Tuyển chọn và hướng dẫn giải 500 bài tập Toán 10: Phần 1

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Phần 1 tài liệu Tuyển chọn 500 bài tập Toán 10 giới thiệu các bài tập theo chuyên đề thuộc phần Đại số bao gồm: Tập hợp và hàm số, phương trình và bất phương trình bậc nhất một ẩn, hệ phương trình bậc nhất hai ẩn, bất đẳng thức,... Mời các bạn cùng tham khảo nội dung chi tiết.

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LCfl N O I DAII MlfC i f f C Phan I SO 500 b a i t o a n loTp 10 dUcrc s o a n t h e o c a c y e u cau : Chuyen de 1 Tap hop va h a m so 05 Bam s d t s a c h giao k h o a h i e n h ^ n h . Chuyen d l 2 Phuong t r i n h v^ bat phucfng t r i n h bac n h a t mQt an 30 Cdc k i e n thufc ca hAn difcfc t a p t r u n g v a o tifng c h u y e n de. Chuyen d§ 3 He phuong t r i n h bac n h a t h a i a n 37 T r o n g m o i chuyen de, chiing toi t o m li/cfc p h a n h' t h u y e t , chi r a c&ch Chuyen de 4 B a t d i n g thufc 43 van d u n g l i t h u y e t v a o cAc b a i t a p tis d& d e n k h o , n a n g cao d a n theo Chuyen de 5 N h i thiic bac n h a t : f(x) = ax + b 63 htfdng tiep can miJc do c a c de t h i t u y e n sinh D a i hoc, t r i c h d a n v a Chuyen de 6 Phifong t r i n h b§c h a i mot an 66 g i a i m o t so de t h i t u y e n sinh D a i hoc c a c n a m 2002, 2003, 2004. Chuyen de 7 He phuong t r i n h bac h a i doi xufng doi v d i x va y 78 D a c b i ^ t , trUdc m o i b a i g i a i , c h i i n g toi d u a ra p h a n hiTdng dSn, phan Chuyen de 8 He phuong t r i n h d i n g cap bac hai. 84 n a y h e t siJc q u a n t r o n g , no giiip b a n doc t i m duoc h i i d n g g i a i quyet b ^ i t o a n , torn t ^ t qua t r i n h g i a i , de k h i g i a i xong b a n c h i c a n n^m Chuyen de 9 Dau ciia t a m thiic bac h a i 89 difoc c a c y c h i n h la dii. Chuyen de 10 B a t phiTOng t r i n h bac h a i 98 Sdch dugc chia lam hai phan : Chuyen de 11 Xac d i n h gia t r i ciia t h a m so m de t a m thilc bac h a i P h a n 1 : D a i so (300 bai). CO dau k h o n g t h a y d6i 109 P h a n 2 : H i n h h o c (200 bki). Chuyen de 12 So sanh so a v d i h a i n g h i e m ciia phuong t r i n h bac h a i 113 Sau m o i p h a n , c h i i n g toi c6 p h a n t o a n t d n g h o p n h ^ m giiip 'hoc s i n h Chuyen de 13 Phirorng t r i n h va bat phuong t r i n h c6 chiJa gia t r i tuyet doi on l a i c a c k i e n thiJc ccf b a n , t a p t r a I d i c a c c a u h o i tr^c n g h i e m . 129 Chuyen de 14 Phifcfng t r i n h v^ bat phucrng t r i n h c6 chufa cSn thufc 137 D&t bi^t : Cuoi s a c h c6 10 de k i e m t r a n h k m giiip hoc s i n h t\i d a n h gid v a k i e m t r a l a i c a c k i e n thiJc d a h o c t r o n g m o i h o c k i . Chuyen de 15 T d n g hop cudi n a m 153 M6i de kiem tra c6 : ^h^n HiNMHOC Cau h o i l i t h u y e t . Chuyen de 1 Vecto • 172 Cau h o i trftc n g h i e m . Chuyen de 2 True - Toa dp t r e n true 188 Toan t u l u a n . Chuyen de 3 He true toa dp Descartes vudng goc 194 M a c dii d a CO g S n g n h i e u k h i b i e n s o a n , n h i t o g c h ^ c k h o n g t r a n h Chuyen de 4 T i so luong giac 209 k h o i t h i e u sot, c h i i n g toi m o n g n h a n dufoc sU d o n g gop y k i e n x a y Chuyen de 5 Tich v6 hudng ciia h a i vectP 222 diTng ciia quy d o n g n g h i e p , c a c e m hoc s i n h de I a n t a i b a n s a u , s a c h se h o a n c h i n h h o n . Chuyen de 6 He thiic lupng t r o n g t a m giac 235 Chuyen de 7 G i a i t a m giac - f n g dung thifc te 244 Chuyen de 8 He thiJc lUcJng t r o n g di/dng t r o n 250 Chuyen de 9 Tdng hop cudi nSm 256 DQ kiem tra 290 De k i e m t r a Hoc k i I 290 De k i e m t r a Hoc k i I I va cuoi n a m 3
ci.H.y6n 1 T6P HOP V6 HflM so K i e n thurc ca ban I) Tap xac djnh c u a ham s o y = f(x) • Tap xac d i n h ciia h a m so' y = f(x) la : D = |x e R \ f ( x ) c6 nghia) • Cach t i m tap xac d i n h ciia h a m so'y = f(x). BU 0 hay y i > y 2 : H a m so nghich bien t r e n k h o a n g (a; b). 5
