Oi Apodeixeis b Lykeiou

7/31/2019 Oi Apodeixeis b Lykeiou

1/48

B

30

4328920 - 4311211

w w w . s t o x o s o m i l o s . g r


2/48

.

22

25 .

1 .

14 .

.

.

. 3 .

.


3/48

x

, , .

, .

.

.

, .

, .

, . .

:

.

6

4

2

2

14

. 2

1

6

4

- 2

2

14

6

3

2

3

14

2

6

4

. & 2

. - . 2

14

2

3 5

:

x = 100%


4/48

30

.: 210 4328920

Fax: 210 4311211

30

210 - 4328920

www.stoxosomilos.gr

. 62

210 - 4314727

www.stoxosnet.gr

. 62

210 - 4314727www.alexander.edu.gr

30

210 - 4328920

www.stoxosnet.gr

. 62

210 - 4314727

210 - 4007938

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5/48

1 1 1 1

----

1 1 1 1 2222

2 2 2 2 8888

3 3 3 3 ---- 11111111

4 4 4 4

---- . . . . 14141414

2222

----

1 1 1 1 18181818

2 2 2 2 27272727

3 3 3 3 33331111

4 4 4 4 38383838

3333

44441111


6/48

~ 2 ~

- x

xo= 100%

;;;


7/48

~ 3 ~

- x

----

1 1 1 1

I II III IV

+ + - -

+ - - +

+ - + -

+ - + -

x 2

x

2

+ x x +

3

2x

3

2x

+ 2 x 2 +

90 90 x+ 180 180 x+ 270 270 x+ 360 x 360 +

x x x x x x x

x x x x

x x x x x x

x x x x x x

() 0

30

45

60

90

180

270

360

(rad) 0

6

4

3

2

3

2

2

0 1

2 2

2

3

2

1 0 -1 0

1 3

2

2

2

1

2

0 -1 0 1

0 3

3

1 3 - 0 - 0

- 3 1 3

3

0 - 0 -

2 2 1 + = R

= , 0 R

= , 0 R

1 = , 0 R

2

2

21

=

+ , 0 R

2

2

1

1

=

+ , 0 R

1

-


8/48

~ 4 ~

- x

f , *+T R

A :i. , , x T A x T A+

ii. ( ) ( ) ( )f x T f x T f x+ = = f .

x ( )rad

: ( )x x rad =

2 : ( )2x x = , R

0

2

3

2

2

x

0 1 0 -1 0

= 1 = 2

= -1 =

3

2

2.

[ ]0,2 .

x ( )x rad

: ( )x rad = 2, : ( )2 x = R

y

( )y f x x= = 1

0

1

2 x


9/48

~ 5 ~

- x

2.

[ ]0,2 :

0

2

3

2

2

1 0 -1 0 1

=1

=0

=-1

=

=1

=2

.: ( )f x x

x

= = { }: 0= A x R x

x : ( ) x = , A . .

x () 2

2x

< x + ,

2x

=

f ( )f x x=

y

( )y f x x= = 1

0

1

2

2

y

( )y f x x= =

0

2

2

3

2

3

2


10/48

~ 6 ~

- x

( ) ( ),f x x = , 0 > ( ) ( ),g x x = , 0 >

1 1x 1 ( ) 1x 0 > :

( ) ( )x f x

f .

< 0 .

f :2

T

=

2x x = = + 2 ,x = +

2 ,x x = =

,x x = = + ,x x = = +

( ) + = ( ) = +

( ) + = + ( ) =

( )1

++ =

( )

1

=

+

1( )

+ =

+

1( )

+ =

2

1. 2 2 =

2. 2 22 = 2. 22 2 1 = 2. 22 1 2 =

3. 22

2 1

=

4.

2 1

2 2

= 5.2

2

2 1

=

+

6.2

2

12

1

=

+ 7. 2

1 2

1 2

=

+8. 2

1 2

2

=

9. 21 2

2

+= 10. 33 3 4 = 11. 33 4 3 =

1 ( ) = +


11/48

~ 7 ~

- x

2 ( ) + =

( ) = + :

( ) ( )

( ) ( ) ( )

= +

= + =

3 ( ) + = +

2

x x

=

2

x

=

:

( )( )2 2

2 2

+ = + = =

= + =

4 ( ) =

( ) + = + :

[ ]

[ ]

( ) ( )

( ) ( ) ( )

+ =

+ = + =

5 ( ) , ( ) 01

++ = +

0 .

