ΤΕΛΕΥΤΑΙΑ ΕΠΑΝΑΛΗΨΗ ΜΑΘ ΚΑΤΕΥΘ Β ΛΥΚΕΙΟΥ

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- - - 2012 2 ENOTHTA 1--3-252 26-543 55-69 e-BOOK , 2012 , .. . , , , , . 2011-2012 , . . 1 : M . 2 3 - ( ) () ( ) ( . ) ( ) ( ) MATHEMATICA ( ) .. , , . vaggelisnikolakakis@hotmail.com6937020032 3 - -1 - 1. : - : . ( ) - : . ( ) - : . - : . - : .( .). - , o| : ( , ) o | ( , ) | o 00ss1800.. OA = o koi OB = | : 00.. || o =00 1800 ..c |+ , =1800 =900 ..=900 u v 0 (00ss1800) 2. : - : AI = BI + ABo | | o + = +) ( ) ( | o | o + + = + +o o = + 00 ) ( = + o o- : IB = I A AB) ( | o | o + = 14- :| o | o + = + ) ( | o | o = ) (o o o + = + ) ( o o o = ) (0 0 = = o 0 = o) ( ) ( ) ( o o o = = ) ( ) ( ) ( o o o = = | o = 0 = | o =o o = 0 = o = 3. : +=2 2 + = 4. : - :(x1, y1)+(x2, y2)=(x1+x2, y1+y2) (x, y)=(x , y) - :22 1x xx+= 22 1y yy+=- ( ) (x1, y1)B(x2, y2) ) , (1 2 1 2y y x x AB =:// yy = (0 , y) ( 0) // = (x , 0) ( 0) - :2 2y x a + =21 221 2) ( ) ( y y x x AB + =22o o =5. : - c|| = =xy- x x' // o=0 - y y' // o 6. : - ) , det( // | o | o =21xx21yy=0 , ( ) x, y (x,y) 5-2 1// | o = - | o | o = //- | o | o | o = //7. : - ,, BI AB// IB AI// IA BI // .8. : - ) , ( | o ouv | o | o = - o | | o = - 0 = | o | o- | o | o | o = || | o | o | o = |+-22o o = 22) ( | o | o = - (2 22 ) | | o o | o + = ) ( ) (2 2| o | o | o + = -2 1 2 1y y x x + = | o9. : -| o| oouvu= 222221212 1 2 1y x y xy y x x+ + + = ouvu10. : v to| o v oo = 11. !!! ) : :) ( ( ) = ) :: = // = ( )) : | | |||| = ) : : 2 22( ) = ) : :2 3 3 23(+) +3 +3 + = AB . OM 1 , : 2 +=6: + = AM OA OM + = BM OB OM .,= + + + = BM OB AM OA OM 2 + = OB OA OM 22 OB + OA= OM ) y , x (1 1A ) y , x (2 2B ) y , x ( .) OB OA (21OM + = ) y , x ( OM = , ) y , x ( OA1 1= , ) y , x ( OB2 2= ) OB OA (21OM + = )] y , x ( ) y , x [(21) y , x (2 2 1 1+ = )] y21, x21( ) y21, x21[( ) y , x (2 2 1 1+ =(x, y) = |.|

\| + +=2y y,2x x2 1 2 1 2x xx2 1 += 2y yy2 1 += ) , (1 1y x A ) , (2 2y x B ) , ( y x AB.:) y , x ( AB = , ) y , x ( OB2 2= , ) y , x ( OA1 1= , : = OA OB AB ) y , x ( ) y , x ( ) y , x (1 1 2 2 = ) y y , x x ( ) y , x (1 2 1 2 = x = x2 x1 y = y2 y1.2 xOy (x1,y2) (x2,y2) . = =, : x=2x x2 1 + y=2y y2 1 +yxA(x1,y1)B(x2,y2)(x,y)3 (x, y) (x1,y1) (x2,y2) : x = x2 x1 y = y2 y1.yxA(x1,y1)B(x2,y2)7 = OA ) , ( y x = o . x y , | x | ) (1 = OA| | ) (2y = OA . : = A A + OA = o21212) ( ) ( | | = OA + OA =2221) ( ) ( .2 2 2 2y x | y | | x | + = + =2 2 2y x | | + = o2 2y x + = o ) y , x (1 1A) y , x (2 2B . ) (AB ) y y , x x ( AB1 2 1 2 = , : 21 221 2) y y ( ) x x ( | AB | ) ( + = = AB ) y , x (1 1= o ) y , x (2 2= | 1 2. : 2 122111 2 2 12 21 1xyxyy x y x 0y xy x// = = = = | o. ) y , x (1 1= o ) y , x (2 2= | o = OA| = OB . : .u OB OA OB + OA = AB ) )( ( 2 ) ( ) ( ) (2 2 2(1)4 = (x,y), : 2 2y x + =.a A1yxA(x,y)25N (x1,y2) (x2,y2) () =21 221 2) y (y ) x (x + yxA(x1,y1)B(x2,y2)6 // 1= 2 1, 2 , ..7 . 