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Chuyn

Tip Tuyn

V d 1:Cho hm s 3 23 2 5 ( ) y x x x C . vit phng trnh tip tuyn ti im c honh x = 1

Bi gii :Vi x = 1 y

- 4 (1, 5)M ( )C

' 2 '3 6 2 (1) 1 y x x y ; vy tip tuyn ti M c dng : 1( 1) 5 4 y x y x V d 2 : (D b D2006)

cho hm s 3 ( )

1xy Cx

. cho m 0 0( , ) ( )M x y C . tip tuyn ti M ct cc tin cn ca th hm s

(C) ti hai im A, B . chng minh rng M l trung im AB .bi gii:

0

0 0

0

3( , ) ( )

1o

xM x y C y

x

, '

2 2

0

4 4

( 1) ( 1)y k

x x

, tip tuyn ti M c dng (d) :

2

0 0 00 0 02 2 2 2

0 0 0 0 0

3 5 34 4 4( ) ( )

( 1) ( 1) 1 ( 1) ( 1)

x x x y x x y y x x y x

x x x x x

Gi A l giao im ca tip tuyn (d) v tim cn ng x = 1 . suy ra ta im A l nghim ca h :2

0 0

2 2 00 0 0

0

0

15 34

7(1, )( 1) ( 1) 71

11

xx xy x x

Ax x xy xxx

Gi B l giao im ca tip tuyn v tim cn ngang y = 1 , suy ra ta ca B l nghim ca h :2

0 002 2

00 0

5 342 1

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

1

x xy x x x

B xx xy

y

Nhn xt :

00

0

0 0

0

1 2 1

2 27

trung diem AB1

1 32 2 1

A B

M

A BM

xx xx x

xM l

x xy y yx

(pcm)

V d 3 : (D2005)

Cho hm s3 21 1 ( )

3 2 3m

m y x x C . cho M ( )mC , bit rng 1Mx , tm m tip tuyn ti M

song song vi ng thng 5x - y = 0Bi gii :

' 2 y x mx h s gc tip tuyn ti M '( 1) 1k y m , tip tuyn song song vi ng thng 5x y = 0 1 5 4k m m

Dng 1 : Vit Phng Tr nh Tip tuyn ti im 0 0M( , ) ( ) : ( )x y C y f x

Cch gii

:* tnh ' '( ) y f x ; tnh ' 0( )k f x ( h s gc ca tip tuyn )

* tip tuyn ti M c dng : 0 0( )y k x x y

V d 4 : (H Thng Mi 2000)Cho hm s 3 3 1 ( ) y x x C , v im 0 0( , )A x y (C) , tip tuyn ca th (C) ti im A ct (C) ti

im B khc im A . tm honh im B theo 0xBi gii : Vi im 0 0( , )A x y (C)

3

0 0 03 1 y x x ,' 2 ' 2

0 03 3 ( ) 3 3 y x y x x

Tip tuyn ca th hm c dng :

Dng 1 : Vit Phng Tr nh Tip tuyn ti im0 0

M( , ) ( ) : ( )y C y f x

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Vy im B c honh 0

2B

x x

Khi s l cc bi ton dng ny thng thng h s gc k cho dng gin tip thng thng bi ton chotip tuyn song vi ng thng :

1y k x m h s gc ca tip tuyn 1k k . Nu bi ton cho tip

tuyn vung gc vi ng thng :2

y k x m h s gc ca tip tuyn 22

1( . 1)k do k k

k

.

Nu bi ton cho tip tuyn to vi ng thng (d) : 'y k x m mt gc l , cc em c th dng cng

thc sau tm k :'

'tan

1k k

kk

( tuy nhin cc em phi chng minh khi s dng , xem cun: gip tr

nh Ton hc , Nguyn Dng 2008)Mt s v D in Hnh

V D 1 : (H Ngoi Ng 2001)

cho hm s 31 2

3 3 y x x , vit phng trnh tip tuyn bit tip tuyn vung gc vi ng thng

1 2( )

3 3 y x d

Dng 2 : Vit tip tuyn ca thi hm s ( ) y f x(C) khi bit trc h s gc ca nNu h s gc ca tip tuyn l k ta c th lp tip tuyn bng 2 cch sauCch 1 :

Tip tuyn (d) c dng y kx m ( k bit )

(d) tip xc (C )'

( ) (1)

( ) (2)

f x kx m

f x k

c nghim

T phng trnh 2 ta gii ra c0

x x ( honh tip im ) th vo (1) ta tm c k tip

tuyn

Cch 2 :Gi

0 0( , )M x y l tip im , gii phng trnh '

0 0( ) f x k x x ,

0 0( ) y f x

n y tr v dng mt ta d dng lp c tip tuyn ca thi : 0 0( )y k x x y

' 2 3 2 3

0 0 0 0 0 0 0 0 0 0( )( ) (3 3)( ) 3 1 (3 3)( ) 2 1 ( )y y x x x y y x x x x x y x x x x d phng trnh

honh giao im ca (d) v (C) :