ni) T i n h c h i n 1^ c u a mQt h a m so • D$c bi^t a) Neu b = 0 t h i y = ax va do t h i 1^ dudng t h i n g d i qua goo tpa do. Gia sii h a m so y = f(x) c6 tap xac d i n h D thoa m a n d i l u k i e n Vx, x e D =>-xeD y (Ta bao D la tap doi x i i n g qua O) \ H a m so y = f i x ) chSn t r e n D H a m so' y = f(x) l e ' t r e n D Dinh nghia Dinh nghIa O f(- X) = f(x), V X e D f ( - X) = - f i x ) , V X e D a < 0 T i n h chat T i n h chat [b = 0 Do t h i ham so chSn n h a n true D 6 t h i h a m so' le n h a n go'e to a t u n g l a m true doi xufng do 0 l a m t a m doi xiifng b) Neu a = 0 t h i y = b la h a m so' h k n g v a do t h i la dudng t h i n g eung y phLfOng v d i true h o a n h Ox (Ta bao h a m so' h i n g c6 do t h i l a dudng t h i n g n i m ngang). y - X ->x O b y = b -y s 0 IV) Ham s o b a c n h a t : y = ax + b (1) Dinh nghia V) Hamsobachai: y = ax + bx + c {a^O) (1) H a m so' bae n h a t la h a m so c6 dang : y = ax + b (a ?^ 0) (1) 1) Dang y = a x % c (a ^ 0) (1) / I • Tap xAc d i n h : D = R • Tap xac d i n h D = R a > 0 => H a m so (1) d6ng bie'n t r e n R • H a m so (1) la h a m so^chin =>D6 t h i h a m so (1) eo true doi xiJng la Oy. • T i n h chat : a < 0 => H a m so' (1) nghieh bie'n t r e n R • B a n g bie'n t h i e n a > 0 a < 0 a > 0 a < 0 X -00 + 00 x -00 +00 X - 00 0 + 00 X - 00 0 + 00 +00 +00..^ y = ax + b -00 — " y = ax + b + GO \ 00 y y c - 00 - 00 Do thi : D6 t h i h a m so' bae n h a t la mot dudng t h i n g . H a m so (1) : H ^ m so (1) : y y • Dong bie'n t r e n k h o a n g (0; + » ) • D 6 n g bie'n t r e n k h o a n g (- oc; 0) • Nghieh bie'n t r e n k h o a n g (- oo; 0) • N g h i e h bie'n t r e n k h o a n g (0; + \ • D 6 t h i la mot parabol c6 : ->x D i n h S(0 ; c) o ~0 - True doi xiJng 1^ Oy. a > 0 a < 0 7
B a i 1. a) Cho t^ip A c6 n phan tiif (n S: 1). Chtfng minh A c6 2" t^p con. b) Cho A = |1, 2, 3, 4, 5, 6|. Gpi B la tgp con ciia A sao cho B chiiTa 1 ma khong chuTa 2. Hoi c6 bao nhieu tap B ? * HUdng dan a) D i i n g phufong phap qui nap. b) H i n h t h a n h tap B : ' • T i m tap A ' = A \ | l , 21 a > 0 a < 0 • T i m cac tap con ciia A' r 6 i t l i e m so' 2 vao mSi tap con nay ta c6 tap B 2) Dang y = a x % bx + c (a ^ 0) (2) GIAI 4ac - a) Xet m e n h de P(n) "A c6 n phan tuf (n > 1) thi A c6 2" t§p con" B i e n doi ve dang y ~ a X + = aX^- '2a 4a 4a • Ta CO P ( l ) diing v i neu A c6 m o t p h a n tilr t h i A c6 2^ = 2 tap con, do b la tap 0 va A. . ,, X = X + vdri 2a (Ta t r d ve dang y = ax^ + c) • Gia sii m e n h de P(n) diing k h i n = k (k > 1 , k e N ) ta chuTng m i n h A = b^ - 4ac P(n) diing k h i n = k + 1. Bang bien t h i e n - T h a t vay, k h i A c6 k p h i n tiJf, ch^ng h a n : A = Au = |ai, a2, , aki thi A CO 2^ tap con, dp la | |, | a i l , |a2l, , |ai, 32, , aiJ b b x - oc + X X - X + X Do do k h i A diioc t h e m mot p h a n tiir (au + i ch^ng han) t h i A c6 t h # m 2a 2a + » + =c A 2'' tap hop con bfing each t h e m p h a n tuf ak + 1 vao cac tap con n6i y y ^ 4 a ^ t r e n , do la |au + 1), |ai, ak + i l , , l a i , a2, , ak, ak + i l 4a - X - X => A CO 2'' + 2'' = 2''' ' tap con • ^ M e n h de P(n) diing k h i n = k + 1 (dpcm) Do t h i la mot parabol c6 : • b) • T i i tap A = I I , 2, 3, 4, 5, 6| ta c6 tap A' = A \ | l , 2| = {3, 4, 5, 61 , J - h -A ( A c A) - Dmh S • T i m t a t ca tap con ciia A ' (cac tap con nay k h o n g chufa 1 va 2) roi t h e m p h a n tii: 1 vao m 6 i tap con ciia A ' ta se dugc tap con B theo yeu cau d l bai. Vay so' tap B = so tap con ciia A ' = 2^ = 16 -a^ B a i 2. Cho bleu thii-c T = J^J—- + - a' - 1 1-2 + a2 Hay tim tat ca gia tri ciia a de T d\X(fc xac dinh va rut gon T ? * HUdng ddn • Chu y : a^ = a ^ va (x - y)^ = x^ - 2xy + y^ T xac d i n h k h i mau ^ 0.
GlAl • l a a - 1- a a -1) - 4 a| + 1 a - 1 a + l) 2 -1 - a l + a^ -1 1-a^ B a i 3. V e d o thi h a m so y = ^ x ^ - 2 x + 1 - ^4x'^ + 4 x + 1 y = X + 2 K HUdng dan . B i e n ddi y = ^ - 7B^ = |A| - |B • T i m d a u ciia A v a B roi l a m m a t d a u tri tuyet doi. u neu u > 0 ) (Chii y : I u I = -u n e u u < 0 V e da thi. GlAl • Bi^dtc 1 : T a CO y = J(x - if - ^{2x + if = |x - l| - |2x + 1 B a i 4. V e d o thi h a m so y = ^Ayi^ + 1 2 x + 9 - 2x. • Bvldc 2 : X e t d a u (x - 1) v a 2x + 1 Hitdng d&n • x - l > t ) o x > l va x - l < O o x < l • Giai giong bai 3 • 2x + l > 0 o x > - 4 v a 2 x + ly = - ( x - l ) + ( 2 x + l ) = x + 2 -y = 3 (x > 2 1^ -1 _2_ 3 ->x 2 10