( )( )

( )

+ ++ = = =

+ ( 0 )

1

++

= =

6 ( )1

=

( ) ,1

++ =

:

[ ]

[ ]

( ) ( )

( )

( ) 1 ( ) 1

= +

+

+ = = +


12/48

~ 8 ~

- x

7 2 2 =

2 ( ) 2 = + = + =

8 2 2 2 22 2 1 1 2 = = =

2 2

2 2 2 2 2 2 2

2 ( )

(1 ) 1 2 1

= + = =

= = + =

2 2 2 2 21 1 2 = =

92

22

1

=

2

22 ( )

1 1

+= + = =

10 21 2

2

+=

:

2 2 2 1 22 2 1 2 1 22

+

= + = =

11 21 2

2

=

2 2 2 1 22 1 2 2 1 22

= = =

12 2 1 21 2

=+

22

2

1 21 22

1 2 1 2

2

= = =+ +

.. ,, 11556611--11662266,,


13/48

~ 9 ~

- x

2222

x vax a R , *v N

R . .

, .. 3 50 ,0x .

, ..0 07 7 , 9 9x x= = .

x 11 1 0... ,

v v

v va x a x a x a

+ + + +

0 1 1, ,..., ,

v va a a a R , x

R .

( ), ( ), ( ), ( )P x Q x f x x ..

: 11 1 0( ) ...

v v

v vP x a x a x a x a

= + + + +

:

3 2( ) 4 5Q x x x x= + + .

00( )P x a x=

: 0.

11 1 0( ) ... , 0

v v

v v vP x a x a x a x a a

= + + + +

: 0,...,v

va x a

: 0

a

: 1 1 0

, ,..., ,v v

a a a a

:

x = : 1 0( ) ...P

= + + +

( )P x .

: R ( )P x , , ( ) 0P p =

.

x :vx

v

vx

-


14/48

~ 10 ~

- x

x , .

11 1 0( ) ...P x a x a x a x a

= + + + +

1

1 1 0( ) ...v v

v vQ x x x x

= + + + +

, :

0 0 1 1

1 2

, ,...,( ) ( )

... 0

aP x Q x

+ +

= = ==

= = = =

1 ( )

( ) ( )x ( ) 0x ( )

( ) : ( ) ( ) ( ) ( )x x x x = + ( )

( ) .

( ) ( ) ( ) ( )x x x x = + :

( ) : ( )x ( ) 0x = .

: ( ) ( ) ( )x x x =

( ) ( )

. :

- ( )x ( ) , ( )x ( )x , ( ) : ( )x x ,

( )x : ( ) ( ) ( )x x x = .

2

( )P x x p ( )P p .

:

: ( ) ( ) ( ) ( )P x x p x P p= +

( )P x x p ( )( ) ( )P x x p x = + , ( )x

x p . p= , ( )( ) ( ) 0 ( )P p p p p p = + = + , ( )P p = .

( ) ( )

( )x

( )x

( )P x x p

( )

( )P p =

( )

( )x

( )

( )x

( )x , ( ) 0x = .


15/48

~ 11 ~

- x

3

( )P x p , , p ( )P x .

x p ( )P x ( )( ) ( )P x x p x= . p= ,

( )( ) ( ) 0P p p p p= = , p ( )P x .

:

p ( )P x ( ) 0P p = , ( )P x p

: ( ) 0P p = =

:

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) 0 ( ) ( )P x x p x P p P x x p x P x x p x = + = + = , x p ( )P x .

4

:1

1 1 0... 0v v

v va x a x a x a

+ + + + = ,

. 0p p

.

0p , : 11 1 0... 0v v

v va p a p a p a

+ + + + =

( ) ( )1 2 1 21 1 0 0 1 1... 0 ...v v v vv v v vp a p a p a a a p a p a p a + + + + = = + + + .

Z

Z p 0a .

0

a


16/48

~ 12 ~

- x

3333

* .

.

1 2 3, , ... , , ...

.

.

.

:

i.

.. 2 1 = , 1 1 = , 2 3 = , 3 5 = ,

ii.

.. 1

1 = , 2 3 = 2 1 + += +

:3 2 1

1 1 2 = + = + =

4 3 22 1 3 = + = + =

5 4 33 2 5 = + = + =

.. 100 99 .

(..)

(..)

.

.

, , :

1 1, *

+ += + =

.

i.

