()

()

1. ; 2. ; 3. ; 4. ; : 1. , . .

2. ,

. .

: = d(, ).

3. .

.0 .

: = 0 = 0.

4.

.

.

1. ; 2. ; 3. ; 4. ; : 1. . .

, , , . :

.

2. :

( )

, ( ).

,

: .

3. .

: .

.

4. ( ) .

1. ; 2. ;

3. , :

;0),(

;),(

;2

),(

1. .

.

:

i)

ii)

2. .

.

,

),(

),(

AOB .

),(0

3. :

,0),(

.

,),(

.

,2

),(

,

, .

1. ;

2. ; 3. ;

1. , .

.

.

:

.

2.

.

,

( ). , , :

, ,

, . 3. :

i) ( )

ii) )()( ( )

iii) 0 ( :0 )

iv) 0 )( ( : )

, .

1. ;

2. .x

1. , .

. :

)(

, , :

.

2. . :

x

0x)]([x

)()()x(x

.xx

. AB . OA OB

.

AB OA OB

BAB

, .

. . .

:

, ,0MN

MN .

,

AB .AB

, . ,

OM ,

. . :

,0M .

,0MM .

:

, .

.

, .

. .

1. ;

2. ; 3. ;

1. 0 ,0

, :

, > 0 , < 0,

, .

= 0 ,0 .0

1

,

0.

2. :

)(

)(

)()(

:

0(0 )0

0,

,0 =

)()()(

)(

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

2211

,1 ,2 , .

, 32 723

, , .

1. () () ; 2. , , ( ) ;

1. , () ()

() () ,

.

2. , , ,AB , A ( ).

.

, 3AB :

, , , ,

.3AB

, ,0

, R

( )

,0

.

.0

,

.

:

,

,

= ,0 0

.

, .

1. ; 2. G ; 1. , :

2

OO

.O2OO

2. G

3

OOOOG

(1)

, G (1) , G . (1) :

OG3OOO (2)

, . :

i) 0yx x, y R;

ii) yx x, y, , R;

i) 0yx (1) x = y = 0.

, x 0, (1)

x

y

, . x = 0, (1)

,0y0y 0

x = y = 0. ii)

)i(

0)y()x(yx

)i(

(x = 0 y = 0) (x = y = )

x = y = .

. , , .

, , ,

, ,

. . . . , , :

, ,

( ), R

R,

, , ,

. .

1. ; 2. ; 3. xx yy; 1. xx

1OI OIi

x, , xx. x Ox .

xx, O .i

x , .xiO x . ,

x (x).

2. xx, yy ,i j

, xy. , , xx yy . x, y , . (x, y). x y .

3. xx , (, 0) R. yy , (0, ) R.

1. ; 2. . 3. ;

1. .

,

jyix21

x, y ).y,x(

ix jy i j

.

x, y .

2. )y,x( ).,(

jyix ji

jy)(i)x(jijyix (1)

x ,

jx

yi

i ,j .

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

1. )y,(x 11 ),y,(x 22

, , R.

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

1. ,)y,(x 11 .jyix 11 , ,)y,(x 22

.jyix 22

)jyix()jyix( 2211

j)yy(i)xx( 2121

: . :

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

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

2

OO

2

xxx

.

2

yyy

(x1, y1) (x2, y2), (x, y)

2

xxx 21

2

yyy 21

1. (x1, y1) (x2, y2), ;

2. y);(x,

3. (x1, y1) (x2, y2), (); 1.

OO (1)

).y,x( )y,(xO 22 ).y,(xOA 11

(1) (x, y) = (x2, y2) (x1, y1) = (x2 x1, y2 y1)

x = x2 x1 y = y2 y1. : .

2. ,y)(x, .yx 22 ,

()2 = ()2 + ()2

22

2

yx 22 yx

3. = (x2 x1, y2 y1).

() = 2122

12 )y(y)x(x

212

212 )y(y)x(x)AB(

, ().

)y,(x 11 )y,(x 22 .

0yx

yx

22

11

, ).,(det

0),(det

1. xx

;

2. y)(x, ;

3. ;

1. x ,

),O(

xx. 0 < 2. :

xx = 0

yy = 2

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

y ,

.

x = 0 yy,

yy .

= 0 xx

xx yy, = .