B a i 5. T r o n g m a t p h d n g t o a d p , c h o t a m g i a c A B C v u o n g c&n t a i A . B i e t ( B C ) q u a M ( 0 ; 3) n e n 3 = - ~ (0) + m » m = 3 A ( l ; 6) v a t r u n g d i e m B C l a M(0; 3). T i m phU'cfng t r i n h c a c c a n h v a t o a dp c u a B , C ? Vay phuang t r i n h ( B C ) l a y = - + 3 (1) • Hi/dng ddn G o i ( x ; y ) l a t o a do B ( h o a c C ) , t a c6 : A B ^ = A C ^ = (x - 1)^ + (y - 6r • T i m phumig trinh (AM). ma A B - 2 Vs nen (x - 1)^ + Cy - 6)^ = 20 (2) • V i e t phircfng t r i n h ( B C ) (1) C h u y : ( B C ) q u a M v a v u o n g goc v d i ( A M ) y = - - X + 3 (1) G i a i he BC • A M J. B C & A M = = M B = M C => T a m g i a c A M B , t a m g i a c (x-1)" +(y-6)' = 20 (2) AMC vuong can t a i M A B = A C = A M ^2 T i r (1) t a CO : x = 9 - 3 y , t h a y x = 9 - 3 y v a o (2) t a dugc : G o i (x; y ) l a t o a do B ( h o a c C ) , t i n h A B ^ t h e o x v a y , t i n h A M ^ r o i (9 - 3 y - 1)^ + (y - 6)^ = 20 lOy^ - 6 0 y + 80 = 0 t h a y vao AB^ = 2AM^ (2) "y = 2 => x = 3 • y - - 6y + 8 = 0 G i a i h e (1) v a h e (2) t a c6 t o a do B , C t i i do v i e t phufcfng t r i n h ( A B ) y = 4 i ^ y : . - 3 va (AC) D a p so : B ( 3 ; 2) v a C ( ~ 3 ; 4) GIAI P h i i c f n g t r i n h ( A B ) c6 d a n g y = k x + b, ( A B ) q u a A d ; 6) v a B ( 3 . ; 2) [6 = k + b k = -2 nen < 2 = 3k + b b = 8 V a y p h i r a n g t r i n h cua ( A B ) l a y = - 2x + 8 T u o ' n g t\J. p h u a n g t r i n h cua ( A C ) l a : y = — x + — -3 2 2 A ( l ; 6), M ( 0 ; 3) A M - = ( 1 - 0 ) - + (6 - 3 ) - = 10 B a i 6. T r o n g m a t p h d n g toa d p , c h o t a m g i a c A B C v u o n g t a i A ( - l ; 2). [AM 1 BC B i e t B ( 0 ; 4) v a A C = 2 A B . T i m t o a dp C (xc > 0). Tam giac A B C vuong can t a i A n e n * HUctng ddn AM = — = M B = M C = VlO 2 • T i n h A B v a v i e t phu'o-ng t r i n h ( A B ) ( G i o n g b a i 5) Tam giac M A B va t a m giac M A C vuong can t a i M • ( A C ) 1 ( A B ) , ( A C ) qua A => P h u a n g t r i n h cua ( A C ) (1) =:> A B = A C = AM^f2 = ^/lO .^2 = 2 ^/5 • G o i (x; y ) l a t o a do C (x > 0). T i n h A C " t h e o x, y r o i cho A C ^ = 4 A B ^ t a se CO m o t phucfng t r i n h bac h a i a n so x, y (2) P h u a n g t r i n h ( A M ) c6 d a n g y = a x + b. Phirang t r i n h (AC) (1) A(l;6) 6 = a + b a = 3 G i a i he (AM) qua nen M(0;3) 3 = a(0) + b b = 3 AC- = 4AB- (2) D a p so : V a y phi/cfng t r i n h ( A M ) l a y = 3x + 3 (he so goc = 3 ) ( A B ) : y = 2 x + 4, ( A C ) : y = - - x + ^ (1) ( B C ) 1 ( A M ) n e n ( B C ) c6 h e so goc - - AC- = 4AB- c=> (X + 1)^ + (y - 2f = 20 (2) 3 f(l) G i a i he x>0) D a p s o ' : C ( 3 ; 0) =:> P h u c f n g t r i n h ( B C ) c6 d a n g y - - x + m 1(2) 3 131
B a i 7. t a i 8. C h o h a m s 6 ' y = ^ 2 + x + ^ 2 - x (1) a) V e do t h i h a m so : y = + 2x + 1 + 2x. + 1 a) Chufng m i n h do t h i h a m s o (1) c 6 t r i i c d o i xu'ng. b) B i ^ n l u ^ n t h e o m s o n g h i $ m phUc(ng t r i n h : b) T r e n k h o a n g (0 ; 2) h a m so' (1) t S n g h a y g i a m . + 2x + 1 + - 2 x + 1 = m (1) HUcfng d&n * HU&ng dan a) D a t f(x) - T i m tap xac d i n h D cua h a m so. a) G i a i giong bai 3. T i n h f ( - x ) theo x r o i so sanh f ( - x ) v^ f(x) b) So n g h i e m ciia phijang t r i n h l a so giao d i e m ciia do t h i h a m so da b) L a y x i , X2 e (0 ; 2), t i n h y i = f(xi) v ^ y2 = f(x2) r o i t i n h y i - y2. ve va dudng t h i n g y = m. Cho x i < X2 r o i t i m dau cua y i - y2. GIAI GIAI a) V e do t h i h a m s o y = | x + l | + | x - l | a) D a tfix)= 72 + X + ^2 - x . - oc - 1 1 fx + 2 > 0 ••f: M a m so xac d i n h o o - 2 < X < 2 X + 1 0 2 - X >0 X - 1 - 0 Vay t a p xac d i n h ciia h a m so 1^ D = [- 2; 2| (day l a tap doi xiifng) X - 2 - 1 Ta CO X e [- 2; 21 - X e [ - 2 ; 21 -2x neu x < - 1 y 4 2 o y= 2 neu - 1 < X < 1 • f ( - x ) = ^2 + ( - x ) + ^2 - ( - x ) = 72 - X + 72 + X = fix) X 1 2 2x neu x > 1 Vay h a m so (1) l a h a m so chSn nen do t h i c6 true doi xu'ng l a true y 2 4 tung (dpcm). 2x y i = f ( x i ) = 72"+"^ + 72 " '^i b) Lay x i , X2 e (0; 2) ta c6 : / 1) y2 = f(x2) = 72 + X2 + 72 - ^2 y = m Ta CO : y i - y2 = (72 + Xj - p + x^] + (p - Xj - p - x^) Xn X i (- 1 < x < 1) Xi — Xn 72 + 72 + X2 72 - ^1 + V2 - ^2 -2-1 I i 2 Xi + 1 1 b) B i $ n l u g n t h e o m so n g h i e m p h i f d n g t r i n h (1) = ( X i - X2) + Xi + 72 + X2 72 + 72 -^2 Phuong t r i n h (1) l a phuong t r i n h hoanh do giao diem ciia do t h i h a m so da ve va dudng thSng (d,„) : y = m - 727^) + ( 7 2 ^ - 72T1^) = ( X i - X2) \ ; w r = r \~ ^ ' m So n g h i e m phifong t r i n h (1) (72 + Xi + ^ T X 2 )(72 - X, + 72 - X2 ) ,,^, . 2 + Xj, 2 - Xi, 2 + X2 , 2 - X2 > 0 m > 2 2 0 < x i < X2 < 2 72 - xi < 72 + x i va 72 - X2 < 72 + "2 m = 2 V6 so n g h i e m x e [- 1 ; 1| x i - X2 < 0 • m < 2 0 => y i - y2 > 0 => H a m so' (1) nghich bien t r e n k h o a n g (0; 2) 15
