Cẩm nang hướng dẫn luyện thi Đại học - Hình học: Phần 1

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516.0076

(Nhom giao vien chuyen luyen thi Dai hoc)

LUYEN TH DAI HOC

JION Th.S NGUYEN TAT THU

1^ (Nhom giao vien chuyen luyen thi Dai hoc)

L6I NOI DAU

Cac em hpc sinh than men!

Trong nhung nam gan day, de thi Dai hpc luon c6 3 cau thupc ve phan mon

Hinh hpc so cap Cac bai toan thupc chu de hinh hpc rat da dang va sang tao

trong loi giai da gay khong it kho khan cho cac thi sinh Hon nua, nhung chu de

Hinh hpc so cap cac em hpc sinh dupe hpc 6 ca ba khoi lop nen luong kien thuc

ciia cac em hpc sinh ciing bj roi rac Voi niem dam me va nhieu nam kinh

nghi^m giang day danh cho bp mon toan, thay Nguyen Tat Thu da danh thoi

gian va tam huye't vie't tap sach "Cam nang luy^n thi Dai hpc Hinh hpc so cap"

nham khoi day niem yeu thich toan hpc, ren luyf n ky nang tir lam bai, tu on tap

Npi dung ciia bp sach dupe chia lam 3 chuong ^ ^ ^

Chuong 1 Phuong phdp toa do trong mat phdng *

Chuong 2 Hinh hoc khong gian

Chuong 3 Phuong phdp toa do trong khong gian

Voi lo'i vie't khoa hpc, sinh dpng, bp sach giiip cac em tiep can mon toan mot

each nh? nhang, tu nhien, khoi nguon cam hung khi tu hpc

Kien thuc trpng tam ngan gpn, day dii bao gom ly thuyet, phuong phap giai

cac dang bai di tu co ban den mo rpng va chuyen sau, qua do giiip cac em hieu

ro ban chat, phan tich, lap luan nhuan nhuyen de chu dpng tim ra phuong

phap giai quyet bai toan

Vi du minh hpa trong tung phan dupe phan loai, s^p xep chpn Ipc tir de den

kho nham dan dat cac em den nhirng dang bai thi Dai hpc Loi giai vira chi tiet

vua gpi mo de cac em tung buoc vira phan tich vua tim toi ra each giai chinh

xac va thu vj nhat Loi binh va nhan xet cua tac gia sau moi bai giai la kinh

nghi^m quy bau cho cac em

Lam nhieu bai tap de nang cao nang luc tu duy, do la mot each hpc toan

hifu qua nhat Cac em se hung thii hon khi dupe thu sue voi nhieu bai t^p

khac nhau va tinh huo'ng da dang c6 kem huong dan giai

De su dung bp sach hi^u qua va mang lai ket qua cao nhat, cac em can kien

tri tim hieu de nam chSc ly thuyet, cham chi ren luy^n ky nang lam bai thong

qua cac vi du, bai tap trinh bay trong bp sach N^y ( ^

Hi vpng, tap sach: " Cam nang luyen thi Dai hoc Hinh hpc so cap" se tiep

tyc la nguon tai li^u bo ich cho cac em hpc sinh trong ky thi D^i hpc - Cao

dang toi

Mac du tac gia da danh nhieu tam huye't cho cuon sach, song sy sai sot la

dieu kho tranh khoi Chung toi rat mong nhan dupe su phan bi^n va gop y quy

bau cua quy doc gia de nhung Ian tai ban sau cuon sach dupe hoan thi^n hon

Nguyen Tat Thu

* Theo cau tnic de thi Dai hpc - Cao dSng ciia Bp Giao d\ic thi trong de thi vao cac truong Dai hpc - Cao dSng c6 3 diem Hinh hpc dupe chia thanh 3 cau, moi cau 1 diem nhu sau: ,

Cau 5 Gom cac van de ve hinh hpc khong gian tong hpp

Noi dung ciia cau 5 trong de thi thuong gom hai y: Tinh the ti'ch khoi da dien (khoi chop va khoi lang try) va y thii hai thuong xoay quanh cac va'n de +) Chung minh quan he vuong goc, quan h^ song song;

+) Tinh goc giiia hai duong thang cheo nhau;