1

: ( )1 1 = +

ii. , , ,2

+=

iii. , , , .. , ,2

+=

iv. .. :

1 2...

v vS a a a= + + + : ( )1

2v v

vS a a= + ( )12 1

2v

vS a v = +

-


17/48

~ 13 ~

- x

(..)

.

.

1

0 0

0

, *

( )

11

++ = = , .

i. 1

:

1

1

=

ii. , 0 , , =

iii. , , , .. , , 2 =

iv. .. , 1 2

...v v

S a a a= + + +

11

1v

S a

=

1

1vS v a= 1= .

1

1

: ( )1 1 = +

:

1 1

2 1

3 2

4 3

1 2

1

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

= = + = +

= +

= +

= +

: ( )1 1 = +

2

( ). , ,

: 2 = + 2

+=

, , . :

= 22

+ = = + = + = + =


18/48

~ 14 ~

- x

3

1

: 11

=

:

1 1

2 1

3 2

4 3

1 2

1

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

=

= =

=

=

=

: 11

=

4

( ). , ,

:2 =

, , 0 . :

= 2

= = =

5

.

1 : 11

1v

S a

=

.

: 2 2 11 1 1 1 1...vS a a

= + + + + + (1)

(1) : 2 3 11 1 1 1 1...vS a a

= + + + + + (2)

1,2 :1 1

v

v vS S a a = ( )1vS

1 :1

1

1v

S a

=

:

1= 1 1vS va= .

(( 11996688))


19/48

~ 15 ~

- x

4444

, 0 > 1 2, ,x x R , :i. 1 2 1 2x x x xa a a + =

ii.1

1 2

2

xx x

x

aa

a

= iii. ( )2

1 1 2x

x x xa a =

iv. ( )x x xa a =

v.

x x

x

a a

=

:

0 *1,a a R=

*

, , N

=

*1

, , R

=

0x > : 0 0x = .

: :f R R ( ) xf x a= , 0 1a< .

: 1a = , ( ) 1f x =

: A R=

: ( )0, +

i. 1a > R

1 2,x x R

:

1 2

x< 1 2x xa a< .

f 'Ox

ii. 0 1a< < R

1 2,x x R

:

1 2

x< 1 2x xa a>

f

+ Ox

( ) xf x a= 0 1a< x R 1 2 1 2x xa a x x= = 1 2,x R .

'x

'y

y

O

( )0,1A

xy a=

1a >

'x

y

O

( )0,1A

xy a=

'y

0 1a< <

-


20/48

~ 16 ~

- x

'y y ( )0,1

'x 0xa > x R .

( ) xf x a= 1

( )

x

g xa

=

x R :

1 1( ) ( )

x

x

xg x a f x

a a

= = = =

'y y 1a > .

e

, 1

1

v

va

v

= +

e 2,718e .

1

lim 1

v

ve

v

= +

.

( ) xf x e=

( ) , 1xf x a a= > ( 2, 718... 1a e= = > ) .

e 0( )

ctQ t Q e=

, .

0

Q Q 0t= . 0c >

Q . 0c < Q

.

, 1 0, 0xa = > > .

( ) xf x a= .

loga

.

: log , 1 0, 0xa x = = > >

:

loga

.

1 0 > , R 0 >

log xa a x= logaa =

1 = log 1a a =

0

1 = log 1 0a

=

y

x O

( )0,1A

( ) xg x a=

'

'y

( ) xf x a=

x = 100%


21/48

~ 17 ~

- x

1 0 > 1 2, , 0 > R :

1. ( )1 2 1 2log log loga a a = +

2. 1 1 22

log log loga a a

=

3. log loga a

=

1

0 > 1

= :1

1log log log

a a a

= = .

2

1 1 2, ,...,

.

: ( )1 2 1 2log ... log log ... loga a a a = + + + .

3

2 :1

log loga a

= .

10 .

10 log , 0x x = = > .

:

1. log10x = log10 =

2. log10 1= log1 0=

3.

( )1 2 1 2log log log = +

4. 1 1 22

log log log

=

5. log log =

6.1

1log log log

= =

1 2, , 0 > R

e.

.

, ln loge .

: ln , 0xe x = = >

:

1. ln xe x= lne = 2. ln 1e = ln1 0=

3. ( )1 2 1 2ln ln ln = +

4. 1 1 22

ln ln ln

=

5. ln ln =

7.1

1ln ln ln

= = 1 2, , 0 > R


22/48

~ 18 ~

- x

6.