3 2 3 3 2 3 2

0 0 0 0 0 0 0

2

00

0

00

3 1 (3 3)( ) 2 1 3 2 0 ( ) ( 2 ) 0

( ) 0( 0)

22 0

x x x x x x x x x x x x x x

x xx xx

x xx x

Bi gii :V tip tuyn vung gc vi ng thng (d) tip tuyn c dng : 3 y x m

iu kin tip xc :3

2

1 23 (1)

3 3

1 3 (2)

x x x m

x

c nghim

3

3

2

1 241 2 14

4 3 3 2,3 3 3

22, 64

2

x x m x x m x m

xx mx

x

Vi14

3m tip tuyn c dng

143

3y x

Vi m = 6 tip tuyn c dnh y = 3x +6

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V d 2 : (H cnh st 1998) Cho hm s

2 3 3

2

x xy

x

; vit phng trnh tip tuyn bit rng tip tuyn song song vi ng thng :

y = -3x +2Bi gii : Tip tuyn song song vi ng thng y = -3x + 2 tip tuyn c dng y = -3x + m

iu kin tip xc

2

2

2

3 33 (1)

2

4 3 3(2)( 2)

x xx m

x

x xx

c nghim 2x (2) 2

3

24 16 15 0

52

x

x x

x

Vi3

32

x m tip tuyn c dng : 3 3y x

Vi5

112

x m tip tuyn c dng : 3 11y x

V d 3 :

Cho hm s 33 4y x vit phng trnh tip tuyn bit rng tip tuyn to vi ng thng (d) :

3 6 0y x mt gc 030 Hng dn gii:

(d)1

2 33

y x c h s gc 11

3k ; tip tuyn c h s gc 2k

p dng cng thc (*) : 0 1 2

1 2

tan301

k k

k k

d dng tnh c 2k

Sau p dng dng 2 lp tip tuyn khi bit trc h s gc ta tm c 3 tip tuyn tha mn yu cu cabi ton l :

1 2 2

11 3 11 3( ) : 4 ; ( ) : 3 ; ( ) : 3

3 3d y d y x d y x

V d 4 : (H Ngoi Thng 1998)Cho hm s 3 23 9 5 ( ) y x x x C . trong tt c cc tip tuyn ca (C ) tm tip tuyn c h s gc

nh nhtBi gii :

TX: D RTa c : , 23 6 9 y x x ; gi 0 0( , ) ( )M x y C h s gc tip tuyn ca (C ) ti M :

20 0 0

' '0 0 0 0

( ) 3 6 9

( ) 6 6 ; ( ) 0 1

k f x x x

f x x f x x

( 1)f -12

Bng bin thin : x 0 -1

f(x 0 ) - 0 +

f(x) -12

+

Da vo bng bin thin ta thy 0 0 0min ( ) 12 1 , 16 f x x y

Vy ti im c ( 1,16)M th tip tuyn c h s gc nh nht ( chnh l im un ca th )Cach khc :

Ta c : 2 20 0 0 0( ) 3 6 9 3( 1) 12 12 min 12,k f x x x x k t c khi

0 01 12x y

Vy ti im c ( 1,16)M th tip tuyn c h s gc nh nht ( chnh l im un ca th)

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p tuyn bit n i quamt im cho tr

V d 1 : (H Ngoi Thng 1999)

Cho hm s2

2

xy

x

; vit phng trnh tip tuyn bit rng tip tuyn i qua im ( 6,5)A

Bi gii :

Tip tuyn i qua ( 6,5)A c dng : ( 6) 5y k x

iu kin tip xc :

2

2( 6) 5 (1)

2

4(2)

( 2)

xk x

x

kx

c nghim 2x

Th (2) vo (1) ta c : 22

02 4( 6) 5 6 0

62 ( 2)

xx x x x

xx x

Vi x = 0 1k tip tuyn c dng : 1y x

Bi ton : cho hm s : ( )y f x v im 0 0( , )A x y vit phng trnh tip tuyn bit rng tip tuyn i

qua im ACch gii :

bc 1 : tip tuyn i qua 0 0( , )A x y c dng : 0 0( )y k x x y bc 2: iu kin tip xc 0 0

'

( ) ( ) (1)

( ) (2)

f x k x x yc

f x k

nghim

bc 3: gii h ny ta tm c k phng trnh tip tuyn ca th hm s

Vi x = 6 14

k tip tuyn c dng : 1 74 2

y x

Nh vy ta k c hai tip tuyn tha mn yu cu bi ton

V d 2 : (H Ngoi Ng H Ni 1998)

Cho hm s : 3 21

2 33

y x x x vit phng trnh tip tuyn bit rng tip tuyn i qua4 4

( , )9 3

A

Bi gii :

Tip tuyn i qua A c dng :4 4

( )9 3

y k x

iu kin tip xc :3 2

2

1 4 42 3 ( ) (1)3 9 3

4 3 (2)

x x x k xc

x x k

nghim

Thay (2) vo (1) ta c :