+) Tinh khoang each tu mot diem den mat phang; i • ' +) Tinh khoang each giiia hai duong thang cheo nhau *

Cau 7a, 7b: Gom cac van de ve phuong phap tpa dp trong mat phang Chii yeu

' xoay quanh cac van de sau +) Lap phuong trinh duong thang, duong tron, Elip, Hypebol; ^ +) Xac dinh tpa dp ciia mot diem

Cau 8a, 8b: Gom cac van de ve phuong phap tpa dp trong khong gian, chii yeu

xoay quanh cac chii de +) Lap phuong trinh duong thang, phuong trinh mat phang, phuong trinh mat cau;

+) Chung minh vi tri tuong doi giCra duong thang, mat phSng va mat cau;

+) Xac djnh tpa dp ciia mpt diem thoa man tinh chat cho truoc

Sau day, chiing ta di phan tich de tim loi giai cac bai toan dupe trich trong cac de thi Dai hpc nam 2013 , ,^,

* Khoi A-2013

Cau 5 Cho hinh chop S.ABC c6 day ABC la tam giac vuong tai A, ABC = 30"

Tam giac SBC deu canh a va mat ben SBC vuong goc voi day Tinh theo a the tich ciia khoi chop S.ABC va khoang each tu C den (SAB)

Phan tich: Yeu cau ciia bai toan c6 2 y: Tinh the tich khoi chop va tinh khoang each tu diem C den mat phing (SAB) , j ^

+) Tinh the tich khoi chop S.ABC :

De tinh the tich ciia khoi chop, truoc he't phai xac dinh dupe chan duong cao ciia hinh chop? Theo de bai, mat ph5ng (SBC) vuong goc voi day nen duong cao ciia tam giac SBC ke tu S chinh la duong cao ciia hinh chop

Hon nij-a, tam giac SBC deu nen chan duong cao ciia hinh chop chinh la trung diem BC

Nen the tich cua khoi chop dugc tinh nhu sau:

Lai giai

Goi H la trung diem canh BC, suy ra SH 1 BC =i> SH 1 (ABC) va SH =

Ta co: AB - BC cos B = , AC = BC sin B = - ,

2 2

c suy ra S ^ B C = -j_ ^^^C = •

-Do do, the tich khoi chop S.ABC la: V = ^SH.S > R C = l l ^ ^ ^ ^ = ^

^ 3 ^^^^ 3 2 8 16 +) Tinh khoang each tu C den (SAB):

De' tinh khoang each tu mot diem M den mat phang (a), ta c6 cac each sau

Cach 1: Dung hinh chie'u H ciia M len (a) va tinh M H

Cach 2: Chuyen tinh khoang each tu M ve tinh khoang each t u diem khac bang

each dua \o tinh chat

Neu duong thSng M N ck (a) t^i I thi d(M,(a)) = ^ d ( N , ( a ) ) Diem I ta|

thuong chon la chan duong cao ciia hinh chop

Cach 3: Sii dung cong thuc the tich: Xet hinh chop M.ABC, khi do

d(M,(ABC)) = ^ ^ ^ ^ M ^

Voi bai toan tren, viec dung hinh chie'u cua C len (SAB) tuang doi kho, nen

ta chuyen ve tinh khoang each tu H den (SAB) Vi CH c3t (SAB) tai B va H

la trung diem cua BC nen d(C,(SAB)) = 2d(H,(SAB)) Do do, ta c6 the tinh

d(C,(SAB)) theo each 1 nhu sau:

Cach 1: Goi M la trung diem ciia cac doan AB va K la hinh chie'u ciia H len SM

Taco: H M / / A C = ^ H M l A B , H M = !AC =

-2 4

Ma AB 1 SH AB 1 (SHM) => AB 1 HK

W Mat khac HK 1 SM, do do HK 1 (SAB)

Trong tarn giac SHN ta c6:

V39

, Cau 7a Trong mat ph^ng voi he tpa do Oxy, cho hinh chi> nhat ABCD co diem

C thuQC duong thang d:2x + y + 5 = 0 va A(-4;8) Gpi M la diem doi xung cua B qua C, N la hinh chie'u vuong goc ciia B tren duong thang MD Tim

toa do cac diem B va C, bie't rang N(5;-4)