1 0 > 1 0 > 0 > :log

loglog

=

4

ln

logln10

=

logln

log e

=

0a > 1a 1 2, , 0 > R :

1

( )1 2 1 2log log loga a a = +

1 1loga x = 2 2loga = . :1

1

xa =

22

xa =

: 1 21 2

x xa = 1 2 1 2

x x + =

: ( )1 2 1 2 1 2log log loga a ax x = + = +

2

11 2

2

log log loga a a

=

1 1

loga

x = 2 2loga = :

1

1

xa = 2 2

xa =

:1

2

1

2

x

x

a

a

= 1 2 1

2

x x

=

: 1 1 2 1 22

log log loga a ax x

= =

3

log loga a =

loga

= xa =

: ( )xa

= kx =

: log loga ax = =

:

0 >

1

= :

11

log log loga a a = =


23/48

~ 19 ~

- x

----

1111

= uuur uuur

.

= = = = uuur uuur uuur uuur uuur uuur uuur uuur

ur

, ur

, r

, R :

1. + = +

ur ur ur ur

2. ( ) ( ) + + = + +ur ur r ur ur r

3. 0 + =ur r ur

4. ( ) 0 + =ur ur r

5. + = + =ur r ur r ur ur

6. 0x x + = =ur r ur r r

7. 0x x + = = ur r r r ur

8. ( ) ( ) ( ) + = + ur ur ur ur

9. uuur uuur uuur

10. + +ur ur ur ur ur ur

: + = +ur ur ur ur ur ur

= = +ur ur ur ur ur ur

11.0 0 =ur r

, 0 0 =r r

12. ( ) + = +ur ur ur ur

13.( ) + = +ur ur ur

14. ( ) ( ) =ur ur

15.1 =ur ur

16. 0 0 = =ur r

0 =ur r

17.( ) ( ) ( ) = = ur ur ur ur

18. ( ) = ur ur ur ur

19.( ) = ur ur ur

20. =ur ur

0

=ur ur

21. =ur ur

0 ur r

=

:

/ / =ur ur ur ur

R , 0 ur r

|| ur ur

r

ur

, ur

, R = +

r ur ur

. r

ur

ur

.

:2

+ =

uuur uuur

uuuur

( ).

-


24/48

~ 20 ~

- x

y ur

,

( )1 1,x y =ur

, 1,

1y

1 1a x i y j= + r r r

. :

1. ( )1 1,x y =ur

, ( )2 2,x y =ur

1 2x = =ur ur

1 2

y y=

2. ( )1 1,x y =ur

, ( )2 2,x y =ur

( )1 2 1 2,x y y + = + +ur ur

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

( )1 1,x y =ur

4. 1 1

( , )A x y , ( )2 2,B x y :

( )2 1 2 1,AB x x y y= uuur

( ) ( )2 2

2 1 2 1AB x x y y= +

uuur

( , )M x y : 1 22

x xx

+= , 1 2

2

y yy

+=

5. ( )1 1,x y =ur

, ( )2 2,x y =ur

:

1 1

1 2 2 1

2 2

|| 0 0x y

x y x yx y

= =ur ur

1

1

y

x =ur , 2

2

y

x =ur 1 0x , 2 0x

||

=ur ur ur ur

, ,

ur ,

ur

( ) ( ), , , 0 .a a a a = r ur r ur r ur r ur

0a =r r

0 =ur r

, 0a =r ur r

( ) ( )1 1 2 2, , ,a x y x y= =r ur

:( )

1 2 1 2

1 2 1 2

2 2 2 2

1 1 2 2

,

a x x y y

x x y yaa

a x y x y

= +

+ = = + +

r ur

r ur

r ur

r ur

1. a a = r ur ur r

2. ( ) ( ) ( )a a a = =r ur r r ur

3. ( )a a a + = + r ur r r ur r r

4.22

a a a = =r r r ur

. .

5. = ur ur ur ur ur ur

6. = ur ur ur ur ur ur

7. 0 =ur ur ur ur

8. 1 2 1 2 0x x y y + =ur ur

9. 1

= ur ur ur ur

,

ur ur

10. 1 = ur r ur ur

( 1 =ur

ur r

) 1ur

r

1ur

ur


25/48

~ 21 ~

- x

1

ar

ur

. ar

ur

. .

=ur ur

=ur ur

.

' ' ' =uuuuur ur

' =uuuuur ur

.

' ' = =uuur uuuuur ur

' ' = =uuuur uuuuuur ur

, ' ' = uuuur uuuur

' ' = uuuur uuuuur

. , ' ' =uuuur uuuuur

,

' ' = uuuur uuuuuur

.