3f(x + l ) - 2g(x + l ) = 12X+1 (1) GIAI B a i 9. T i m h a i h a m so f(x) v a g(x) b i e t f(x) + 2g(x) = - 4x - 1 (2) T a t h a y d i , d 2 , da c6 he so goc I a n liigt l a a i = - 1 , aa = 1 va as = 2. a i , 3 9 , as k h a c n h a u d o i m o t => d i , d 2 , ds cSt n h a u tiTng d o i mot. • Hiidng ddn T o a do g i a o d i e m A ciia d] va da l a n g h i e m h e phucfng t r i n h : • D o i v d i ( 1 ) , d a t t = X + 1 t a c6 3 f ( t ) - 2g(t) = sau do t h a y t bdi X ta CO 3f(x) - 2g(x) = (3) y = - X + 1 X = - 1 • G i a i h e (3) v a (2). y = X y = 2 GIAI V a y d i n d^ = A ( - 1; 2) • D o i v d i (1), d a t t = x + l o x = t - l , taco: • d i , da, ds t a o t h a n h t a m g i a c A B C ds k h o n g qua A . 3 f ( t ) - 2 g ( t ) = 1 2 ( t - 1) + 1 = 1 2 t - 11. 2 ^ 2 ( - 1) + m o m ^ 4. T h a y t b d i x t a c6 3 f ( x ) - 2g(x) = 12x - 1 1 (3) • T a CO ai.aa = - 1 => d i 1 da => T a m g i a c A B C v u o n g t a i A . . C o n g (2) v a (3) t a diTOc 4 f l x ) = 8 x - 12 h a y f ( x ) = 2 x - 3, t h a y f(x) KB't l u a n v a o (2) t a CO : 2x - 3 + 2g(x) = - 4x - 1 o g ( x ) = - 3x + 1 Vdri m * 4 t h i d i , da, ds t a o t h a n h t a m g i a c A B C v u o n g t a i A ( - 1 ; 2) f(x) - 2g(x) = 11 B a i 10. X a c d i n h h a m so f(x) v a g(x) b i e t f ( l - 2x) + g ( l ~ 2x) = - Gx + 5 • HU&ng ddn G i a i g i o n g b a i 9. D a p so : f(x) = 2x + 5 va g(x) = x - 3 B a i 11. T r o n g m ^ t p h d n g t p a d p , g p i d i , da, da I a n lu'
Cho (P) qua d i e m (- 4; 5) => => a = b) S(2; - 1); C(0; 3), A(3; 0) Toa dp B C : ys = yc = 2 => = 2 (B, C e (P)) T a CO S C = 74 + 16 = 2V5 , SA = N/2 , A C = 3N/2 T i n h OA, OB, A B . D u n g d i n h l i Pitago ta duge t a m giac S A C vuong t a i A va c6 chu vi GlAl 2p = 4V^ + 2^5 = 2(272 + (P): y = + ax + b, (a ^ 0) b = 0 b -A^ b = O v a c = l B a i 15. Trong mftt ph^ng tpa dp Oxy, cho ba diem : y . Dinh S = S(0; 1) o 2a' 4a = 1 4a A d ; 2), B ( - 3 ; - 1) va C(3 ; i ) V 6 i b = 0 va c = 1, t a c6 (P) : y = ax^ + 1 ^ a) Tim diem M tren true hoanh sao cho MA + MB ng^n nha't. M a (P) qua d i e m (- 4; 5) nen 5 = 16a + 1 A B (hang so) M OB = 0 0 = 2N/2 Vay m i n ( M A + M B ) = A B M = MQ B BO = 4 (Mo la giao d i e m ciia A B va (d)) OB = 00 = 2V2 • Tri/cfng hpfp 2 : A va B of cung mpt phia doi vdri (d). [OB^ + 0 0 ' = BO' • Gpi A ' la diem do'i xiJng ciia A qua (d). => T a m gi^c OBO vuong can t a i 0 ( 0 ; 0) • M 6 (d) o MA = MA' B a i 14. Trong mSt p h i n g tpa dp Oxy, cho parabol (P) : y = ax^ + bx + c co Ta CO: M A + M B = M A ' + M B > A'B (h^ng so) dinh S(2; - 1) va cdt true tung tai diem C c6 tung dp yc = 3. Vay m i n ( M A + M B ) = A ' B a) T i n h a, b, c. M = Mo (Mo la giao d i e m ciia A ' B va (d)). b) T r e n (P) lay diem A c6 hoanh dO XA = 3. GIAI Chtfng minh tam giac S A C vuong va tinh chu vi tam giac S A C . a) MA + MB ngSn nha't • HU&ng d&n y a) * 0(0; 3) e (P) nen • » c = 3 _b_ - b ' + 4ac' Dinh S = S(2; - 1) nen : 2a 4a -3 O ^x b - b ' + 4ac ^ = 2 va = - i 2a 4a B' -1 Dap s6': a = 1; b = - 4 va c = 3 => (P) : y = x ' - 4x + 3 19 18 •
T a t h a y A v a B or h a i p h i a do'i v d i t r u e h o a n h ( v i y A v a y a t r a i d a u ) Bai I V . T r o n g m a t p h ^ n g t o a dp O x y , c h o p a r a b o l (P) : y = i va dUdng M Ox : M A + M B > A B ( h a n g so) t h a n g (d) : y = X - 2. Vay m i n ( M A + M B ) = AB M = Mo (Mo l a g i a o d i e m c u a A B va t r u e h o a n h ) . T i m d i e m M t r e n (P) sao c h o k h o a n g e a c h iH M d e n (d) ngsin n h a t . - Phucfng t r i n h ( A B ) c6 d a n g ; y = k x + m . • HUcing d6n f 2 = k + m ( A B ) qua A ( l ; 2) va B ( - 3 ; - 1) n e n • Lay diem M(m; ) e (P) -1 = - 3 k + m • V e M H 1 (d), H e (d), t i m p h i / o n g t r i n h (MH). 3 . 5 k = — va m = —. V a y phiTcfng t r i n h eiia ( A B ) l a y = — X + — . 4 4 • ^ 4 4 • T i m t o a do H : G i a i h e phucfng t r i n h (d) v a (MH). DuTorng t h a n g A B e a t t r u e h o a n h t a i M o ( — ; 0) • T i n h do d a i M H . GIAI s D a p so : M e O x va min(MA + MB) = AB o M(- - ; 0) o • M(XM; yw) 6 (P), ta CO : y M = -^XM , a ; ; , , ' ' b) T a m g i a c l A C c6 c h u v i n h o n h a t Chu v i eua t a m g i a c l A C l a 2 p = A C + IA + IC ma AC k h o n g doi nen 2p nho n h a t cs. I A + I C n h o n h a t . D a t XM = m , t a cd : yM = — m' M ( m ; -^m") • T a t h a y A v a C d eCing m o t p h i a do'i v d i t r u e t u n g ( v i XA v a xc e i i n g d a u ) • V e M H 1 (d), ( H ) e (d). • A ' ( - 1 ; 2) l a d i e m d o i xuTng c i i a A q u a Oy. P h u ' o n g t r i n h ciia ( M H ) c6 d a n g : • I e O y