chuyen đe tiep tuyen ( rat hay )

8/14/2019 chuyen e tiep tuyen ( rat hay )

1/14

Luyn thi H cht lng cao ths. Ng Dng 093 252 8949

Bi 1 : 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 s3

( )

1

xy C

x

. 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:

00 0

0

3( , ) ( )

1ox

M x y C yx

, '

2 20

4 4

( 1) ( 1)y k

x x

, tip tuyn ti M c dng (d) :

20 0 0

0 0 02 2 2 20 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 :20 0

2 2 00 0 0

00

15 347

(1, )( 1) ( 1) 7

111

xx xy x x

Ax x x

y xxx

Gi B l giao im ca tip tuyn v tim cn ngang y = 1 , suy ra ta ca B l nghim ca h :20 0

02 200 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 AB

11 3

2 2 1

A BM

A BM

xx xx x

x M l

x xy yy

x

(pcm)

V d 3 : (D2005)

Cho hm s 3 21 1

( )3 2 3 m

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 5xy = 0 1 5 4k m m http://chuyentoan.wordpress.com Nha Trang 8/2009

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


2/14


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)

im B khc im A . tm honh im B theo 0x

Bi gii :

Vi im 0 0( , )A x y (C) 30 0 03 1 y x x , ' 2 ' 20 03 3 ( ) 3 3 y x y x x Tip tuyn ca th hm c dng :

' 2 3 2 30 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 trn

honh giao im ca (d) v (C) :3 2 3 3 2 3 2

0 0 0 0 0 0 0

200

000

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

Vy im B c honh 02Bx 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 : 2y 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

1

k 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

http://chuyentoan.wordpress.com Nha Trang 8/2009

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 c 0x 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


3/14


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

33

2

1 241 2 14

4 3 3 2,3 3 32

2, 642

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 +6V d 2 :(H cnh st 1998)

Cho hm s

2 3 3

2

x x

y x

; vit phng trnh tip tuyn bit rng tip tuyn song song vi ng thng y = -3x +2

Bi 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 33(2)

( 2)

x xx m

x

x x

x

c nghim 2x (2) 2

3

24 16 15 05

2

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 2tan30 1

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 cbi 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 nht



4/14

Luyn thi H cht lng cao ths. Ng Dng 093 252 8949Bi gii:

TX: D R Ta 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)

Mt s v D in HnhV 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)

24

(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 http://chuyentoan.wordpress.com Nha Trang 8/2009

+

Dng 3: vit phng trnh tip tuyn bit n i qua mt im cho trcBi ton : cho hm s : ( ) y f x v im 0 0( , )A x y vit phng trnh tip tuyn bit rng tip tuyn iqua im A

Cch 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


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Vi x = 61

4k tip tuyn c dng :

1 7

4 2y 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 2

12 3

3 y x x x vit phng trnh tip tuyn bit rng tip tuyn i qua

4 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

01 4 4 8

2 3 ( 4 3)( ) 3 11 8 03 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 5

3 9k tip tuyn l :

5 128

9 81y x

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

3

Vy t A v c ba tip tuyn ti th hm sV 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 h

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)



6/14


Mt s v d in hnh :

V d 1 : (D2007)Cho hm s

2( )

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

4

Bi 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 0

0 0 0 02 2 20 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 :20 2

202 200 0

22( ,0)( 1) ( 1)

00

xy x x x

A xx xy

y

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

2 2 0 00 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 20 0

2 20 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 21

( ; 2) ; (1,1)

2

M M

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 l

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


Dng 4 : Mt S Bi Ton Nng Cao V Tip Tuyn Luyn Thi i


7/14


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 mm

y x C x

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

C 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 tuyphn 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 24

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

(2) 24 12 0 24 12 0 32

m

m m m mm

g m m

m

Vy M (m , -4) vi4

( , 4) ( , ) & 23

m m l im cn tm

V d 5 : ( hc vin BCVT 199)Cho hm s 3 23 2 ( ) y x x C . Tm c im thuc th (C) m qua k c mt v ch mt ti

tuyn vi th hm s (C )