2 1 2 1y y x x + = | oa |yx(x1,y1)(x2,y2)8 , , . 21 221 22) y y ( ) x x ( ) ( + = AB ,21212y x ) ( + = OA 22222y x ) ( + = OB, (1) : | o + + + = + 2 y x y x ) y y ( ) x x (2222212121 221 2| o + + + = + + +2 y x y x y y 2 y y x x 2 x x22222121 2 12221 2 12221| o = 2 y y 2 x x 22 1 2 1 2 1 2 1y y x x + = | o ) y , x (1 1= o, ) y , x (2 2= | ) y , x (3 3= , : i) ) ( ) y y x x ( y ) y ( x ) x ( ) y , x )( y , x ( ) (2 1 2 1 2 1 2 1 2 2 1 1| o = + = + = = | o ) ( ) y y x x ( ) y ( y ) x ( x ) y , x )( y , x ( ) (2 1 2 1 2 1 2 1 2 2 1 1| o = + = + = = | o.,) ( ) ( ) ( | o = | o = | o ii) ) y y ( y ) x x ( x ) y y , x x )( y , x ( ) (3 2 1 3 2 1 3 2 3 2 1 1+ + + = + + = + | o) y y x x ( ) y y x x ( ) y y y y ( ) x x x x (3 1 3 1 2 1 2 1 3 1 2 1 3 1 2 1+ + + = + + + = o + | o = .iii) 1 1xyxyx x y y 0 y y x x 02 122112 1 2 1 2 1 2 1 = = = = + = | o | o ) y , x (1 1= o) y , x (2 2= |,o | o | ouvu =| | | | | | | || o | o= u . 2 1 2 1y y x x + = | o,2121y x | | + = o 2222y x | | + = | :222221212 1 2 1y x y xy y x x+ ++= u8 : i) | o= o) ( | =( | o) ii) o) ( + | = | o+ o iii) o12 1 = | 9o, | , =222222212 1 2 1y x y xy y x x+ ++9 1. ; 2. AB; ;3. AB; ABIA ; ; 4. ABIA ; ; 5 ABIA ; ; 6. : = OA OB AB7. : - AB= - - AB= . AI= .- A ... M =8. , , :- + + = [] -( ) ( ) + + + = + [] - +0 = [ ] -( )+ - 0 = [ ] 9. N =( x , y ) 2 2x y o = +10. N (x1, y1 ) (x2 , y2 ) ( ) ( ) ( )2 22 1 2 1AB x x y y = + 11. +12. : ) ( ) = .) = =0 ..) ( ) = )=0 .) () = ) (-) = (- ) = -()) = =0 . 13. , .14. , 0 = = ( R e );1015. . 16. ; 17. y) (x, =, : 2 2y x + =18 . ( ) ( )1 1 2 2= x ,y, = x ,y . : - = . - + = ( , )- = ( , )- + = ( , )19. j , i .20. ; () 21. 22. ; 23. ; 24. 0 = ;25 . : = ....... = ............... =- .............. =0 ................... =.....=...... 26. 0 > 0 < ; 27. . 28 . : ( )( )( )( ) =-1 (, yy) = = = 29. ; 30. : = 11-2 - 1. (x0,y0) , : y-y0 = (x-x0) (x1,y1) (x2,y2) , : 2 11 12 1y yy y (x x)x x = 2. :- yy (0,): y=x+ - : y=x - (xo,yo) (//yy)x=xo- xx (xo,yo)y=yo: y=x y=-x.3. : ( ) ( xo , yo)..y-yo= (x - xo )4. ..: - . 2 1 2 1 // c c = - ) , ( y x = ooc o c = //(: xy=o )- : 12 1 2 1 = c cA(x ,y )00M(x,y)A(x ,y )1 1B(x2 y2) ,- - . 12- ) , ( y x = o1 = oc o c- 2 (x1,y1) (x2,y2 )1 21 2x xy y= - xx: c|e =- A x x' // c 0 =c- y y' // c . 