t a c6 I A = I A ' . T a eo I A + I C = I A ' + I C > A ' C ( h a n g so) y = - X + b, ( M H ) qua M ( m ; — m^) n e n : Vay m i n d A + IC) = A'C c=> I = lo d o l a g i a o d i e m c i i a ( A C ) v a t r u e t u n g ) 3 13 1 2 , , 1 2 • PhUcfng t r i n h c i i a ( A ' C ) : y = - - x + — (hoc s i n h t y g i a i ) - m = - m + b o b = —m + m 8 8 2 2 ( T u o n g t i r e a c h t i m p h u o n g t r i n h d u d n g t h a n g ( A B ) t r o n g c a u a) V a y ( M H ) c6 p h u o n g t r i n h : y = - x + - m' + m 13 ( A ' C ) eat t r u e t u n g t a i d i e m Io(0 ; (MH): y = - x + - m ^ + m D a p so : T o a do H l a n g h i e m h e phi/Ong t r i n h I e Oy v a c h u v i t a m g i a c l A C n h o n h a t (d):y = x - 2 G i a i h e n a y t a c6 x - 2 = - x + - m " + m 2 x = ^ n r + m + 2 • 2 2 1 •> 1 , 1 2 1 1 x = — m " + — m + 1 => y = ~ m + - m - 1 4 2 • ' 4 2 • • Bai 16. T r o n g m a t p h S n g t o a dp, c h o b a d i e m : M ( - l ; 1), N(4; - 2 ) , P ( l ; 3). f l 2 1 , 1 2 1 1 Vay H —m U a) T i m d i e m I t r e n t r u e t u n g sao c h o I M + I N n g a n n h a t . + — m + 1; ~ m + —m - 1 2 4 2 b) T i m d i e m J t r e n t r u e h o a n h sao c h o J M + J P n h o n h a t . • Hu6ng dan G i a i g i o n g b a i 15 T a CO Mir = ( x i i - X M ) - + (yn - y.y) 2' 1 . 1 T 1 •> I 1 J -'-;0 —m - —m + 1 — m' + - m - 1 D.-ip so : 10;-- 4 2 2 4 2 20
f i m ^ - Am . 1 U 2 4 2 i ^1 2 1 = 2 = i(m^-2m.4f — m'^ ni + 1 4 2 1 i2 (m - i f + 3 > - X 9 = - o M H > (hang so) 8L 8 8 2\j2 K e t luan : m i n ( M H ) = m = 1 o M ( l ; - ). 2V2 2 B a i 1 8 . Tim diem A G (P) : y = - sao cho khoang each tiif A den drfcfng B a i 2 0 . Khao sat ham so y = x^ - 1 1 + 2x I - 1 thSng y = 2x + 4 ngdn nhat. • HUdng dan • Hiidng dan y G i a i giong bai 19 G i a i giong bai 17 Dap so : Do t h i . Dap so : A ( - 1 ; - 1) va m i n ( A H ) = -75 B a i 1 9 . Khao sat stf bie'n thien va ve do thi ham so y = - + I 2x - 4| + 1 ( 1 ) • HUdng d&n a neu a > 0 • Dung cong thilc I a I = de l a m m a t dau t r i tuyet doi ciia (1) - a neu a < 0 Bai 21. a) Khao sat s6' y = x^ - 4x + 3. • K h a o sat h&m so' y = ax + bx + c t r o n g m o i triTdng hop. b) Suy r a do thj ham so y = x^ - 4| x I +3 GIAI • HUdng dan • 2x - 4 > 0 X > 2, luc do 2x - 4 = 2x - 4 y = -x^ • H a m so bac h a i : y = ax^ + bx + c (a 0) • • 2x - -4 < 0 X < 2 , l u c do 2x - 4 = - 2 x + 4 = > y = -x^ x neu x>0 , X I = va I - x I = i - X neu X < 0 X < 2 x > 2 GIAI y = - 2x + 5 y - x^ + 2x - 3 a) Khao sat ham so y = - 4x + 3 b * X = - 1 =>y = 6 x = - — = 1 (loai) 2a 2a • Tap xac d i n h D = R ^ ^ ~ ^ • B a n g bie'n t h i e n -00 -1 2 +00 -00 1 2 +00 X - 00 2 + 00 + 00 + 00 y -1*. -1 ^ 22
• D o t h i l a p a r a b o l c6 d i n h S(2; - 1), t r u e d o i xufng l a ducJng t h S n g l i a i 2 3 . T i m t r e n do t h i ( P ) cvia h a m so y = - + 4x h a i d i e m A v a B s a o X = 2, c a t t r u e O y t a i d i e m ( 0 ; 3) v a c f i t t r u e O x t a i h a i d i g m : ( 1 ; 0) c h o A B c u n g phifofng vtJi t r u e h o a n h v a dp d a i d o a n t h ^ n g A B = 6. v a (3 ; 0) y • HUcfng d&n 4^ • A B ciJng phuang v d i O x nen y A = y n = m : A ( X ] ; m), B(X2 ; m) • A , B G ( P ) nen m = - x^ + 4 x i . ni = - x.^ + 4 x 9 => X i , X2 l a nghiem phuang trinh m = - x^ + 4x h a y x~ - 4 x + m = 0. • A B " = ( x i - X2)'^ + (m - m)^ = ( d u n g dinh l i V i e t do'i v d i phiWng b) S u y r a do t h i h a m so y = - 4| x I+3 trinh b a c h a i ) GIAI Dat y i = X" - 4 x + 3 ( d a v e do t h i ) • V i A B cijng phycfng v d i O x nen y ^ = y g = m : A ( x i ; m), B(X2 ; m) y 2 = x^ - 4l x I + 3 = f(x) • A, B e ( P ) nen m = - x ^ + 4 x ] v a m = - x^ + 4 x 2 • X > 0 IxI = X o y 2 = y ] , v a y do t h i h a m so ya = d o t h i h a m so => x i * X 2 la nghiem phuang trinh m = - x " + 4 x h a y x " - 4 x + m = 0 (") yi k h i X > 0 • D i e u kien de phifong trinh C*) c6 h a i nghiem phan biet X i , X2 l a • ft- X) = ( - x)2 - 4| - X I +3 = x ' - 4i X I +-3 = f ( x ) A' = 4 - m > 0, « m < 4 V a y h a m so y2 l a h a m so c h a n n e n d o t h i h a m so y2 d o i xufng n h a u • T a CO A B ^ = ( x i - X 2 ) ' + (m - m)^ = ( x i + X2)^ - 4xi.X2 qua t r u e t u n g . M a x i , X2 l a h a i nghiem cua phuong trinh C') nen : y X] + X2 = 4 v a X1.X2 = m. V a y A B ' - = 16 - 4 m = 36.(vi A B = 6) 0 m = - 5 < 4 - T h a y m = - 5 v a o phUcfng trinh (^^') : x^ - 4 x - 5 = 0 ^ o x = " l V x = 5 , Bai22. . D a p so : A ( - 1 ; - 5) v a B ( 5 ; - 5) a) K h a o sat b i e n t h i e n v a ve do t h i h a m so y = - - 2x. B a i 2 4 . C h o h a m so y = + 4x + 2 c6 do t h i ( P ) . T i m h a i d i e m M v a N t r e n S u y r a do t h i h a m so y = - + 2| x I. (P) s a o c h o M N s o n g s o n g v
