0 x 4 y 0+ + + = + = + + =

( ) : x y 4 0 + = ( ) : x y 4 0+ + = .

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46/136

54. 3

ii. ( ) ( )

1 11

1 1

11

= = = = =

iii. ( )1 4,=G

1

4=G . :

11

// 1 4

4 = = =G

G

( )2 8,2= G

2

2

8 4= = G .

:2

2 // 1 44 = = =G

G. ( ) 4,4 .

iv. , :

2 2 4 4 32 4 2= = + = =

JJJG

: ( )2

2 2x y 4 2+ =

5. : : 3x y 10+ =

: x y 6 =

: 13x 3y 8 = :

. .

. .

. .

. .

:

. B :

x y 6

13x 3y 8

= =

:x y 6

13x 3y 8

= =

... x 1

y 7

= =

. ( )B 1, 7 .

.

( ) BM A 1 3 1 = =

1

3= -

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47/136

55. 3

( )1

y 7 x 1

3

+ = + 3y 21 x 1 x 3y 20 0+ = + + + =

:

x 3y 20

3x y 10

+ = + =

3x 9y 60

3x y 10

+ = + =

10y 50

x 3y 20

= = +

y 5

x 5

= =

, (5,-5).

(,) 1 7

N ,2 2

, (5,-5),-

1 52 = 7 5 11

2 = = 3= , (11,-3).

. (

) .

: 1 1 1 1= = =

: ( )y 3 x 11 y x 8+ = = +

. :

3x y 10x y 8

+ = + = 3x y 10x y 8

+ = = 2x 2y 8 x

= = x 1y 7

= = (1,7).

: y=x+8, (,-+8)

1 15

,2 2

+ +

. :13x 3y 8 =

: 1 15

13 3 82 2

+ = 13 13 45 3 16+ + = 16 48 3= = .

: (3,5).

6. (2,3), : 3x 5y 6 0 + = : x 11y 2 0 + = .

.

:

3x 5y 6

x 11y 2

= =

3x 5y 6

3x 33y 6

= + =

28y 0 y 0

x 2 11y x 2

= = = + =

, (-2,0)

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48/136

56. 3

( )

3 0 32 2 4= = ( )

3A : y x 24= + .

:

3 1 1

5 = =

5

3

= ( )

5 : y 3 x 2

3 = .

5 19B ,

3

+

2 28 5 ,

2 6

+

: x 11y 2 0 + = :

2 28 511 2 0

2 6

+ + = ( ) ( )3 2 11 28 5 12 0+ + =

3 6 308 55 12 0+ + + = 58 290= 5= (5,-2).

( )

2 0 2

5 2 7

= =

, ( )

2AB : y x 2

7= + .

7. :

( )1 : y 2x 3= ( )2 : y x 3 0+ + =

. () (3, 0)

1,

2 , -

.

. 1,

2 -

.

:

. .

1 : y 2x 3= (,23). -

2 : y x 3= , (,3)

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49/136

57. 3

2 6

,

2 2

+

(3, 0) :

3

6 3 12 42

2 6 2 6 2 6 20

2

+ = + = = = = = = =

, : (4, 5) , (2,5)

( )

5 5 10 5

4 2 2

= = =

, : ( )y 5 x 3 y 5x 15= = .

. :

1

2

: y 2x 3 : y x 3= =

, x 0

y 3

= = , (0, -3).

( )A 4, 8= JJJG

( )AB 2, 10= JJJG

:

( ) ( ) ( )4 81 1 1

AB det A, 40 16 12 ..2 102 2 2

= = = = =

JJJG JJJG

8. (2, -3) (4, 1) : x y 2 0+ = . () -

().

. ().

. () = 10.. .

. : y x 2= + (, +2), R

2 4 3 1

3, 12 2

+ + = =

3 1

,2 2

+ +

.

M(x, y)

3x

2x 32

1 2y 1y

2

+ = = + = =

: 2x 3 2y 1 y x 1 = = () ( ) =

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50/136

58. 3

. : ( )( )

22

3 12

3 1 4 32 2d M, 2 2 21 1

+ + +

+ += = =+

.

. : ( )( ) ( )

( )( ) ( )

B 4 2, 1 3 2,4

BA 2, 2 3 2, 5

= =

= + = +

JJJG

JJJG

: ( ) ( ) ( )2 4

det B, BA 2 5 4 2 2 18 2 5

= = + = + +

JJJG JJJG

: ( ) 10= 1 2 18 102 + = 2 18 20 + =

2 18 20 2 2 1 + = = = 2 18 20 2 38 19 + = = =: (1,0) (11,10)

9. (): ( ) ( ) 2 x 2 1 y 2 0 + + = .i. R

, .

ii. (2, 2) ()

3

5

iii. R () xx

yy

0(0, 0)

iv. 1,

2

1

2= 1,

1,

2 ()

1,

2 .