,

:

=uuur ur

=uuur ur

,

+ur ur

.

2

.

1. + = +ur ur ur ur

( )

2. ( ) ( ) + + = + +ur ur r ur ur r

( )

1. :

+ = + = ur ur uuur uuuur uuuur

+ = + =

ur ur uuur uuuur uuuur

.

+ = +ur ur ur ur

.

ur

ur

ur

ar

A M

'M

O

'O

a +r ur

a +r ur

ur

ar

'A

ur

ur

ur

ur

ur

ur

+

ur ur

ur

ur

ur

ur

+

ur ur

ur

ur


26/48

~ 22 ~

- x

2. :

( ) ( ) + + = + + = + = ur ur r uuur uuur uuur uuur uuur uuur

( ) ( ) + + = + + = + = ur ur r uuur uuur uuur uuur uuur uuur

( ) ( ) + + = + +ur ur r ur ur r

.

3

) .) :

.

) .

uuur

,

.

) .

uuur

+=uuur uuur uuur

=uuur uuur uuur

.

4

ur

ur

+ +ur ur ur ur ur ur

,

:

( ) ( ) ( ) ( ) ( ) +

+ +ur ur ur ur ur ur

5

, ur ur

, 0 ur r

, // , .R = ur ur ur ur

ur

ur

, 0 ur r

, =ur ur

,

. ( ). .

, ur

ur

0 ur r

,

:

=

ur

ur , =ur ur

:

, ur ur

=ur ur

, ur ur

= ur ur

0 =ur r

, 0 = ur ur

= ur ur

ur

ur

r

+ +ur ur r

+ur r

+ur ur

ur

ur

ur

ur

+

ur ur

=ur ur

ur


27/48

~ 23 ~

- x

6

uuuur

:

2

+ =

uuur uuur

uuuur

uuur

.

uuuur

:

= + uuuur uuur uuuur

(1) = + uuuur uuur uuuur

(2)

, (1) (2)

:

2 = + + + = + uuuur uuur uuuur uuur uuuur uuur uuur

. 2

+ =

uuur uuur

uuuur

7

, ur

,

ir

jr

.

Oxy ur

.

OA a=uuur r

.

1

A 2

A 'x

'y y , : 1 2OA OA OA= +uuur uuur uuuur

(1)

,y ,

: 1OA xi=uuur r

2OA y j=uuuur r

.

(1)

a xi y j= +r r r

(2)

ur

ir

jr

.

y .

ur

ir

jr

.

, ur

: ' 'a x i y j= +r r r

(3)

(2) (3) : ' 'xi y j x i y j+ = +r r r r

( ) ( )' 'x x i y y j = r r

'x , ' 0x x , '

'

y yi j

x x

=

r r

, , / /i jr r

, , ir

jr

.

'x x= , 'y y= .

//

//

ar

ar

A

x

y

O ir

jr

1A

2A

xo= 100%


28/48

~ 24 ~

- x

8

ur

ur

,

+ur ur

, ur

, R

ur

ur

ur

ur

.

1 1( , )y =ur

( )2 2,y =ur

, :

( ) ( ) ( ) ( )1 1 2 2 1 2 1 2x i y j x i y j x x i y y j + = + + + = + + +ur ur r r r r r r

( ) ( ) ( )1 1 1 1x i y j x i y j = + = +ur r r r r

( )1 2 1 2,x x y y + = + +ur ur

( )1 1,x y =ur

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

, +ur ur

:

( ) ( ) ( )1 1 2 2 1 2 1 2, , ,y x y x x y y + = + = + +ur ur

9

( )1 1,A x y ( )2 2,B x y

( ),y . 1 22

x xx

+=

1 2

2

y yy

+= .

1 1

( , )A x y ( )2 2,B x y

( ),y . :

( )12

OM OA OB= +uuuur uuur uuur

, ( ),OM x y=ur

, ( )1 1,OA x y=uuur

, ( )2 2,OB x y=uuur

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

, , , ,2 2 2

x y yx y x y x y

+ + = + =

1 2

2

x xx

+= 1 2

2

y yy

+= .

10

( ),x y 1 1( , )A x y ( )2 2,B x y

:2 1

x x x= 2 1y y y= .

1 1

( , )A x y ( )2 2,B x y

( ),x y

Buuur

.