, yy, 2

,

2

3 .

3. 1, 2 1 , 2

1 2 1 = 2

,1 2 , 1 yy,

2 yy, 1 .2

. , . :

1. ;,

2. .

1. , ,

.

0 ,0 .0

2. :

( )

0

,22

2

1jjii 0ijji

)()(

,

,

2

,

)(

2

22

,

222 )(

)y,(x 11 )y,(x 22 ,

i) 2121 yyxx

ii) )()()(

iii) )(

iv) 1,

, yy

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

.

()2 + ()2 2() ()

AOB = ()2 (1)

2)yx()yx( 2222

21

21

2122

12 )yy()xx(

2121 yyxx

(1) , , , . ii) :

21212211 yyxx)y,x()y,x()( )()yyxx( 2121

)()yyxx(yyxx)y,x()y,x()( 212121212211

.)()()(

iii) ).y,x( 33

)( = (x1, y1) (x2 + x3, y2 + y3) =

= x1(x2 + x3) + y1(y2 + y3) =

= (x1x2 + x1x3) + (y1y2 + y1y3) =

= (x1x2 + y1y2) + (x1x3 + y1y3) =

=

iv) , yy, x1x2 0.

0

x1x2 + y1y2 = 0 0xx 21

0x

y

x

y1

2

2

1

1

1

1

1 x

y .

x

y

2

2

, :

2)(22

2

2)(22

2

22

)()(

)y,(x 11 );y,(x 22

(1)

2121 yyxx (2)

21

21 yx

22

22 yx (3)

22

22

21

21

2121)3(),2()1(

yxyx

yyxx

(4)

, ,

,O OM O .

.O

.O

O)O(

, .0

.

. 2

x y (C) (C), , .

. . , : . , , . .

. (C) : f(x, y) = 0 (x0, y0) (C). (x0, y0), (x0, y0) (x0, y0) yy, xx , :

(C1) : f(x, y) = 0 (C) yy.

(C2) : f(x, y) = 0 (C) xx.

(C3) : f(x, y) = 0 (C) . , (C) , x x y y x, y x, y, yy xx .

. xy () xx . : xx () () xx. 0 < .

, () xx, () xx = 0.

. : (), xx

.2

()

= .

= ,2

. ,

yy . :

0 < < ,2

> 0 .

2

< < , < 0 .

= 0, = 0 .

= ,2

.

. ().

() xx,

= = + . = .

: ()

().

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

yy, x1 x2, . = (x2 x1, y2 y1)

.xx

yy

12

12

. (1), (2) ,1 2 (1),

(2) . ,1 2 1, 2

. :

(1) (2) 1 2 1 = 2

(1) (2) 1 2 12 = 1

:

( (1), (2) yy). 1

0 ( (1) yy (2) xx ).

B. :

, , , .

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

, ,1

.

(x0, y0),

2

yy,

2

xxM .

, . .

, . , , . :

1. . , , , .

2. . , , 1 ( 2 ..). , .

3. , , .. .

(x0, y0) x = x0 y y0 = (x x0) x + y + = 0 0 0. , x + y + = 0 0 0 . :

0, ,xy

.

= 0, 0, .x

yy. : x + y + = 0 0 0. 0 0 2 + 2 0 || + || 0.

. :

(). , . .

. , . , .

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

. (), (x0, y0) . (x, y)

)xx(yyxx

yy00

0

0

= (x x0, y y0). x = x0, y = y0. : (x0, y0)

y y0 = (x x0).

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

x1 x2. ()

.xx

yy

12

12

()

)xx(xx

yyyy 1

12

121

x1 = x2. () yy () x = x0, x0 = x1. () x = x0.

() yy . x = x0 .

. () (x0, y0) , () :

y y0 = (x x0), (), ,

x = x0, . () , .

, . :

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

() yy y = x, .

() (x0, y0) xx, y = y0. yy x = x0.

xOy yOx

y = x y = x, 1 = 45 = 1 2 = 45 = 1 .

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

0, .1

yx

.

() ( ).

() yy, . , () : x = x0,

AB

A0B

AB

A00B

yy

xx2x

yy

xxxx

(2x0 xA, yA).

() yy. ()

, .1

() : y yA = (x xA).

() () .

BA x

2

xx

BA y2

yy

(x, y).