GlAl B a i 27. Trong mat phftng toa dp Oxy, cho hai parabol : a) Khao sat sir bien t h i e n va ve do t h i (P) ciia h ^ m so y = x"^ + 4x + 3 (Pj) : y = '^^-^ c5 dinh S i . 3 • D = R x = - — =-2 => y = - 1 (P2) : y = + 2X + 11 jjjj^^ • Bang bien t h i e n 3 Gpi A va B la giao diem cua ( P i ) va (P2). X - 00 - 2 + 00 ChuTng minh rSng A, B , S i , S2 la bon dinh cua mpt hinh vuong. + -oo ^ _ + 00 y - 1 ^ • Hiic/ng dan • T i m toa do Si va S 2 DCing cong thufc S 2a 4a, Do t h i la parabol (P) c6 d i n h S{- 2; - 1), true doi xufng x = - 2, cSt cac true toa do t a i (0; 3), (- 1 ; 0), (- 3; 0) [(Pi) T i m toa do A va B : G i a i h? phUcfng t r i n h [(Pa) Chu-ng m i n h tut giae A S 1 B S 2 c6 h a i dudng cheo A B va S 1 S 2 : b^ng nhau, vuong goc nhau, eo chung m p t t r u n g diem. GIAI ^ b D i i n g cong thiJc S ( d i n h ciia (P) : y = ax + bx + c) 2a 4a • Ta CO S i ( l ; - 2) va Sad; 4). x^ - 2x - 5 (Pi):y = b) B(XB; YB) va S ( - 2; - 1) doi xdng nhau A ( - 3; 1) nen A la t r u n g d i e m G i a i he phucfng t r i n h doaa t h a n g S B - x ^ + 2x + 11 (P2): y = _ Xg + X B XA = 2 XB = 2XA - XS = ~ 4 Ta CO : x^ - 2x - 5 = - x^ + 2x + 11 2x^ - 4x - 16 = 0 < Vay B ( - 4; 3) ys = 2yA - ys =3 'x = 4 o x - 2 x - 8 = 0 « ^ y = 1 X = -2 Ta t h a y 3 = (- if + 4 ( - 4) + 3 hay ye = x^ + 4XB + 3 ^ B e (?) Vay giao d i e m ciia (Pi) va ( P 2 ) la A ' 4 ; 1), B ( - 2 ; 1) Dap so : B ( - 4; 3) thoa m a n b a i toan. y B a i 26. .P2 a) Khao sat suf big'n thien va ve do thi ( P ) cua ham so' y = - - 4x. b) Goi S la diem cijfc dai cua (P). Tim diem M e ( P ) sao cho I ( - 3; 2) la trung diem doan thSng SM. B jl Ta t h a y : ^x * Hitdng ddn ;1 AB = S1S2 = 6 (1) -2 G i a i giong bai 26. ^Si AB 1 S1S2 (2) Dap so : M(-- 4; 0) A B va S 1 S 2 CO chung m o t t r u n g d i e m la 1(1 ; 1) (3) Ti^ (1), (2) va (3) => A S 1 B S 2 la h i n h vuong. 26 27
x'^ + 4x b) Difdng thftng t r u n g triTc cua doan OA (x = 2) cat dudng t h a n g ( S i S v ) (P,):y = B a i 28. Trong mat phang toa dp Oxy cho hai parabol tai 1(2; 2). X - 4x S], S2, O va A tao thanh mot hinh thang can (day OA va S1S2) (P2):y = [ l l a t r u n g d i e m SjS., (1) n h u n g k h o n g p h a i h i n h chuf n h a t < " • Chu"ng minh rang hai dinh ciia ( P i ) va ( P 2 ) cung vofi hai giao diem S1S2 ;^ OA (2) ciia ( P i ) va ( P 2 ) tao thanh mpt hinh thoi. y , X = 2 ^ Hii&ng ddn ' G i a i gio'ng bai 27 Si 2 I S2 V Dap so : S i ( - 2 ; - 1), S 2 ( - 2 ; 1), A ( - 4 ; 0), 0 ( 0 ; 0) A / ^ B a i 29. Trong mat phSng toa dp Oxy, cho hai parabol : -2 0 21 4 6 ' (P,): y = - 2mx + +2 - Tir>(l) ta CO X I = | ( X H + Xj; ) o 2 = i ( m - 3m) = - m » m = - 2 (P2): y = - x ^ - 6mx - 9m^ + 2 - V d i m = - 2, ta CO : S i ( - 2 ; 2) va 82(6 ; 2) 8,82 = 8 a) Chii'ng minh rang cac dinh S i va S 2 ciia ( P i ) va ( P 2 ) cung (i trc^n mpt duong thiing co dinh. r:> S1S2 * OA (vi OA = 4). b) Xac dinh m de S i , S 2 va 0(0; 0), A(0; 4) la bo'n dinh ciia mpt hinh Dap so : m = - 2. thang can day OA va S 1 S 2 nhifng khong phai la hinh chu" nhat. Bai 30. • HU&ng ddn a) Khao sat sii bien thien va ve do thi ( P ) ciia ham so y = x^ + 4x - 5. • T i m toa do S i , S2 ta se t h a y k e t qua. b) ( P ) cat true hoanh tai A va B (XA < 0). DUcfng thSng y = - 5 cat (P) • T r u n g trirc ciia doan t h f i n g OA cat duo'ng thSng (SiS^) t a i I , de bai tai C va D ( X D < 0). Chiing minh rang A B C D lal hinh thang can. thoa m a n k h i I la t r u n g d i e m ciia 8182 dong t h d i 8182 ^ OA HU&ng ddn GIAI • T i m toa do A va B (giai phiTcfng t r i n h + 4x - 5 = 0) Parabol y = ax'^ + bx + c c6 hoanh do d i n h la x = - 2a • T i m toa do C va D (giai phuong t r i n h x" + 4x - 5 = - 5) a) ( P i ) : y = x^ - 2mx + m^ + 2 c6 d i n h S i ( m ; 2) C h i l n g m i n h A B va CD co chung mot dUdng t r u n g trUc va A B 1^ CD. (P2) : y = ^ x^ - 6mx - 9m^ + 2 c6 d i n h 8 2 1 - 3 m ; 2) Dap so :. A ( - 5; 0), B ( l ; 0), C(0; - 5), D ( - 4; - 5). Ta thay y^ = y,s =2 PhUo'ng t r i n h dudng t r u n g trUc chung cua A B va CD la x = ~ 2. S i va 82 ciing d t r e n ducfng t h a n g co d i n h y = 2 (dpcm) y / s, 2 82 -y = 2 X = - 2 i i ^ m 0 -3m 00 28