:

i. 2 2 0

12 1 0

2

= = + = = , () -

R .1

= 2 : y = 0

1

2

= : x = 1

(1, 0).

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51/136

59. 3

(),

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

( )0 0 x , y :

R :

( ) ( )0 0 2 x 2 1 y 2 0 + + = 0 0 0 0x 2x 2y y 2 0 + =

( ) 0 0 0 0 00 0 0 00 0 0 0 0

x 2y 1 x 2y 1 x 1x 2y 1 2x y 2 0

2x y 2 4x 2y 4 y 0

= = = + = + = + = =

(1,0).

ii. : ( )( ) ( )

( ) ( )2 2

2 2 2 2 1 2 3 3d A,

5 5 2 2 1

+ + = =

+ +

2 25 3 4 3 5 5 3 4 3 1+ = + + = +

( )2 2 2 2 73 4 9 1 9 24 16 9 9

24+ = + + + = + =

iii. () :

i. xx , :

2 2 0

212 1 0

2

= = = +

ii. yy , :

2 2 0 1

12 1 0 2

2

= + = =

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

iv. 1

: ( ) ( )1 1 1 2 x 2 1 y 2 0 + + =

: ( )1 1 2 1, 2= + G

1.

(2) : ( ) ( )2 2 2 2 x 2 1 y 2 0 + + = -

(2) ( )2 2 2 2 1, 2= +

JJG.

: ( )( ) ( )( )1 2 1 2 1 2 2 1 2 1 2 2= + + + =JJG JJG

1 2 1 2 1 2 1 24 2 2 1 2 2 4= + + + + + =

( )1 25 5 5 1 5 0= + = + = 1 2

JJG JJG

1 2 .

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60. 3

10. : i. x y 1 0 + = ii. ( ) ( ) 1 x 1 y 4 0+ + =

. R .

. i. -

: 1 : 4x 2y 3 0 + = .. ii.

: 2 : x 3y 4 0 + = .. . .

.

. ii.

.

. i. ii. R

: 0 1 0

1 0 1 0

= + = = =

.

. i. y = x + 1 (1)

3y 2x

2= +

= 2.

. ( ) 1 , 1= +G ii.

( ) 2 3,1 =G

& :

( ) 0 3 1 1 0 = + + =G G G G

3 3 1 0 2 + + = =

. = 2 i. 1 : 2x y 1 0 + = ( ) 1v 1, 2 =

G& .

= 2 ii. 2 : 3x y 4 0+ + = ( ) 2w 1, 3 =

JJG&

:

n

( )vw 1 6 5 5 50 50 5 2 2

v, w 50 10 10 25 10 50v w

+

= = = = = = =

G JJGG JJG

G JJG

450.

. (x0, y

0) ii.

( ) ( )0 0 1 x 1 y 4 0+ + = 0 0 0 0x x y y 4 0+ + + =

( )0 0 0 0x y x y 4 0+ + + = 0 0 0

0 0 0

x y 0 x 2

x y 4 y 2

+ = = = =

ii (-2, 2).

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61. 3

11. :

1 1 1 1 : x y 0+ = , ( )1 1 0 0 ,2 2 2 2 : x y 0+ = , ( )2 2 0 0 1 2 2 1A B A B

( ) ( )1 2 1 2 1 2 : x y 0+ + = . :

. 1

2

.

. .

. 1,

2 .

. xOy

1 1 2 2A B A B= + .

. 2 2 2 21 1 2 2

A B A B= +

n( )1 2

, .

:

.1 .

1

2

:

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

1 11 2 2 1

2 2

A BD A B A B 0

A B= = , -

, 1

2

.

2 .

( )1 1 1 ,= JG

( )2 2 2 ,= JJG

1

2

.

( ) 1 11 2 2 1 1 2

2 2

det , 0

= = +

JG JJG

1 2 ,

JG JJG

-

, 1

2

, -

.

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

1 2

A A 1 2B B .

, 1 2

A A 0 = 1 2

B B 0 = 1 2

A A= 1 2

B B=

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62. 3

1 2 2 1A B A B= , .

1 2A A 0 1 2B B 0 .. ( )0 0M x , y 1 2.

1 0 1 0 1A x B y 0+ + = 2 0 2 0 2A x B y 0+ + = .

( ) ( )1 2 0 1 2 0 1 2 x B B y 0 + + = x

0, y

0 ,

1

2.

. ( ) ( )1 2 1 2 1 2 1 2 ( B ), A B B ,A A= = +

G

( ) 1,1 =JG

, : y x x y 0= = . , :

( ) 1 2 1 21 2 1 2

B A || ' det , ' 0 0 B B A 0

1 1

+ = = + + =

G JG G JG

1 1 2 2A B A B+ = + .

. M(x,y) n( )1 2 , , , :

( ) ( ) 1 1 1 2 2 21 2 22 2 2 2

1 2 2 2

x y x y d , d ,

+ + + += =

+ +()

2 2 2 21 1 2 2A B A B+ = + , () :

1 1 1 2 2 2 x B y x y + + = + + .

