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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
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
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
.
.......................................................................................................................................................................................................................................................
...........................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
( 25)
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-
:
[ .[ -
, .
[ (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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 .............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
2. () x 135 yy (0,4).
i. ().
ii. () ()
(2,6).
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
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64. 4
3. (3,1), (5,1) (7,2). - - .
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
4. ( ) ( ) ( )2 2 2 x 5 6 y 1 0 1+ + + + = .i. R ;
ii. R xx;
iii. R
(0,0).
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
5. (1,1), (2,0) : x y 1 0 = .i. ()
.
ii.
.
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65. 4
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
6. , - ,
: 2 22x 2y 4x 8y 40 0 + = . (2,10)
-
(2,14). -
-
; ( .)
(1,4)
. ,
;
( (2,10).
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
7. :
( )1
: x 3 y 2 = ( )2 : 3 x y =i. (1) (2) R.
ii. .
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66. 4
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
8. (+2, 2) (,), , R .i. ()
.
ii. () .
iii. (3,3) ().
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
9. (3,6) : 4x 3y 9 0 = . (3,1)
45 .. :i. d(A,),
ii. ,
iii. .
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
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67. 4
............................................................................................................................
............................................................................................................................
............................................................................................................................
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.
............................................................................................................................
............................................................................................................................
( 10)
B.
(2,1) 1.
............................................................................................................................
............................................................................................................................
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69. 5
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
( 15)
20
A (2, 1) (4, 6). 2 ..
B. (4, 0), (2, 0) (1, 4).
(x, y) : ( ) ( )1
MB 2
= .
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
( 25)
30
. ( )x 1 y 2 + = ( ) 1 x y 2 1+ + = + R .
. (2,1)
, . .
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
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70. 5
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
( 25)
40
xy
.
1 (t - 30,2t - 40), t 0 .. ;
. ;
. ;
. 2 (t - 20, t 10), t 0+ ;
. t = 10;
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
............................................................................................................................
( 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