() x + y + = 0.

0, () .

()

= (, ) = (, ).

= 0, () yy, ()

= (, ) ( = (0, ) yy)

= (, ). :

() : x + y + = 0

= (, ) = (, ).

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

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

:

(1) (2) 1 2 0BA

BA

22

11

(1) (2) 1 2 21 = 0 12 + 12 = 0

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

=

0.

()

() = .

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

(x0, y0) () : x + y + = 0

22

00

yx)d(M,

(x1, y1), (x2, y2), (x3, y3) ,

),det(2

1),det(

2

1),det(

2

1E

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

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

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

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

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

(x, y) () d(, ) = d(, ) 22

2

22

1 yxyx

(x + y + 1 = x + y + 2 x + y + 1 = x y 2) 21

21

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

21 = 0

()

x + y + 2

21 = 0

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

() d(, ) = d(, ) 2222

yxyx

, , . . , (1) (2). .

1. (x0, y0) ; 2. (0, 0) ;

: 1. (C) ( ) (x, y) , (x0, y0) . (x, y)

220

20

20

20 )yy()xx( )yy()xx()K,M(d

(C) (x0, y0) 22

02

0 )yy()xx(

2. (0, 0) x

2 + y

2 = 2.

(C) : x2 + y2 = 1 .

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

. 5. P , . 6. . 7. . . 8. . 9. .

(C) : x2 + y2 = 2.

1

y

x y x

22

222

[0, 2)

x

y

x = y = . , (x, y) x = y = ,

x2 + y

2 = 22 + 22 = 2(2 + 2) = 2

(x, y) (C) : x2 + y2 = 2. , (C) : x2 + y2 = 2

x = y = [0, 2)

(C1) : x2 + y

2 = 2 (x1, y1)

xx1 + yy1 = 2.

(x, y) () (C1) (x1, y1)

0

= (x1, y1) = (x x1, y y1).

:

0 x1(x x1) + y1(y y1) = 0 xx1 + yy1 = 21

21 yx

xx1 + yy1 = 2

(C1), .yx22

121

() : xx1 + yy1 = 2.

(x0, y0) ,

(C) : x2 + y

2 = 2. xx0 + yy0 =

2.

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

() : xx1 + yy1 = 2. (), x0x1 + y0y1 =

2 (1)

() : xx2 + yy2 = 2. (), x0x2 + y0y2 =

2 (2)

xx0 + yy0 =

2

, x0 0 y0 0, , (1), (2). (3) .

x2 + y2 + x + y + = 0 , , R. :

x2 + y

2 + x + y + = 0 (x2 + x) + (y2 + y) =

42

y

22y

4

A

2

Ax

22x

222

222

4

4

2

By

2

x

2222

(1)

2 + 2 > 4,

2

B,

2

K

.2

422

2 + 2 = 4, .2

B,

2

K

2 + 2 < 4, .

x2 + y2 + x + y + = 0 2 + 2 > 4.

2

B,

2

K .

2

422

(C) : (x x0)2 + (y x0)

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

. () (C), () (C),

22

00 yx) d(,

. (C)

(1) : x = x0 (2) : x = x0 + yy.

1. (); 2. ;

:

1. (C) (), .

2. xx (),

yy ,

0,

2

p (x, y) :

(C) d(, ) = d(, ) 2

pxy

2

px 2

2

y2 = 2px

xx ,0,2

p

(C)

() : x = 2

p y2 = 2px.

yy, ,2

p,0

() : y = 2

p (C) x2 = 2py.

.

, .

p y2 = 2px x2 = 2py .

y2 = 2px p x x 0 .

(C1) : y2 = 2px (C2) : x

2 = 2py

(x1, y1);

(C1) : y2 = 2px (x1, y1)

yy1 = p(x + x1)

(C2) : x2 = 2py (x1, y1)

xx1 = p(y + y1)

(C1), (C2) , :

,y

p

1

(C1) : y2 = 2px

,p

x1 (C2) : x2 = 2py

(x0, y0) (C) : y2 = 2px .

() ; (x1, y1) (x2, y2),

() : yy1 = p(x + x1) y0y1 = p(x0 + x1) (1) .

() : yy2 = p(x + x2) y0y2 = p(x0 + x2) (2) .

yy0 = p(x + x0) , , (1) (2).