3 2 2 3 2

0

1 4 4 82 3 ( 4 3)( ) 3 11 8 0

3 9 3 3

1

x

x x x x x x x x x x

x

Vi x = 0 3k tip tuyn c dng : 3y x

Vi x = 8 53 9

k tip tuyn l : 5 1289 81

y x

Vi x = 1 0k tip tuyn c dng : y =4

3

Vy t A v c ba tip tuyn ti th hm s

Dng 3: vit phng trnh tic

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Tip Tuyn

V d 1 : (D2007)

Cho hm s2

( )1

xy C

x

tm im M ( )C sao cho tip tuyn ca th hm s ti M ct hai trc ta

ti A, B sao cho tam gic OAB c din tch bng1

4Bi gii :

00 0 0

0

2( , ) ( )

1

x M x y C y

x

,2

2'

( 1)

y

x

Tip tuyn ti M c dng :2

0 00 0 0 02 2 2

0 0 0 0

2 22 2'( )( ) ( ) ( )

( 1) 1 ( 1) ( 1)

x xy y x x x y y x x y x d

x x x x

Gi ( ) oxA d ta im A l nghim ca h :2

0 2

202 2

00 0

22

( ,0)( 1) ( 1)0

0

xy x x x

A xx xy

y

Gi ( ) oyB d ta im B l nghim ca h :20 2 2

2 2 0 0

0 0 2 2

0 0

220 2 2

(0, )( 1) ( 1)( 1) ( 1)

0

xy x x x x

Bx xy x x

x

Tam gic OAB vung ti O ; OA = 2 20 0x x ; OB =

2 2

0 0

2 2

0 0

2 2

( 1) ( 1)

x x

x x

Din tch tam gic OAB : S =1

2OA.OB

=

2 240 0 0 0 0 04 20

0 02 2 20 0 0 0 0

0 0

12 1 2 1 0 221 1

. 4 ( 1) 22 ( 1) 4 2 1 2 1 1( ) 1 1

x x x x x yxx x

x x x x x vn x y

Vy tm c hai im M tha mn yu cu bi ton :1 2

1( ; 2) ; (1,1)

2M M

Dng 4 : Mt S Bi Ton Nng Cao V

V d 3 : (d b B 2005)

Cho hm s :2 2 2

( )1

x xy C

x

, chng minh rng khng c tip tuyn no ca (C ) i qua giao im I

ca hai ng tim cn ca th hm s (C )Bi gii:

2 2 2 11

1 1

x xy x

x x

tim cn ng x = -1 ; tim cn xin y = x +1 . gi I l giao im ca hai

ng tim cn trn ( 1, 0)I

ng thng (d) qua I c dng : ( 1)y k x

(d) l tip tuyn ca (C )

2

2

2

2 2( 1) (1)

1

2(2)

( 1)

x xk x

x

x xk

x

c nghim 1x

Thay (2) vo (1) ta c :2 2

2

2 2 2( 1) 2 0

1 ( 1)

x x x xx

x x

(v nghim ) vy t I khng k c tip

tuyn no ti th hm s (pcm)

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(C

V d 3 : (d b D 2007)

Cho hm s1

xy

x

(C ) ; vit phng trnh tip tuyn (d) ca (C ) ; sao cho (d) ct hai ng tim cn

ca (C) to thnh mt tam gic cn

V d 4: (d b B2007)

Cho hm s 1 ( )2

m

my x C

x

tm m hm s c cc i ti A v tip tuyn ca ( )

mC ti A ct

trc oy ti B m tam gic OAB vung cn

V d 5: (hc vin BCVT 1998)Cho hm s 3 12 12 ( )y x x C . tm trn ng thng y = - 4 nhng im m t v c 3 tip tuyn

phn bit ti th ( C)Bi gii :im M nm trn ng thng y = -4 nn M( m , - 4)

Tip tuyn qua M c dng : ( ) 4y k x m

iu kin tip xc :3

2

12 12 ( ) 4 (1)

3 12 (2)

x x k x m

x k

c nghim

Th (2) vo (1) ta c :3 2 3 2

2 2

2

12 12 (3 12)( ) 4 12 16 (3 12)( )

( 2)( 2 8) 3( 2)( 2)( ) ( 2) 2 (4 3 ) 8 6 0

2

( ) 2 (4 3 ) 8 6 0

x x x x m x x x x m

x x x x x x m x x m x m

x

g x x m x m

t M c th k c 3 tip tuyn ti th (C) th g(x) = 0 phi c hai nghim phn bit 2

2 2

4

(4 3 ) 8(8 6 ) 0 3 8 16 0 4

(2) 24 12 0 24 12 0 3

2

m

m m m mm

g m m

m

Vy M (m , -4) vi4

( , 4) ( , ) & 2

3

m m l im cn tm

Vi D 2 : (A2009)

Cho hm s2

(1)2 3

xy

x

1. Kho st s bin thin v v th ca hm s (1).2. Vit phng trnh tip tuyn ca th hm s (1), bit tip tuyn ct trc honh, trc tung ln lt

ti hai im phn bitA,B v tam gic OAB cn ti gc to .

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