C h u y e n d« 2 B a i 3 1 . T u y t h e o m , g i a i v a b i $ n l u $ n phttctng trinh : ( x - 1)^ + m ^ x + 3 = ( m + 1 ) ^ + - X (1) * HUdng dan PhiTofng t r i n h v a b a t phufoTng t r i n h b a c n h a t mpt a n « B i e n d6i phiTcfng t r i n h ve d a n g Ax = B r o i xet h a i triTdng h o p : Phiidng trinh bac n h a t : Bat phadng trinh bac nhat ; • A = 0 (luc do k h o n g c a n x e t B) ax + b = 0 ax + b > , > , < , < 0 . A ;^ 0 due do x6t B (B = 0 h a y B ^ 0) C a c h giai GIAI C a c h giai • B i e n d6'i phuong t r i n h * a 0 Phirang t r i n h c6 B i e n doi ve m p t t r o n g bo'n dang : n g h i e m duy n h a t x = — (1) x^ - 2x + 1 + m^x + 3 = ( m + l)'^ + x^ - x Ax>B;Ax>B;Ax, < 0 Sai B a i 3 2 . G i a i v a b i ^ n lu^in t h e o m phUorng t r i n h 5^ '-—^ = 1 0.x > < 0 Dung +4 a ^ 0 Tuyy * Hudng ddn 0.x >, > so dacfng Sai G i a i giong bai 31 t a c6 : n i ( m + 2 ) x = ( m + 2 ) ( m - 2 ) 0.x , > so a m Dung b ^ 0 S = 0 0.x
Bai 3 3 . C h o phu-cfng t r i n h (2m - x)^ + (1 - m)^ = (x - m)^ + (1 - x) (1) Chii y : X a c d i n h m de phu'cfng t r i n h c 6 n g h i e m x t h o a m a n d i e u k i # n - 1 < x < 0. 1 3 - m + 1 = (m f + — > 0, V m e R n e n - m + 1 ?i 0, V m e R + Hiidng d&n b) T h e o de b a i t a c6 X i = m i + 2 va X2 = m 2 + 2 • B i e n d o i phUcfng t r i n h d a cho ve d a n g A x = B r o i t i m d i e u k i e n d§ phufcfng t r i n h c6 n g h i e m . o mi = xi - 2 va m 2 = X2 - 2 . • T i n h n g h i e m x t h e o m r o i cho - 1 < x < 0. nij + m2 = (xj + Xg) - 4 • Ta CO : A * 0 Can nh&: Phuang trinh Ax = B c6 nghiem o m^.mg = (xj - 2)(x2 - 2) = Xj.Xg - 2(xi + X2) + 4 A = B = 0 GIAI Vay ( m i + m2) = 2mi.m2 o ( X i + X2) - 4 = 2 [ x i . X 2 - 2 ( x i + X2) + 4 | = 0 B i e n d o i phUorng t r i n h (1) 5 ( x i + X2) - 2 x i . X 2 = 12 (dpcm) (!) 4 m " - 4 m x + x^ + 1 - 2 m + m ' = x^ - 2 m x + m ^ + 1 - x B a i 3 5 . C h o phiforng t r i n h (x + m)^ + 6 ( - m + 1) = 5(m + 1)* + x(x - 1) CO (1 - 2m)x = 2 m ( l - 2m) (^•') 3 a) X a c d i n h m de phifofng t r i n h c 6 t ^ p n g h i e m S R va S 0. • 1 - 2m = 0 m = ^ , l u c do phiTcfng t r i n h (*) t r a t h a n h 0.x = 0 b) G o i X i , X 2 l a h a i n g h i e m c u a phtfoing t r i n h xi'ng vdri h a i g i a t r i m i , m 2 CO n g h i e m x G R n e n c6 n g h i e m x e (- 1; 0 ) c i i a m (mi, m 2 ^ - i ) . T i n h P = xf + X 2 theo m i . m 2 biet m i + m 2 = - 2 • 1 - 2 m 7i 0 m ? i — , l i i c do phifcfng t r i n h (*) c6 n g h i e m d u y n h a t • Hi^otng d i n 2 a) B i e n d o i p h i r a n g t r i n h d a cho v g d a n g A x = B r 6 i c h o 0. X - 2m, d o d 6 - l < x < 0 o - 1 < 2m < 0 o - 4 < n i < 0 b) T i n h x i t h e o m i v a X2 t h e o m 2 r 6 i t i n h V = x\ x\ D a p so : D e b a i t h o a m a n o m = — V - — < m < 0. ( C h u y : a" + b^ = (a + b)^ - 2ab). 2 2 GIAI ' Bai 3 4 . C h o phi^omg t r i n h (x - l ) ( m ^ - m + 1) = m'* + 1 (*). a) P h u a n g t r i n h d a cho a) Chufng m i n h r a n g phifoTng t r i n h (*) l u o n c 6 n g h i e m . x^ + 2 m x + m ^ + 1 0 m + 6 = 5 ( m ^ + 2 m + 1) + x^ - x b) Got X ] , l a n g h i e m c i i a phifotng t r i n h tifofng i^ng vdfi h a i g i a t r i nil v a c i i a m. o (2m + l)x = 4m^ - 1 = (2m + l)(2m - 1) C h i J n g m i n h r S n g (m, + m^) = 2mi.ma « 5(xi + X 2 ) - 2 x i . X 2 = 12. Phifcfng t r i n h c6 t a p n g h i e m S ^ R v a S # 0 ( n g h i a l a phUcfng t r i n h d a c h o CO d u n g m o t nghiem) ^f- HUdng dan a) B i e n d o i p h u o n g t r i n h d a cho ve d a n g A x = B r o i chufng m i n h k * 0, 2m + 1 * 0 c:> m 9^ - —, l i i c d o n g h i e m c i i a p h u o n g t r i n h la : Vm [ C o n g thijrc : a^ + b'^ = (a + b)(a^ - a h + b~)| X = 2 m - 1 T a p n g h i e m c i i a p h u c f n g t r i n h l a S = |2m - 11 b) T i n h m i t h e o x i v a m2 t h e o x-z, s a u do b i e n d o i ( n i j + m2) = 2 m i . m 2 Xj = 2mi - 1 o b) T h e o de b a i t a c6 : GIAI X2 = 2m2 - 1 a) P h u o n g t r i n h (*) (x - l ) ( m ^ - m + 1) = ( m + l ) ( m ^ - m + 1) Ta CO : P = + x^ = ( 2 m i - if + ( 2 m 2 - if ~ Vi m - m + 1 = (m - - )^ + — > 0, V m n e n phUcfng t r i n h d a c h o = 4 ( m i + mg) - 4 ( m i + m2) + 2 2 4 c:> x - 1 = m + 1 w> x = m + 2. V a y p h u o n g t r i n h d a c h o c h o l u o n = 4 [ ( m i + m2)^ - 2mi.ni2l - 4 ( m i + m2) + 2 •' CO n g h i e m x = n i + 2. = 4 ( 4 - 2 m i . m 2 ) - 4 ( - 2) + 2 = 26 - 8mi.m2. 32 33
B a i 36. G i a i v a b i ^ n l u ^ n theo m b a t phi^oTng t r i n h : « Tri^cfng hgtp 2 : - 1 < 0 I a I < 1, liic do bat phifcrng t r i n h {*) a + 2 (X + 1 ) ^ - ( m + l)"'^ > x(x + m) - 2(m + 2) (1) CO n g h i e m x > • HU&ng dan a +1 • B i e n d6i bat phirang t r i n h da cho ve dang Ax > B r o i xet ba tracing • Tru"dng h