() : yy0 = p(x + x0)

P (C) .

() (C) : y2 = 2px (C)

1(x1, y1) M2(x2, y2). y1y2 = p2.

:

p2

yxpx2y

21

1121

p2

yxpx2y

22

2222

1

21

1 y,p2

yM

2

22

2 y,p2

yM

12

21

22

21 yy,p2

yyMM

1

21

1 y,2

p

p2

yEM

1, 2, ,

21MM 0)EM,MMdet(EM 1211

0

yp2

py

yyp2

)yy)(yy(

1

221

122112

y1(y1 + y2) (21y p

2) = 0

21y + y1y2 21y + p

2 = 0 y1y2 = p

2

; .

. .

= 2, , 2

+ > 2 > 2 >

, , .

(, 0) (, 0),

1

y

x2

2

2

2

2 = 2 2.

y , (0, ) (0, )

1

x

y2

2

2

2

2 = 2 2.

1

y

x:)(C

2

2

2

2

1 1;

x

y:)(C

2

2

2

2

2

:

x x, yy ,

y y, xx ,

x x y y, .

1

y

x:(C)

2

2

2

2

, ;R*

. :

> , (C) xx 2.

< , (C) 2 yy.

= , (C) (C) : x2 + y2 = 2 . (C) .

, ; ;

(C) : 1

y

x2

2

2

2

> ,

. = .

2 = 2 + 2, 2

222

2

2

2

2

2

2

11

> ,

< 1, < 1.

.

1

y

x:)(C

2

2

2

2

1 1

x

y:)(C

2

2

2

2

2

(x1, y1);

(C1) : 1

y

x2

2

2

2

(x1, y1)

1

yy

xx2

1

2

1

(C2) : 1

x

y2

2

2

2

(x1, y1)

1

xx

yy2

1

2

1

, , xx x = x = , .

2x2 + 2y2 = 22 2x2 + 2y2 = 22, (x1, y1)

2xx1 + 2yy1 =

22 2xx1 + 2yy1 =

22 .

(x0, y0) (C) : 1

y

x2

2

2

2

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

() : .1

yy

xx2

1

2

1 (), :

1

yy

xx2

10

2

10 (1)

() : .1

yy

xx2

2

2

2 (),

1

yy

xx2

20

2

20 (2)

1

yy

xx2

0

2

0 ()

, (1) (2). () , .

;

(C) () (C) . () () ,

.

, . () . , , .

; .

.

= 2 | | = 2,

| | < 2 < 2 <

= 2 .

(, 0)

(, 0), ,1

y

x2

2

2

2

2 = 2 2.

xy yy ,

,1

x

y2

2

2

2

2 = 2 2.

(C) : .1

y

x2

2

2

2

:

xx yy .

.

xx (, 0) (, 0) y = 0.

,1

y1

x2

2

2

2

x2 2 |x| (x x )

x = x = . .

x = 0, ,1

y2

2

.

yy.

= , .

,1

x

y2

2

2

2

.

1

y

x:)(C

2

2

2

2

1 1;

x

y:)(C

2

2

2

2

2

.

x

y x

y

(C1).

x

y x

y

(C2).

1. (x1, y1) :

i) 1,

y

x:)(C

2

2

2

2

1

ii) 1;

x

y:)(C

2

2

2

2

2

i) (1) (C1) (x1, y1)

1

yy

xx2

1

2

1

ii) (2) (C2) (x1, y1)

1

xx

yy2

1

2

1

2. ( ); , .

(C1) (C2) , y

0

y

x2

2

2

2

0

x

y2

2

2

2

, .

(, ), (, ), (, ) (, )

.1

y

x:C

2

2

2

2

.

,1

x

y2

2

2

2

(, ), (, ), (, ), (, ). () , . , , (, 0), (, 0) (0, ), (0, ).

1. i) 1

y

x:C

2

2

2

2

;

ii) .

i) () 1

y

x:C

2

2

2

2

.

ii) > ,

> 1. > 1.

2. 1

y

x:)(C

2

2

2

2

1 1

y

x:)(C

2

2

2

2

2 1 2

. 1 < 2 ; .

1 < 2 21 <

22

2

1

<

2

1

<

xy

y .

, .