, , = uuur uuur uuur

( ), ,y =uuur

( )2 2, ,x y =uuur

( )1 1, ,y =uuur

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

y

O

( )2 2,B x y

( ),M x y

( )1 1,A x y

( )2 2,B x y

( )1 1,A x y

y

x O


29/48

~ 25 ~

- x

11

( ),a x y=r

.

: 2 2a x y= +r

.

( ),a x y=r

OA a=ur r

.

1

A 2

A

'x x 'y y .

y ,

( )1OA x= ( )2OA y= .

1

OA A :

( ) ( ) ( ) ( ) ( )2 2 22 2 2 2 2 2 2

1 1 1 2a OA OA A A OA OA x y x y= = + = + = + = +r

.

: 2 2a x y= +r

12

1 1

( , )A x y ( )2 2,B x y

( ) ( ) ( )2 2

2 1 2 1AB x x y y= + .

1 1

( , )A x y ( )2 2,B x y . ( )AB

( )2 1 2 1,AB x x y y= ur

,

:

( ) ( ) ( )2 2

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

uuur

:

1 1

( , )A x y ( )2 2,B x y

( ) ( ) ( )2 2

2 1 2 1AB x x y y= +

13

ar

, ur

, ( )1 1,x y =ur

, ( )2 2,x y =ur

, 1, 2 0

1

2

: 1 2/ / =ur ur

( )1 1,x y =ur

( )2 2,x y =ur

1

2

,

:1 1 1 2

1 2 2 1 1 2 2 1 1 2

2 2 1 2

// 0 0x y y y

x y x y x y x yx y x x

= = = = =ur ur

.

ar

( ),A x y

x

y

1A O

2A

1 1,A x y

2 2,B x y

y

O

x = 100%


30/48

~ 26 ~

- x

14

( )1 1,y =ur

( )2 2,y =ur

.

=uuur ur

=uuur ur

.

:

( ) ( ) ( ) ( ) ( )2 2 2

2 AOB = +

, , .

: ( ) ( ) ( )2 2 2

2 1 2 1AB x x y y= + , ( )

2 2 2

1 1OA x y= + ( )

2 2 2

2 2OB x y= + .

, :

( ) ( ) ( ) ( )2 2 2 2 2 2

2 1 2 1 1 1 2 2 2x y y x y x y OA OB AOB + = + + +

( ) ( )2 2 2 2 2 2 2 21 2 1 2 1 2 1 2 1 1 2 22 2 2x x x y y y y x y x y OA OB AOB+ + + = + + +

( ) ( )OA OB AOB a = r ur

, : 1 2 1 2x y y = +ur ur

15

:

( ) ( ) , R = = ur ur ur ur ur ur

( ) + = + ur ur r ur ur ur r

1 2 1 = ur ur

, 1

= ur 2 =ur, , // /

ur ur

( )'y y

( )1 1 ,a x y= r

( )2 2,y =ur

( )3 3,y =r

, :

( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 2 2 1 2 1 2 1 2 1 2, ,x y x y x x y y x x y y = = + = + = ur ur ur ur

( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 2 2 1 2 1 2 1 2 1 2, ,x y x y x x y y x x y y a = = + = + = ur ur r ur

.

: ( ) ( ) ( ) = = ur ur ur ur ur ur

( ) ( )( ) ( ) ( )1 1 2 3 2 3 1 2 3 1 2 3, ,x y x x y y x x x y y y + = + + = + + +ur ur r

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

a a = + r ur r r

.

1 21 2 1 2 1 2 1 2 1 2

1 2

0 0 1 1y y

x x y y y y x xx x

= + = = = = ur ur ur ur

y

x O

ar

ur

( )2 2,B x y

( )1 1,A x y


31/48

~ 27 ~

- x

16

( )1 1,y =ur

( )2 2,y =ur

.

: 1 2 1 22 2 2 2

1 1 2 2

x x y y

y x y

+=

+ +.

a a =r ur r ur

=

ur ur

ur ur .

1 2 1 2a x x y y = +r ur

, 2 21 1a x y= +

r

2 22 2y = +

ur

.

,

1 2 1 2

2 2 2 2

1 1 2 2

x x y y

x y x y

+=

+ +

17

,a vr r

0a r r

. a v a v

= urr r r r

.

=uuur ur

=uuuur r

.

uuur

1

.