:

1 1 1 2 2 2 x y x y + + = + + ( )1 1 1 2 2 2 x y x y + + = + + :

( ) ( )1 2 1 2 1 2 x y 0 + + = ( ) ( )1 2 1 2 1 2 x y 0+ + + + + = .

( ) ( )1 2 1 2 1 2 : x y 0 + + =

n( )1 2 , .

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63. 4

1. :i. (1,2) 3 .

ii. (1,2)

( ) : x y 1 0+ = .iii. (1,4) ( ) : x 2 .............................................................................................................................

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2. () x 135 yy (0,4).

i. ().

ii. () ()

(2,6).

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64. 4

3. (3,1), (5,1) (7,2). - - .

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4. ( ) ( ) ( )2 2 2 x 5 6 y 1 0 1+ + + + = .i. R ;

ii. R xx;

iii. R

(0,0).

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5. (1,1), (2,0) : x y 1 0 = .i. ()

.

ii.

.

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65. 4

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6. , - ,

: 2 22x 2y 4x 8y 40 0 + = . (2,10)

-

(2,14). -

-

; ( .)

(1,4)

. ,

;

( (2,10).

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7. :

( )1

: x 3 y 2 = ( )2 : 3 x y =i. (1) (2) R.

ii. .

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66. 4

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8. (+2, 2) (,), , R .i. ()

.

ii. () .

iii. (3,3) ().

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9. (3,6) : 4x 3y 9 0 = . (3,1)

45 .. :i. d(A,),

ii. ,

iii. .

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67. 4

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10. (x,y) :2 2

x 4y 3x 6y 4xy 2 0 + + =i. 1, 2.

ii. (1)//(2).

iii. .

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68. 5

10

. ()

().

1. 2x y 0 ( 0) = x + y =1 .

2. 2 (1, ) = G

.

3. (1, 4), (4, 7):

. 1

. x +2y =

. x (1, 0).

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

5.

x y

1, , 0+ = (0, ) (, 0).6. u ( , )=

G v ( , )=

G :

x + y + = 0

7. u ( , )= G

: x + y + = 0, 0 .8. (0, 0) x + y =1 (0, 1).

9. (4, 2) 135

x - y = 2.

10. 45 x x + y = 4.

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( 10)

B.

(2,1) 1.

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69. 5

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( 15)

20

A (2, 1) (4, 6). 2 ..

B. (4, 0), (2, 0) (1, 4).

(x, y) : ( ) ( )1

MB 2

= .

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( 25)

30

. ( )x 1 y 2 + = ( ) 1 x y 2 1+ + = + R .

. (2,1)

, . .

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70. 5

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( 25)

40

xy

.

1 (t - 30,2t - 40), t 0 .. ;

. ;

. ;

. 2 (t - 20, t 10), t 0+ ;

. t = 10;

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( 25)

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o

:

[ .

-

+ + + + =2 2x y Ax By 0 .

[ -

.

[

.

[ - . -

.

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72. -

.

2 2x y x y 0+ + + + = (1), A, B, R

2 2 4 0 > : (1) :

,2 2

2 2 4

2

+ = .

2 2 4 0 = : (1) ,2 2

.

2 2 4 0 < : (1) .

.

()

.

.

(0,0) (x1

,y1

):

21 1

xx yy+ =

( 0 , 0 )

:2 2 2

x y+ =

( )0 0K x , y :

( ) ( )2 2 2

0 0x x y y + =

: -

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73. -

1. ( )0,0 ,p

E ,02

, p: x2

= 2y 2px=

2. ( )0,0 , p

E 0,2

, p

: y2

= 2x 2py=

2y 2px= ( )1 1 1M x , y : ( )1 1yy p x x= +

2x 2py= ( )1 1 1M x , y : ( )1 1xx p y y= +

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74. -

-

t, -

,

.

: C

. 2, 2.

: ( ) ( )ME ' ME 2+ =

-

xy xx -

yy -

:

2 2

2 2

x y 1

+ = , 2 2 2 = ( 2)

O BB 2.

-

xy yy -

xx -

:

2 2

2 2

x y1

+ = , 2 2 2 = ( 3)

C 2 2

2 2

x y1+ =

, > .

H C

(x1,y

1) :

1 1

2 2

xx yy1+ =

x

y

y

x

y

x

4

1

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75. -

y

x

C

2 2

2 2

x y

1+ = , > .

H C

(x1,y

1) : 1 1

2 2

xx yy1+ =

C 2 2

2 2

x y1

+ = .

C

:2

2

= = .

0 1< < : 2

1

=

.

:

. -

2

2. : ( ) ( )ME ' ME 2 =

.

-

xy xx

yy -

:

2 2

2 2

x y1 =

, 2 = 2 2.

-

xy yy

xx

Mathimatika Kateuthinsis b Lykeiou

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