Bi gii : 3 2 20 0 0 0 0( , ) ( ) 3 2 ; ' 3 6M x y C y x x y x x

Tip tuyn qua M : 3 20 0 0 0 0( ) ( ) 3 2y k x x y k x x x x

iu kin tip xc :

3 2 3 20 0 0

2

3 2 ( ) 3 2 (1)

3 6 (2)

x x k x x x x

x x k

( c nghim )

Th (2) vo (1) ta c :

Nha Trang 8/2009


8/14


3 2 2 3 2 3 2 3 20 0 0 0 0 0 0

0(2 )20 0 0

3 2 ( 3 6 )( ) 3 2 2 3( 1) 6 3 0 (1)

( )(2 3) 0 3

2

x x x x x x x x x x x xx x x

x x

x x x x xx

t M v c 1 tip tuyn ti th hm s th phng trnh (2) phi c mt nghim

00 0 0

31 0

2

x x x y

vy im M cn tm c ta l : M ( 1, 0)

Li bn :( v tip tuyn c ta tip im l M nn (1) lun c nghim kp 0x x , dng s horney chia r

ta c phng tr nh (2) thi )

V d 6 :Cho hm s : 3 3 ( ) y x x C . tm trn ng thng : x = 2 nhng im m t c th k c 3 tip

tuyn vi th hm (C )Bi gii :

im M ng thng x = 2 (2, )M a Tip tuyn qua M c dng : ( 2) y k x a

iu kin tip xc :3

2

x 3 ( 2) (1)

3 3 (2)

x k x a

x k

c nghim

Thay (2) vo (1) ta c phng trnh :3 2 3 2 3 23 (3 3)( 2) 2 6 6 6 0 2 6 6x x x x a x x x a a x x (*)

Xt hm s : 3 2( ) 2 6 6 , f x x x TX : D = R ; 2'( ) 6 12 f x x x

2 0'( ) 0 6 12 0 2

x f x x x

Bng bin thin:x 0 2 y - 0 + 0 -

y2

-6 t M k c 3 tip tuyn ti th hm s th phng trnh (*) phi c 3 nghim phn bit . da vobng bin thin ta thy (*) c 3 nghim phn bit 6 2a

Kt lun : vy im M (2, a ) vi ( 6, 2)a l im nm trn ng thng x = 2 c th k c 3 tiptuyn ti th hm (C )V d 7 : (H Y dc TP HCM 1998 )

Cho hm s 4 22 1 y x x (C ) . tm trn trc tung nhng im m t c th k c 3 tip tuynn th hm s (C)V d 8 : (H S Phm TP HCM 2001 )

Cho hm s2

( )2

xy C

x

, cho im A ( 0 , a ) t m a t A c th k c hai tip tuyn n

(C) ng thi 2 tip im nm v trc oxNha Trang 8/2009


9/14


Bi gii :

Phng trnh tip tuyn qua A c dng : y = kx + a

iu kin tip xc :

2

2(1)

1

3 (2)( 1)

xkx a

x

kx

c nghim 1x

Thay (2) vo (1) ta c : 222 3

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

x xa g x a x a x a x

x x

(*)

t A k c 2 tip tuyn ti (C ) th (*) phi c hai nghim phn bit khc 1 :2( 2) ( 1)( 2) 3 6 0

11 0

2(1) 3 0

a a a aa

aa

g

(**)

Gi s hai tip im ln lt l : 1 1 2 2( , ) ; ( , )A x y B x y l ta hai tip im th 1 2,x x l nghim ca (*)

V 1 21 21 2

2 2;

1 1

x xy y

x x

; p dng nh l Vi-et ta c :