Phan tich: Day la bai toan xac djnh ^

toa do cua mpt diem Ve mat dai

so, de xac djnh tpa dp cua mpt diem ta can tim hai an, tuc la can^

thie't lap hai phuong trinh Trong hai diem can tim B va C thi diem C thupc duong thang d nen tpa dp

ciia C CO dang C ( x ; - 2 x - 5 ) Do

do, de tim tpa dp diem C ta chi can tim 1 an nen can thie't lap m p t , phuong trinh Goi I la giao diem ciia AC va BC, ta co tpa dp diem

Loi giai. Vi C € d => C(x;-2x - 5) Goi I la giao diem cua AC va BD, ta co I

la trung diem AC nen I x - 4 -2x + 3

Trong tam giac vuong BND, ta c6 IN = IB = lA => IN^ = lA^

Cau 7b Trong mat phang voi he toa dp Oxy, cho duong thing A : x - y = 0

Duong tron (C) c6 ban kinh R = %/lO cat A tai hai diem A va B sao cho

AB = 4V2 Tiep tuyen cua (C) tai A va B cat nhau tai mpt diem thupc tia

Oy Vie't phuong trinh duong tron (C)

Phan tich: De vie't phuong trinh (C) ta can di xac dinh tpa dp tam I Gpi M

la giao diem cua hai tiep tai A va B Khi do, gia thiet cua bai toan gom c6:

lA = IB = VlO, AB = 4V2, M e tia Oy Do do, ta se di xac dinh tpa dp diem

M(0;a), a >0 De xac dinh tpa dp diem M ta can thiet lap mpt phuong

trinh, gpi H la giao diem ciia AB va IM, ta xet tam giac MAI vuong tai A va

AH la duong cao, AH = — = 2V2

Tij- day, su dung cong thuc duong cao trong

tam giac vuong ta tim dupe AM, suy ra MI

va MH Mat khac MH chinh la khoang each

tu M den AB (tuc la duong thang d) Tu day

j ta tim dupe a va tpa dp diem M, do do ta c6

phuong trinh MI, hon niia dp dai MI tinh

dupe nen ta tim dupe tpa dp diem I Vay ta

I CO loi giai nhu sau:

Lai giai Gpi I la tam ciia duong tron can tim, H la trung diem ciia AB, M la

giao diem cua hai tiep tuyen tai A, B

Hay d(M,A) = 4V2o-^ = 4^y2=>a-8(do a>0)

Do do, phuong trinh IH la: x + y - 8 = 0 Taco IeMH=>l(b;8-b), IM = 5V2 <:> V2b^ = 5V2 ci> b = ±5

+ Voi b = 5, phuong trinh ciia (C) la (x - 5)^ + (y - 3)^ = 10 + Voi b = -5, phuong trinh ciia (C) la (x + 5)^ + (y -13)^ = 10 Cau 8a Trong khong gian voi he tpa dp (Oxy), cho duong thang

A: x - 6 _ y + l _ z + 2 _2 _2 ^ va diem A (l; 7; 3) Viet phuong trinh mat phSng (P)

di qua A va vuong goe voi A Tim tpa dp diem M thupc A sao cho

AM = 2V30

Phan tich: Npi dung ciia de bai gom hai y: Viet phuong trinh mat phang va

tim tpa dp diem M ; ,„ t

+) De vie't phuong trinh mat phJing, ta can tim VTPT va mpt diem di qua: Trong de bai, diem di qua da eo va VTPT cua (P) chinh la VTCP cua duong thang A

+) Tim tpa dp diem M: Vi M thupc A nen tpa dp cua M chi c6 1 an (an t) Do

AM = 2V3O nen tu day ta tim dupe t, tu do suy ra M Vay ta c6 loi giai sau: Loi giai Vi (P) 1 A => n(-3;-2;l) la vec to phap tuyen ciia (P) do do:

Phuong trinh mat phang (P) la : -3(x -1) - 2(y - 7) + l(z - 3) = 0

Lum m»s mym m vn mnn IH)L isguyen n n n m

-Cau 8b Trong khong gian voi he tpa dp Oxyz, cho mat phang:

+) De tim tpa dp tiep diem H ciia mat cau (S) va mp(P) ta viet phuong trinh I H

di qua I vuong goc voi (P) Khi do, tpa dp H la giao diem cua duong thang

IH voi (P) Ta CO loi giai nhu sau:

Lai giai Mat cau (S) c6 tam la l ( l ; - 2 ; l ) , R = >yi4

2(1)+ 3(-2) +1-11 Taco: d(I,(P)) = = 714 =R

Vay (P) tiep xuc voi (S)

Phuong trinh duong thSng d di qua I va vuong goc voi (P) la :

Cau 5 Cho hinh chop S.ABCD c6 day la hinh vuong canh a, mat ben SAB la

tam giac deu va nam trong mat phSng vuong goc voi mat phang day Tinh

theo a the tinh cua khoi chop S.ABCD va khoang each tu diem A den m|t

phing (SCD)

Phan tich:

+) De tinh the tich khoi chop, trudc het ta

phai di xac djnh chan duong cao ciia

hinh chop Vi mat phling (SAB) vuong

goc voi mat phang day nen chan

duong cao ciia hinh chop chinh la

chan duong cao cua tam giac SAB ve

tu S hay do chinh la trung diem canh

AB (do tam giac SAB deu)

+) Ta tha'y A H song song voi (SCD) nen

d(A,(SCD)) = d(H,(SCD)) Ta c6 loi

giai nhu sau

Lai giai Gpi H la trung diem canh AB, suy ra

SH1AB=> S H I (ABCD) va SH = ^

Cty TNHH MTV DVVH Khang Vi^t

The tich khoi chop S.ABCD la: V = IsH.S.orn = a^

-Gpi M la trung diem CD va K la hinh chieu vuong goc ciia H len SM Ta c6

CD 1 (SHM) => CD 1 H K

Dodo H K l ( S C D )

Mat khac AH//(SCD) => d(A, (SCD)) = d(H, (SCD)) = HK = SH.HM ^ aV21

VSH^+HM^ 7

Cau 7a Trong mat phMng voi h? tpa dp Oxy, cho hinh thang can ABCD c6 hai

duong cheo vuong goc voi nhau va AD = 3BC Duong th^ng BD c6

phuong trinh x + 2y - 6 - 0 va tam giac ABD c6 true tam la H(-3;2) Tim tpa dp cac dinh C va D rs

Phan tich: Gpi I la giao diem cua AC va BD Tu de bai, ta tha'y A I vuong goc ' voi BD nen H thupc A I va I la hinh chieu ciia H len BD, tu day ta tim dupe diem I va viet dupe phuong trinh AC va BD, khi do tpa dp ciia C va D chi

CO 1 an Lai c6, tam giac IBC vuong can tai I va tam giac CBH vuong tai B

nen ta c6 IC = IH = IB, tu do ta tim dupe tpa dp C, B Dua vao ID = -316 ta xac djnh dupe tpa dp diem D

Vay ta eo loi giai nhu sau:

Loi giai Gpi I la giao diem ciia hai duong cheo AC va BD, ta c6 A I vuong goc voi BD nen H thupc AI

Phuong trinh HI: 2x - y + 8 = 0

Tpa dp diem I la nghiem ciia h^ < ^ ^ ~ ^ o • ^ ~ => lf-2- 4)

[2x-y + 8 = 0 [y = 4 ^ ' '

Ta CO Tam giac BIC vuong can tai I nen goc CBI = 45" Mat khac, ACBH

vuong tai H nen BI phan giac ciia goc CBH I la trung diem CH, do do

Cau 7b Trong mat phMng vdi tpa dp Oxy, cho tarn giac ABC c6 chan duong

, chan duong phan giac trong cvia goc A la cao ha tu dinh A la H 17

D (5; 3) va trung diem ciia canh AB la M (0; 1) Tim tpa dp dinh C

Phan tich: Tir de bai ta c6 phuong trinh

BC, suy ra phuong trinh ciia AH Do

do, tpa dp ciia A chi c6 1 an, lay doi

xung qua M ta c6 tpa dp diem B Dua

vao A H vuong goc vdi BH ta tim dupe

tpa dp diem A va B Lay doi xung M

qua AD ta dupe dir'm N thuoc duong

thang AC, tu do ta viet dupe phuong

trinh AC Dua vao C la giao diem ciia

BC va AC ta tim dupe C Vay ta c6 loi

giai nhu sau:

Loi giai Ta c6, phuong trinh BC: 2x - y - 7 = 0; phuong trinh A H : x + 2y - 3 = 0

Vi A € A H ^ A ( 3 - 2 a ; a ) ^ B ( 2 a - 3 ; 2 - a )

Vi AH.HB = 0 nentaco =^a = 3 ^ A ( - 3 ; 3 ) , B ( 3 ; - 1 )

Phuong trinh AD : y = 3 => N (0; 5) la diem doi xung ciia M qua AD

eAC

=> Phuong trinh AC : 2x - 3y + 15 = 0 va phuong trinh BC : 2x - y - 7 = 0

Vay C (9; 11)

Cau 8a Trong khong gian vdi h^ tpa dp Oxyz, cho diem A (3; 5; 0) va mat

phSng (P) : 2x + 3y - z - 7 = 0 Viet phuong trinh duong thing di qua A

vuong goc vdi (P) Tim tpa dp diem doi xung ciia A qua (P)

Phan tich: Duong thang d can viet phuong trinh vuong goc vdi (P) nen

nhan VTPT ciia (P) lam VTCP, tu do ta viet dupe phuong trinh d

Gpi A' doi xung vdi A qua (P) va H la giao diem ciia d va (P), ta cd H la

trung diem AA' Do do, xac dinh dupe H ta se cd diem A' Vay ta cd loi giai

nhu sau:

Loi giai Ta cd n = (2;3;-l) la VTPT ciia (P) Suy ra duong thSng d di qua A

va vuong gdc vdi (P) nhan n lam VTCP

x = 3 + 2t

y = 5 + 3t, t e R

z = - t Phuong trinh d :

Gpi H la giao diem ciia d vdi (P) va A' la diem doi xung vdi A qua (P)

•-ty uvvH Khang Vi^t

Ta cd H(3 + 2t;5 + 3t;-t)e(P)=^2(3 + 2t) + 3(5 + 3t) + t - 7 = 0 ^ t = - l = > H ( l ; 2 ; l )

Vi H la trung diem A A' nen A ' ( - l ; - l ; 2 ) Cau 8b Trong khong gian vdi h? tpa dp Oxyz, cho cac diem A ( l ; - 1 ; l ) ,

^ _ y ^ _ z - 3

B (-1; 2; 3) va duong thSng A:

Viet phuong trinh -2 1 3

duong thing di qua A, vuong gdc vdi hai duong thang qua AB va A

Phan tich: De viet phuong trinh duong thang ta can tim mpt diem di qua va mot VTCP De tim VTCP, ta thudng tim hai vecto khong cimg phuong va cung vuong gdc vdi dudng thang dd Khi dd , tich cd hudng ciia hai vecto

dd la VTCP ciia dudng thang

Trong de bai, dudng thang d can viet phuong trinh vuong gdc vdi AB va A nen AB A U^^ la VTCP ciia d Vay ta cd loi giai nhu sau:

Loi giai Gpi d la dudng thang can lap phuong trinh

Ta cd AB = (-2; 3; 2) va u = (-2;1;3) la VTCP ciia dudng thing A

Vi d vuong gdc vdi A va AB nen a = AB A u = (6; 2; 4) la VTCP ciia d Vay phuong trinh d : -• j - l _ y + 1 _ z - 1

3 1 2

* KhoiD-2013 Cau 5 Cho hinh chdp S.ABCD cd day ABCD la hinh thoi canh a, canh ben SA vuong gdc vdi day, BAD = 120" M la trung diem canh BC va SMA = 45*^ Tinh theo a the tich khdi chdp S.ABCD va khoang each tu Dden (SBC) Phan tich: De bai cho SA la dudng cao ciia hinh chdp nen ta di tinh SA dua vao tarn giac SAM vuong tai A

Vi AD//(SCB) nen d(D,(SBC)) = d(A,(SBC)) Vaytacdldi giai nhu sau:

Loi giai

Loi giai Vi B'AD = 120" => ABC = 60"

=> AABC deu, suy ra AM 1 BC va A M = Tarn giac SAM vuong tai M va SMA = 45°

Cant nang luyen thi DH Hinh Hoc - Nguyen latThti

ri D o

The tich khoi chop S.ABCD la: V = - S A S A B C D " 2 • ~ 2 ~ ~ 2 ~8~'

V i AD//BC ^ AD//(SBC) d(D,(SBC)) = d(A,(SBC))

AB, diem H(-2;4),I(-1;1) Ian luot la chan d u o n g cao ve t u B va tam ducmg

tron ngoai tiep tam giac ABC T i m toa dp d i n h C

Phan tich: V i M va I la trung diem canh AB va tam d u o n g tron ngoai tiep

tam giac A B C nen I M vuong goc v o i AB, t u day ta c6 p h u o n g trinh cua AB

Ta goi toa do cua A (chi c6 1 an) lay doi x u n g qua M ta c6 toa do ciia B

D u a vao A H va B H vuong goc ta t i m dupe tpa dp ciia A va B

Co A, H nen ta c6 p h u o n g trinh A C Dua vao l A = IC ta t i m dupe tpa dp

diem C

7 1 2'2

P h u o n g t r i n h A C : 2 x - y + 8 = 0=>C(c;2c + 8 ) ' " ' '

Taco: I C ' = I B 2 =>(c + l ) +(2c + 7) = 25 « c^ + 6c + 5 = 0 c = - l , c = - 5

Suy ra C ( - l ; 6 )

Vay C ( 4 ; l ) hoac C ( - l ; 6 ) j , , ,

Cau 7b Trong mat phang v o i he tpa dp Oxy, cho d u o n g tron (C): : r <

(x -1)^ + (y - 1)'^ = 4 va d u o n g thang A : y - 3 = 0 Tam giac M N P c6 true

tam trung v o i tam cua (C), cac dinh N va P thupc A , d i n h M va t r u n g diem ciia canh M N thupc (C) T i m tpa dp diem P « -Phan tich: Ta thay (C) va A tiep xuc voi nhau tai T, ma tam I la true tam nen M la giao cua T I voi (C)

Goi J la t r u n g diem M N , suy ra IJ la d u o n g trung binh nen IJ song song voi

A va J thupc (C) nen ta t i m dupe tpa dp diem J, lay doi x u n g ta c6 diem N

Vi P thupc A nen tpa dp cau P chi c6 1 an, dua vao N I v u o n g goc voi M P ta

tim dupe P Vay ta c6 loi giai n h u sau:

L a i giai. D u o n g tron (C) c6 tam 1(1; 1), R = 2

Ta CO d ( l , A ) = K nen suy ra

A tiep xuc (C) tai T

Do 1 la true tam tam giac

P M N nen M I vuong goc A , suy ra x^^ - x, =^1

Ma M thupc (C) nen M ( l ; -1) Gpi J la t r u n g diem M N suy ra

IJ la d u o n g trung binh cua tam giac M T N , suy ra y j = y, = 1

Ma J thupc (C) nen J(3; 1) hay J(-l; 1)

+) Voi J(3; ] ) thi N(5; 3) Gpi P(t; 3) thupc A Ta c6 M 4/MP =^ t = -1 =^ P ( - l ; 3)

+) Voi J(-]; ] ) thi N ( - 3 ; 3) Gpi P(t; 3) thupc A Ta c6 N I 1 M P =^ t = 3 => P(3; 3)

Cau 8a Trong khong gian voi h^ tpa dp Oxyz, cho cac d i e m A ( - l ; -1; -2), B(0; 1; 1) va mat ph^ng ( P ) : x + y + z - l = 0 T i m tpa dp h i n h chieu

Ldm ridHg lU^fH THI UH HMH Hl?e - Nguyen i ai nur

vuong goc cvia A tren (P) Viet phuong trinh mat phang di qua A, B va

vuong goc voi (P)

Phan tich: De lap phuong trinh mat phang ta can tim mot diem di qua va

VTPT De tim VTPT ta thuang tim hai vecto khong cimg phuong c6 gia

song song hoac nam trong mat phing do Voi bai toan tren, mat phang can

viet di qua A,B va vuong goc voi (P) nen AB A np la VTPT Vay ta c6 loi

giai sau:

