ΤΕΛΕΥΤΑΙΑ ΕΠΑΝΑΛΗΨΗ ΜΑΘ ΚΑΤΕΥΘ Β ΛΥΚΕΙΟΥ

  • Published on
    29-Jul-2015

  • View
    579

  • Download
    5

Embed Size (px)

Transcript

<p> - - - 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) = |.|</p> <p>\| + +=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 &gt; 0 &lt; ; 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' |.|</p> <p>\| IB, 0 . 0 B = , , , A = 0 AxI = , ' x x PA|\</p> <p>|.|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&gt;0 ii) H x2+y2+x+By+=0, 2+2-4&gt;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 + &lt; 1 2R R o + = 1 2R R o =19 1 2R R o &gt; 1 2 1 2R R R R o &lt; &lt; +20 1. : - (p2,0) : x=- p2 y2=2px- (0,p2) : y=- p2 x2=2py2.: 1(x1,y1) :- y2=2px yy1=p(x+x1)- x2=2py xx1=p(y+y1)3.: - y2=2px : yy p&gt;0 , yy p0 , p0p0 : x=- p22p,0)E( p0(0,: y=-p2p2 )p ye ) x=-,x==Cx2-y2=2( I )1. : (-,0) (,0) 22 22 2y x- =1 2x2- 2y2=22 2 2 - = C :2 22 2y x=1 , yy (0,-) (0,). ()=2 (x,y) C (ys - xe ) (y &gt; xe ) y=-,y==Cy2-x2=2( I , yy (0,0) , (-,0) (,0)(x,y)=2 ,|- |=2 1234x=- x=(x,y)(0,-)(0,)=2 ,|- |=2123 4y=y=-242. - 2 22 2y x=1 : = - &gt; 1 ( &gt; ) : - 2= 1 : 3. - 2 22 2y x=1 :y= x y= - x- 2 22 2y x=1 :y= x y= - x4. 2 22 2y x=1 (,) , (,-) , (-,-) , (-,) + y= xy=- xy= xy=-x CK --255.: 1(x1,y1) : -2 22 2y x=1 1 12 2xx yy=1 2x2-2y2=22 2xx1-2yy1=22-2 22 2y x=1 1 12 2yy xx=1 2y2 - 2x2=22 2yy1- 2xx1 =22(x1 y1 M1,) . : - , (=2 2 -)- ( yy) 2 ( yy) ( ) . C C: - A(x1,y1) , o - - (x1,y1), : i) ii) H . - C , . - C !!! , :- B , - B - .. - 26 Oxy C O( , ) 0 0 . Mxy ( , ) C, ,, : ( ) OM = (1),( ) OM x y = +2 2. , (1) x y2 2+ = x y2 2 2+ = . (2), , (2). , O( , ) 0 0 x2+ y2 = 2. 2 2 2: y x C = + Ax y ( , )1 1. Mxy ( , ) ) , (1 1y x OA= ) , (1 1y y x x AM = Mxy ( , ) e AM OA 0 = AM OA x x x y y y1 1 1 10 ( ) ( ) + = xx yy1 12+ = ,x y1212 2+ = ., 2 2 2 y x = + ) , (1 1y x A 21 1yy xx = + Oxy C ) y , x ( K0 0 . ) y , x ( M C, : = ) KM ( (1)1 (0,0) x2+y2=2. ; 2 x2+y2= 2 (x1,y1) , xx1+yy1= 2.31. (xo, yo) : (x xo)2+ (y yo)2 = 2CM(x,y)(0,0)xy (x1,y1)M(x,y) xy27, 2020) y y ( ) x x ( ) KM ( + = ., (1) : = + 2020) y y ( ) x x ( 2 2020) y y ( ) x x ( = + ) y , x ( K0 0 , 2 2020) y y ( ) x x ( = + : x2 2xxo + xo2+ y2 2yyo + yo2 = 20 ) y x ( y y 2 x x 2 y x2 2020 0 02 2= + + + 0 By Ax y x2 2= I + + + + 0x 2 A = ,0y 2 B = 2 2020y x + = I . x2+ y2+ Ax + By + = 0(1) : x2+ y2+ Ax + By + = 0 I = + + + ) By y ( ) Ax x (2 24B4A4By2B2 y4Ax2A2 x2 2 2222+ + I =||.