1uuuuur

r

ur

v

urr

:

1v

= ur

uuuuur r

( r

ur

).

ur

r

:

( )1 1 1 1 1 = + = + = = urur ur uuuuur uuuuur ur uuuuur ur uuuuur ur uuuuur r

: v

= urur r ur r

vr

1

ur

;;;


32/48

~ 28 ~

- x

1111

1. () 0 0

( , )A x y

:0 0

: ( )y y x x =

2. () 1 1

( , )A x y 2 2

( , )B x y 1 2

x x :

( )2 11 12 1

:y y

y y x xx x

=

, 2 1

2 1

AB

y y

x x

= =

uuur 2 1

x x

1

x= 2 1

x x=

3. () 'y y ( )0, :y x = + , ()

4. () ( )0,0 'y y ::y x =

5. () 0 0

( , )A x y

'x x :0

:y y =

6. () 0 0

( , )A x y

'y y : 0: x =

7. 0Ax By+ + = , 0A 0B

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

(1) .

: 0B :

1. (1) :A

B = .

2. (1) ( )1 , = uur

( )2 , = uur

.

3. (1) ( ) ( )1 2, , = =

uur uur

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

() 0, 0Ax By A B+ + = +

:

( ) 0 002 2

,Ax By

d d M + +

= = +

-

d

( )0 0 0,x y

x

y

O


33/48

~ 29 ~

- x

2. , :

( ) ( )1

det ,2

AB A = uuur uuur

( )det ,AB A

uuur uuur

ABuuur

Auuur

.

: ( ) ( ) ( )1 1

det , det ,2 2

= = uuur uuur uuur uuur

1

, .

ur

.

ur

'x , = = + .

= . :

, .

2

: ( )1 1,x y

( )2 2,B x y , 1 2x 2 1

2 1

y y

x

=

.

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

( )2 1 2 1,AB x x y y= uuur

, 2 1

2 1

y y

x x

.

y

x

ur

=

y

ur

= +


34/48

~ 30 ~

- x

3

Oxy ( )0 0,x y . A .

Oxy

( )0 0,y .

. ( , )M x y

( )0 0,x y ,

, AMuuuur

,

, AMuuuur

.

( )0 0,AM x x y y= uuuur

, 0

0

A

y y

x x

=

uuuur .

( , )M x y , , : 0

0

y y

x x

=

( )0 0y y x x = .

( )0 0,y .

: ( )0 0y y x x =

4

Oxy ( )1 1,x y ( )2 2,B x y

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

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

1 2

x ,

2 1

2 1

y y

x

=

( )0 0y y x x =

: 2 11 1

2 1

( )y y

y y x x

x x

=

1 2 0

x x x= =

,

.

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

0.

y

( , )M x y

0 0( , )A x y

y

x

2 2( , )B x y

1 1( , )A x y

. .


35/48

~ 31 ~

- x

5

:

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

, (1) .

.

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

( 1) 0x y + + =

0 0

( , )P x y , 0

x= ,

:

00 ( ) 0x y x+ + = .

, ,

0Ax By+ + = 0A 0B

, :

0Ax By+ + = 0A 0B

0B , A

y xB B

= ,

=

'yy 0,

.

0 = , , , 0A x

=

,

'x ,0P

.

0Ax By+ + = 0A 0B .

y

0 0

( , )P x y

-

xo = 100%


36/48

~ 32 ~

- x

1111

.

.

.

(0,0)

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

1 1

( , )x y : 21 1xx yy+ =

0 0

( , )y

: ( ) ( )2 2 2

0 0x x y y + =

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

.

()

.

.

y

x ( )0,0

y

x ( )0,0

( )1 1,y

( )0 0,x y

y

-


37/48

~ 33 ~

- x

1. (0,0) , ,02

p

, :

2

px = 2 2y px=

: y2 = 2x ( > 0 x 0) ( 0 0) (


38/48

~ 34 ~

- x

2 = 2 2 > > : 1

=

: 1

xx2

1

2

1 =

: 1

x

2

2

2

2

= , 2 = 2 + 2

: (-,0) (,0)

: (0,-) (0,)

: x

-yx,

y ==

: 1

>=

: 1

x

2

1

2

1 =x

x2 y2 = 2 y2 x2 = 2( = )


39/48

~ 35 ~

- x

1

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

Oxy

C (0,0)O .

( , )M x y C, ,

, , : ( )OM = (1)

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

x y+ = , 2 2 2x y + = .

2

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

1 1xx yy+ = .

C: 2 2 2x y+ =

1 1

( , )x y .

( , )M x y ,

OA AM , ,

: 0OA AM =uuur uuuur

. (1)

1 1( , )OA x y=uuur

( )1 1,AM x x y y= uuuur

.