5. : 0 = I + + By Ax0 = A 0 = B :BA = ( ) 0 = B- A 0 = B yy ( ) , 0BI- =0 AI = x: - x+y+=0 ). , ( A B = o- x+y+=0 ). , ( B A n =6. (xo ,yo ) : x+y+=0: 2 2) , (B + AI + +=o ooBy AxM d c7. : ( x1, y1) ,B(x2, y2 ) (x3 ,y3):1 31 221) , det(21) (x xx xA AB AB= I = I1 31 2y yy y 2 32 121) , det(21) (x xx xB BA AB= I = I2 32 1y yy y 3 23 121) , det(21) (x xx xAB= IB IA = I3 23 1y yy y8. : 1: y = x + 1 2 : y = x + 2 22 12 11) , (| |c c+= d =. : : x=0, yy y=0, 13 A(x1,y1) B(x2,y2) () xx. () ) y y , x x ( AB1 2 1 2- - = , = AB=1 21 2x xy y--. = 1 21 2x xy y-- Oxy ) y , x ( A0 0 . ) y , x ( M) y , x ( A0 0 ) , (0 0y y x x AM = 00x xy yAM= : AM // , AM =c =00x xy y ) x x ( y y0 0 = . ) y , x ( A0 0. : y yo = (x xo) A(x1,y1) B(x2,y2) () 1 ) y , x ( A1 1 ) y , B(x2 2, 2 1x x = 1 21 2x xy y= .2 xy () (xo, yo) . () y - yo= (x - xo)3) y , x ( A1 1) y , x ( B2 2 ) x x (x xy yy y11 21 21= xy(x1,y1)B(x2,y2) xy M(x,y)(x0,y0)14 2 1x x = , 1 21 2x xy y= ) x x ( y y0 0 = :) x x (x xy yy y11 21 21= . - y y ' ) , 0 ( | E , | + = x y , 0 y ) 1 ( x = | + + - Px y ( , )0 0, 0x x = , 0 ) x ( y 0 x0 = + + . 0 By Ax = I + +0 A =0 B = . , 0 By Ax = I + + 0 A = 0 B = .0 B = , BxBAyI = , BA = yy' |.|

\| IB, 0 . 0 B = , , , A = 0 AxI = , ' x x PA|\

|.|I,0 . Ax By + + = I 0 A = 0B = 0 . 4N x+By+=0 =0 =0 , x+By+=0 =0 =0 xy(x1,y1)B(x2,y2)xy(0,)xyP(x0,y0)15 0 = + + I By Ax ) A , B ( = o- 0 = B , BA = BA = o. . - 0 = B ,oyy' . 0 ) , ( ) , ( = = = AB AB B A A B n ) , ( A B = ) , ( B A n =. o 0 By Ax = I + + , ) , ( B A n = 1. () xx ; ; 2 . () ; ; 3 . N : () //=( x , y ) , (x=0) o=4 . : - 1//2 1 2 = - 1 2 1 2 1 = 5. 1,2 -1,2 1 = 2 1 . 2 = - 1. 6. y = 1x y = - x = 0.7. x+y = 1 , = 0 (, 0) (0, ).8.N A(x0 , y0) :y y0 = (x x0)5 0 By Ax = I + + ) A , B ( = o.6 x + By + = 0 ) , ( B A n =169. N A(x1 , y1) B(x2, y2 ) x1 = x2 :y y1 =2 12 1y yx x(x x1)10. yy (0 , ) y= x + 11 . : - ( yy) - xx A(x0 , y0)- yy A(x0 , y0)- 1 3 . - 2 4 . 12. y=3 x. . 30 . 60 . 45. 90 . 135yxBA013. (), (x1, y1) (x2, y2) . y1 = y2. x1 = x2 y1 = y2. x1 = - x2 y1 = y2. y1 = y2 x1 = x2. x1 = x214. . . y =x . y =x . y= x. y = x . y = xyAxB015.. : Ax+By+ = 0 =0 = 0 (1) (1) . 16 . Ax+By+ = 0 - ( ) = B , -A- ( ) = A , B17 . - (x0 , y0) Ax+By+ = 0 . - A(x1 , y1) , B(x2, y2 ) , (x3 , y3) .- 1 : y= x + 1 2 : y= x + 217-3 - 1. : - :x2+y2=2- (x0,y0) :(x-x0)2+(y-y0)2=22. : - x2+y2=2 (x1,y1) : xx1+yy1=2- (x-x0)2+(y-y0)2=2 ()=+ : 0 KA AM = (0 , 0) , (1,1) (,) ( )0 02 21d ,1o_ + |K c = = o +3. : i) x2+y2+x+By+=0, 2+2-4>0 ii) H x2+y2+x+By+=0, 2+2-4>0 4. : (x0,y0) , d(K , )= d(K , ) (x,y)xyCx,(x,y)yC K(x y )0 0(x,y)xyCA(x1,y1)x yK(-A/2,-B/2) C: - A(x1,y1) , xx1+yy1=2- (x1,y1) , i) C ii) H x2+y2=2d(K , )= d(K , ) :185. : 2 :2 21 1 12 22 2 2( 2 )x y x y x y x yotqo tev cio ocev tev k kev + + A + B + I E`+ + A + B +I )1) 2 . 2 2 . . 2) 1 . . 3) R .: = 1 2R R o + < 1 2R R o + = 1 2R R o =