|</p> <p>\|+ + +||.|</p> <p>\|+ + 44 B A2By2Ax2 22 2I += |.|</p> <p>\|+ + |.|</p> <p>\|+ .: - A B2 24 0 + &gt; I ,(1)KA B |\</p> <p>|.|2 2, =+ A B2 242I.-A B2 24 0 + = I , (1) , KA B |\</p> <p>|.|2 2, .- 0 42 2&lt; + I B A ,(1),) , ( y x M .4 x2+ y2+ Ax + By + = 0 5 :x2+ y2+ Ax + By + = 0 2+ 2- 4 &gt; 0 . (x0,y0)M(x,y) xy28 12222= +yx , 1 &lt; = 2 2 =:o= co | o=2 2222 221 |.|</p> <p>\|o| =o | o= c 221 |.|</p> <p>\|o| = c 21 = . 12222= yx, 1 &gt;o= c . 2 2 + = , : o= co | + o=2 2221 |.|</p> <p>\|+ =12 = . C 1y x2222=|o, = 1y x2222=|o 1y x2222=oo = 2 2y x a2 o= co o + o=2 2oo=22oo=2= 26 1y x2222=|+o. : 21 =7 ;. 12 c =o|8 ; =229 1. O(0,0) 2 2 2x + y = 2. C :2 2 2x + y = A(x1,y1) :21 1xx+yy =3. K(x0,y0) ( ) ( )2 220 0x-x + y-y = 4. 2 2 2 2x+ y+ Ax + By + = 0 4 0 () + B I &gt; () 2 2 4, =2 2 2A B + B I | |K |\ ..5. ; 6. c p p( ,0) (): = -2 2 2y=2px7. xx 2y=2px8. 2y=2px A(x1,y1) .9. ; 10. E ;11 . (,0) (-,0) 2 : 2 22 22 2x y+ =1 , = - 12 . xx,yy 2 22 22 2x y+ =1 , = - 13 . 2 22 22 2x y+ =1 , = - (x1,y1) .14 . () ; - 0 &lt; &lt; 1-2= 1-15. ; 16 . ; 17. ; 18. :2 22 22 2x y+ =1 , = - . - x s s , - y s s3019 . E ;20 . (,0) (-,0) 2 2 22 22 2x y- =1 , = - 21 . xx,yy 2 22 22 2x y- =1 , = - 22. 2 22 22 2x y- =1 , = - . x o &gt;23 . 2 22 22 2x y- =1 , = - (x1,y1) .24. ; 25. () ; : 1 &lt; 2= 126 . 2 22 22 2x y- =1 , = - .27. ; ; 28. ; 29. x2+ y2 + 2x + 3y - 1 = 0 x2+ y2+ 2x + 3y + 2= 0 . 30. (- 2, 2) (4, 2) (x - 1)2+ (y - 2)2= 9 . 31. y = x. (x - )2 + (y - )2 = 2. 32. (x1, y1) (x0, y0) . : (x1 - x0)2+ (y1 - y0)2 &lt; 2. 33. (x - 1)2+ y2 = 1 y2 = - 2x . 34. 32x2+51y2=23 . 35. 22x-22y = 1. &gt; . 36. C: 22x-22y = 1 yy . 31 10B2 =o,3 =|,1 = = + 0 3 2 | o (1), + = A o | | o 4 2. (1) = | o2321 : + = | | o2349412 2 2316261669694641699616469616 9 42 2 2 2 2 2= = + = + = + = + = |o | | | | o(2)(1) + = o | 3 2 : + + = o o | 12 9 42 2 23412161129412491211291241212 2 2 = = = = o | o(3)(1) = o | 3231 : + = | o o | 9494912 2 24494 94149412 2 2= + = + = o | | o(4) (2), (3), (4) : 33238834314 4 2 = + = + + = A20 = = | o 0 = | o, 9 e , 0 22 2 2 2&gt; +|.|</p> <p>\| + | | o o ; (), 0 22 2 2 2&gt; +|.|</p> <p>\|+ | | o o.: =||.|</p> <p>\||.|</p>...