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

2 2

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

1 1 ,xx yy+ = 2 2 2

1 1x y + = .

3

, 0 0

( , )x y : ( ) ( )2 2 20 0x x y y + =

xy C

0 0

( , )x y .

( , )M x y C,

,

, , :

( )KM = (1)

, ( ) ( ) ( )2 20 0x x y y = + . (1) :

( ) ( )2 2

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

0 0x x y y + = .

y

O

( ),y

( )1 1,A x y

y

x ( )0,0

( ),y

C

y

x

( )0 0,y

( ),y


40/48

~ 36 ~

- x

4

, ( ) ( )2 2 2

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

(2) (2) ;

(1) :

( )

2 2 2 2 2

0 0 0 02 2 0x y x x y y x y + + + =

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

:0 0

2 , 2x B y = = 2 2 20 0x y = + .

, (2) : ( ) ( )2 2x Ax y By+ + + = 2 2 2 2

2 22 22 4 2 4 4 4

A A B B A Bx x y y

+ + + + + = + +

2 2 2 2 4.

2 2 4

A B A Bx y

+ + + + =

:

2 2 4 0A B+ > , (2) ,2 2

2 2 2 24 4

4 2

A B A B

+ + = = .

2 2 4 0A B+ = , (2) , ,2 2

.

2 2 4 0A B+ < , (2) , ( , )x y

.

:

2 2 0x y Ax By+ + + + = , 2 2 4 0A B+ > ()

() .

1

'x

.

'x : 2 2y px= p x ( 0x ) . ,

'y y .

.

2

.

1 1 1( , )M x y 2 2y px= , , 21 12y px= ,

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

2 2

1 1 12y y px = = .

'x . , ( ).


41/48

~ 37 ~

- x

1

,a x a= = ,y y = = .

2 2

2 21

x y

a + = . ,

2 2

2 21 1

x y

a =

2 2 0x a a x a . y .

, ,x a x a= = ,y y = = .

2

) ;

) 21

= . ;

) .

2 2

2 21

x y

a + = , 1

= < .

) 2 2 = , 2 2

= ,

22 22

21

= =

21

= .

,

.

,

1

. , , ,

0

.

,

.

1

'x .

2 2

2 21

x y

a = . , :

2 2

2 21 1

x y

a = + ,

2 2 0x a a x a .

, a= a= ,

.


42/48

~ 38 ~

- x

2

i) ;

ii) 2 1

= .

iii) ;

iv) ;

i) .

2 2

2 21

x y

a = , , 1

= > .

ii) 2 2 = + , 2 2

+= ,

2

2 1

= +

, 2 1

= .

iii) , ,

. .

1,

, ,

0. ,

.

iv) = , : 2 21 1 2 2 = = = .

;;;


43/48

~ 39 ~

- x

3333

( )P v .

) 1 (1)P

) ( )P v ( 1)P v + .

( )P v .

, 0 ,

: 0 , :

, 0 = + <

, .

, 0 = + < ,

.

, . 0.

.

.

,

.

.

.

.

.

.

*

,

. *, .

.

.

.

, ,

.

, ,

.

-


44/48

~ 40 ~

- x

, 0 . |

. = .

:

: | ,

0 , .

| |

*1| , , | ,

| 0 *

| *| ,

, , . :

| | : =

| | : |

| |

| | : ( )| +

| 0 :

:

/ / / ( ) + , ,

.

1

.i. x .

ii. x .

2

, , x, y . | ( )| a y |y .

\/ a


45/48

~ 41 ~

- x

1

, , . :

i. | | , = = .

ii. | | , | .iii. | , | .

iv. | , | , ( )| + .

v. | 0 , .

i. | | , , , , = = ,

= , 1= 1 = = , = = .

ii. | | , , , , = = ,

= | .

iii. | , , = , = | .

iv. | | , , , = = ,

( ) + = + ( )| +

v. | 0 , 0 = . , = ,

1 .

;;;


46/48

~ 42 ~

- x

. . . .


47/48

~ 43 ~

- x

, ,

, ,

, ,

, ,

, .

xo

, x

e,

.

e.

, ,

. :

,

.

,

.

x = 100%


48/48

- x

5

3

2

3

2

15

:

. ,

. , ,

.

, ,

4

4

8

3

4

7

4

2

6

3

4

7

:

:

.

A

Documents

Oi Apodeixeis b Lykeiou