B a i 3 9 . X a c d i n h m de p h i / d n g t r i n h 4x - 3 m - 2 = m^(x - 1) - 4 v 6 n g h i $ m . Cliu.y^n d e 3 • HUcfiig dan HE PHCrONG TRINH B^C NH^T Hfil SO B i e ' u d o i p h i / d n g t r i n h v e d a n g A x = B r o i c h o A = 0 v a B ;t 0 GIAI ax + b y + c = 0 • D a n g \a v a b , a' v a b ' k h o n g d o n g t h 6 i t r i e t t i e u ) • B i e n d o i phifcfng t r i n h d a c h o ( m ^ - 4 ) x = - 3m + 2 a'x + b ' y + c' = 0 C5- ( m - 2 ) ( m + 2)x = ( m - l ) ( m - 2) (*) • C a c d i n h thvlc ( m - 2 ) ( m + 2) = 0 P h u o n g t r i n h (*) v 6 n g h i e m c=> m = - 2 a b b c c a ( m - l ) ( m - 2) ^ 0 D = = a b - a b , Dx = = be' - b'c, Dy = = c a ' - c'a a' b' b' c' B a i 4 0 . X a c d i n h a de b a t phu"c(ng t r i n h : Cdch nhd a a x ( a - 1) + > (a - 1)^ + (x + 1)^ - 3 + 2 a v 6 n g h i e m . a' b' c' a' % Hiidng dan V - ^ \ D D y • B a t phucfng t r i n h d a c h o GIAI » a^x - a x + > a - - 2 a + 1 + x'^ + 2 x + 1 - 3 + 2 a D He phirang t r i n h c=> (a^ - a - 2 ) x > a^ - 1 o ( a + l ) ( a - 2)x > (a - l ) ( a + 1) (a + l ) ( a - 2) = 0 (i) [ B a t phuang t r i n h v6 n g h i e m » X = — - D # 0 Tuyy Co n g h i e m d u y n h a t ( x ; y ) < D ( a - l ) ( a + 1) > 0 (ii) Dy y = —^ a = - 1 => ( i i ) s a i i D - V d i (!) t a CO : D a p so : a = 2 a = 2 => ( i i ) d u n g Dx * 0 h a y V6 nghiem Dy;^0 B a i 4 1 . C h o phUofng t r i n h 2(1 x I + m + 1) = m I x I + + 3 (1) D = 0 X a c d i n h m de phrfotng t r i n h (1) c 6 n g h i e m . Co v 6 so' n g h i e m (x ; y ) v d i x e R v a Dx = Dy = 0 ax + b y + c = 0 h o a c * Hifdng dAn y e R v a ax + by + c = 0 B i e n d o i p h i W n g t r i n h da c h o (2 - m ) l x I = ( f D a p so : Phifcfng t r i n h d a c h o c6 n g h i e m : 2 - m > 0 h o a c = 0 B a i 4 3 . T u y t h e o m , g i a i v a b i § n l u a n h ^ phUofng t r i n h mx - y - m + 2 = 0 B S i 4 2 . X a c d i n h a d e b a t p h i f d n g t r i n h s a u c 6 t|ip n g h i e m S = R : X - my = - m a ^ x - 1) + (X - 2f < 5(2 - a ) + x^ (*) • Htfdng d i n • HU&ng dan • L a p cac d i n h thiJc D , Dx, D y . ' • (*) O ( a - 2)(a + 2)x < ( )( ) (a - 2 ) ( a + 2) = 0 • X e t h a i trUcrng h o p .D ^0,1) = 0 • B a t p h i f o n g t r i n h (''') c6 t a p n g h i e m S = R GIAI ( )( ) > 0 mx - y - m + 2 = 0 Dap so': a = - 2 T a CO h e p h u c f n g t r i n h x - my + m = 0 37 36
C a c d i n h thufc (k + l ) x + (3k + l ) y + k 2 m -1 - m +2 m B a i 4 5 . C h o h§ phi^orng t r i n h 2x + (k + 2 ) y 4 = 0 a) X a c d i n h k de h$ p h U d n g t r i n h c 6 nghi(>m. b) K h i he phufofng t r i n h c 6 n g h i e m d u y nlia't (x; y ) t i n h k de x, y e Z D = 1- = (1 - m ) ( l + m ) D * 0 • Ta CO = - + m = m ( l - m) ax + by + c '= 0 D = 0 Can nh& : He phuang trinh CO nghiem Dy = - - m + 2 = (1 - m ) ( m + 2) a'x + b'y + c' = 0 = 0 = 0 Bien iuan • Trifdng hdp I : D Q c : > n i ; t ± l , he phiTOng t r i n h c6 n g h i e m • HUdng dan m X = * T i n h D, D , , Dy D 1 + m duy n h a t a) He phi/ong t r i n h cd n g h i e m m = ±l y = - Vdi n i = 1 => Dx = Dy = 0 => He phuong t r i n h c6 v6 so' nghiem GIAI ( X ; y) vdi X e R va y = X + 1 hoac y e R v a x = y - l a) (V^n t a t ) D = k ( k - 3), Dx = - k ( k + 12), Dy = 6k - V d i m = - 1 => Dx 5^ 0 He phiicfng t r i n h v6 n g h i e m k 0 va k 3 D a p so : o k 3 Tom tit k = 0 m Tap n g h i e m ciia he phi^ong t r i n h b) Pihi k 0 va k ;^ 3 t h i he phi/crng t r i n h c6 n g h i e m duy n h a t (x; y) „ m m + 21 m ^ ± 1 D x _ - k ( k + 12) r k + 12^ r k - 3 + 15^ [1+ m 1+ m ] X = D k(k~3) I k-3 k-3 k-3 S = |(x ; y)/x e R va y = x + 11 hoac vai : m = 1 6k 6 S = |{x ; y)/y e R va x = y - 1| D k(k-3) k-3 m = - 1 S = 0 • X nguyen o 15 i (k - 3) I k - 3 I e |1, 3, 5, 151 6ax + ( 2 - a ) y = 3 • y nguyen o 6 ! (k - 3) o I k - 3 I e |1, 2, 3, 61 B a i 4 4 . T u y theo a, g i a i v a b i $ n l u $ n h§ phUcfng t r i n h (a - l ) x - ay =2 "k - 3 = ± 1 Vay X va y nguyen I k - 3 I e 11, 31 >«• HUcfng d&n k -3 = ±3 6ax + (2 - a ) y - 3 = 0 o k = 4 V k = 2 V k = 6 V k = 0 (loai) • He phuang t r i n h = 0 D a p so : • D = (a + 1)(2 - 5a) ; Dx = - (a + 4) ; Dy = 3(1 + 3a) • k = 4 =:>x=-16vay = 6 • k = 2 =i>x = 1 4 v a y = -6 Bi^n lu^n • k = 6 = > x = - 6 vay = 2 • Trircfng hop 1 : D 0 (khong xet D^, Dy) • k = 0 (loai) • Trifdng hcfp 2 : D = 0 (xet D^, Dy) 39 38