1 2

1 2

2( 2)1

2

1

ax x

a

ax x

a

A, B nm v 2 pha trc ox th y 1 2 1 2 1 21 21 2 1 2 1 2

( 2) ( 2) 2( ) 40 . 0 0

( 1) ( 1) ( ) 1

x x x x x xy y

x x x x x x

2 2( 2)2. 4 21 1 0 3 2 0

2 2( 2) 32. 1

1 1

a a

a a a aa a

a a

Di chiu vi iu kin (**) th ta tm c :2

; 13

a a

Li bn :Nu bi ton yu cu tm a t a k c 2 tip tuyn ti th , sao cho tip im nm v 2 pha trc

oy th ch cn phng trnh (*) c 2 nghim phn bit 1 2,x x sao cho 1 20x x

V d 9 :Cho hm s : 3 23 2 ( ) y x x C . tm trn ng thng : y = -2 nhng im m t c th k n th hm s (C ) hai tip tuyn vung gc vi nhau ?Hng dn gii :

Dim M

ng thng : y = -2 ( , 2)M m

Sau cc em lp phng trnh tip tuyn qua M , s dng iu kin tip xc ta a ra c phng trnsau :

2

2

2( 2) 2 (3 1) 2 0

( ) 2 (3 1) 2 0

x x x m x

g x x m x

Vi x = 2 th tip tuyn : y = 2 ; khng tm c tip tuyn no ca (C ) vung gc vi ng thng trVy t M c th k c 2 tip tuyn vung gc vi nhau th g(x) = 0 phi c hai nghim phn bit

1 2, 2x x v 1 2 1k k lm tng t nh v d trn ta tm c55 55

( , 2)27 27

m M

Nha Trang 8/2009


10/14


Ta bit tip tuyn ca th hm s ti mi im bt k trn khong li lun nm pha trn th v tip tuyn ti mi im trn khong lm lun nm pha di th, cn ti im un ca th th tituyn xuyn qua nn ta c nhn xt sau

Nhn xt: Nu l tip tuyn ca th hm s ti im

( A khng phi l im un) , khi tn ti mt khong I cha im sao cho hoc. ng thc xy ra khi

By gi ta vn dng nhn xt ny chng minh mt s bt ng thc

Bi ton 1: Cho v . Cmr :

Li gii:

*Nhn xt: Ta thy ng thc xy ra khi v Bt cn chng minh c dng :

Trong vi .Nn ta nh gi f(x) v tip tuyn ti im c honh

Tip tuyn ca th hm s ti im c honh l:

Ta c:

pcm

Ch : Nu l tip tuyn ca th hm s ti im th ta lun phn tch

Bi ton 2 :Cho v . Cmr :

Li gii: Ta thy ng thc xy ra khi v Bt cho c dng

trong vi Nha Trang 8/2009

Dng 5 : Ngha Hnh Hc ca Tip Tuyn ( tham kho thm)


11/14


Tip tuyn ca th hm s ti im c honh l:

Tip theo ta s nh gi . Tht vy

iu ny lun ng vi mi x v ng

thc xy ra khi . Vy ta c:

Mt khc : nn suy ra pcm

Ta thy rng yu t quan trng nht chng ta c th s dng phng php ny l ta chuyn c Bt vdng

hoc v tha mn iukin no .

Bi ton 3: Cho .Cmr : Cmr: 2 2 29

( ) ( ) ( ) 4( )

a b c

b c a c a b a b c

Li gii: V nu Bt ng vi b s th cng ng vi b s nn ta c th gi s.Khi Bt cho tr thnh

vi

Tip tuyn ca th hm s ti im c honh l

Ta c:

Suy ra : pcm

Bi ton 4:Cho . 2 2 2 2 2 2( ) ( ) ( ) 6

( ) ( ) ( ) 5

a b c b a c c b a

a b c b a c c b a

Nha Trang 8/2009


12/14


(Trch thi Olympic 30-4 Lp 11 nm 2006)

Li gii: Khng mt tnh tng qut ta gi s

Khi bt cho tr thnh:

Hay

vi vi 0


13/14


cn ca ( )C ti hai im A, B sao cho tam gic OAB vung ti O ( Khi A 2009)

Bi 6: cho hm s 3 22 3 5 y x x vitphng trnh tip tuyn ,bit rng tip tuyn i qua19

( , 4)12

A ( H Quc Gia Thnh Ph HCM 2001)

Bi 7: cho hm s

2

2

x

y x

( )C . vit phng trnh tip tuyn tuyn vi th hm s , bit tip tuyn iim A ( -6,5) ( H Ngoi Thng TPHCM 1995)

Bi 8 : cho hm s y =1

1x

x

, CMR c th k t A( 1,-1) ti th hai tip tuyn vung gc vi nhau

Bch Khoa 1996)

Bi 9: cho hm s sau : 4 21 3

32 2

y x x . vit phng trnh tip tuyn bit tip tuyn i qua (0A

Bai10: cho hm s : 3 21

2 33

y x x x . qua im4 4

( , )9 3

A k c bao nhiu tip tuyn ti th

(H Ngoi Ng 1998)

Bi 11 : cho hm s2 2 2( 1)

x xy

x

. chng minh rng t giao im ca hai ng tim cn khng k

tuyn no ti th ca hm s (B 2005)

Bi 12: cho hm s2 3 3

( )2

x xy C

x

. Vit phng trnh tip tuyn bit tip tuyn vung gc vi

thng (d) : 3 6 0y x (H Cnh St 1998)

Bi 13 : cho hm s 33 4 ( ) y x C . vit phng trnh tip tuyn ,bit rng tip tuyn to vi ng t

3 6 0y x mt gc 030

Bi 14 : cho hm s : 3 23 9 5 ( ) y x x x C . trong cc tip tuyn vi th tm tip tuyn c h s

nh nht ( H Ngoi Thng 1998)

Bi 15 : cho hm s2 1

1

xy

x

, gi I l tm i xng ca th . tm im M thuc th ,sao cho tip

M vung gc vi ng thng IM( D b B2003)

Bi 16: cho hm s : 31 2

( )3 3

y x x C tm trn th nhng im m t k c tip tuyn v

vi ng thng1 2

3 3y x (H Ngoi Ng 2001)

Bi 17: cho hm s1

( )1

xy C

x

. Tm m ng thng ( ) : 2d y x m ct th hm s ( )C ti hai

phn bit A, B sao cho tip tuyn ti A, B song song vi nhau (CSP 2005)

Bi 18: cho hm s2

( )1

xy C

x

, tm im ( )M C , bit tip tuyn ca th hm s ti M ti hai dim A

tam gic OAB c din tch bng1

2 (D2007)

Bi 19: cho hm s ( )1

xy C

x

, Vit phng trnh tip tuyn ( )d ca th ct hai ng tim cn t

cho tam gic IAB cn , I l giao im ca hai ng tim cn( D2007_d b)http://chuyentoan.wordpress.com Nha Trang 8/2009


14/14


Bi 20: cho hm s

3

12 12 ( ) y x x C

tm trn ng thng y = 4 m t c th k c ba tip tuynbit ti th ( )C ( HV Bu Chnh Vin Thng 1998)

Bi 21: cho hm s 3 23 2 ( ) y x x C Tm trn ( )C nhng im m t k c duy nht mt tip th hm s ( )C ( HV Bu Chnh Vin Thng 1999)

Bi 22: cho hm s2

( )1

xy C

x

. v im A(0, a ) . tm a t im A c th k c hai tip tuyn n

sao cho hai tip im nm v hai pha trc ox (H S Phm TP HCM 2001)Bi 23 : cho hm s 3 23 2 ( ) y x x C tm trn ng thng 2y cc im m t c th k n hm s (C) hai tip tuyn vung gc vi nhau

Bi 24 cho hm s 3 23 ( ) y x x C Tm trn ng thng 2x nhng im m t c th k ng 3tuyn ti th hm sBi 25: cho hm s 4 22 1 ( ) y x x C tm tt c cc im thuc trc tung sao cho t c th k tuyn ti th ( )C (H Y Dc TP HCM 1998)


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