Loi giai Gpi (a) la mat phang can lap Ta c6 n = (l;l;l) la VTPT cua (P)

Vi (a) di qua A, B va vuong goc voi (P) nen n'= AB A n = (-l;2;-l) la

VTPT ciia (a)

Phuong trinh (a) la: x - 2y + z -1 = 0

Cau 8b Trong khong gian voi he tQa do Oxyz, cho diem A(-l; 3; -2) va mat

phang (P): X - 2y - 2z + 5 = 0 Tinh khoang each tu A den (P) Viet phuong

trinh mat ph^ng di qua A va song song voi (P)

, - l - 6 + 4 + 5| 2 Khoang each tu A den mat phang (P): d( A,(P)) = — , = — \/l + 4 + 4 3

Goi (Q) la mat phang can tim

(Q) di qua A va c6 mpt vecto phap tuyen la n = (l;-2;-2)

= > ( Q ) : X - 2y - 2z +3 = 0

De thi thu truang THPT Chuyen Luong The Vinh nam 2014

* KhoiA

Cau 5, Cho hinh chop S.ABCD c6 day A B C D la hinh thoi canh a va

B A D = 60^' Hinh chieu ciia S len mat phing ( A B C D ) la trpng tam tam

giac A B C Goc giiia mat phSng ( A B C D ) va (SAB) bSng 60° Tinh the tich

khoi chop S.ABCD va khoang each giua hai duong thang SC va AB

Loi giai

CQ'I H la trong tam tam giac ABC, suy ra SH 1 ( A B C D )

Ke MH vuong goc voi AB, M thuoc AB

Ta CO SMH la goc giua hai mat phSng (SAB) va ( A B C D ) , do do

Cau 7a Trong mat ph^ng Oxy cho hinh vuong ABCD c6 A(1;1), AB = 4 Goi

M la trung diem canh BC, K - ; — la hinh chieu vuong goc ciia D len

.6

AM Tim toa do cac dinh con lai ciia hinh vuong, biet Xg < 2

Loi giai

Gpi N la giao diem ciia DK va AB

Khi do ADAN = AABM AN = BM

=> N la trung diem canh AB

- — r 4 8 ^ Taco AK= , phuong trinh

V ^ ^ J

AM:2x + y - 3 = 0, D K : x - 2 y - 3 = 0

Vi N G DK ^ N(2n + 3;n) =:> AN = (2n + 2;n -1)

Ma AN = 1 AB = 2 =^ AN^ = 4 « (2n + if + (n -1)^ = 4 » Sn^ + 6n +1 = 0

Cau 7b Trong mat phSng Oxycho tarn giac ABCco true tarn H(-6;7), tarn

duong tron ngoai tiep l ( l ; l ) va D(0;4) la hinh chieu vuong goc ciia A len

duong thSng BC Tim tpa do dinh A

Loi giai

Ta CO HD = (b;-3), suy ra phuong trinh B C : 2 x - y + 4 = 0

Phuong trinh DH : x + 2y - 8 = 0

Goi M la trung diem canh BC, ta c6 I M = d(l,BC) = N/S

Kc duong kinh BB', khi dcS AHB'C la hinh binh hanh

non A H = B ' C - 2 1 M - 2 N / 5

Vi A G D H A ( 8 ~ 2a;a) ^ A H = {2a -14;7 - a)

Suy ra (2a -14^ + ( a - 7 ^ = 2 0 ^ ( a - 7 ^ = 4 => a = 9,a = 5

Vay A (2; 5) hoac A (-10; 9)

^ y-2 z - 3

Cau 8a Trong khong gian Oxyz cho duong thang ^ ' — ^ =

mat phang (u): x + 2y + 2z + 1 = 0, (p): 2x - y - 2z + 7 = 0 Viet phuong trinh

mat cau (S) ccS tarn nam tren duong thing d va (S) tiep xuc voi hai mat

phang (a) va ((i)

Lai giai

Goi I la tam cua mat cau (S), l e d nen l(-t;2 + t;3 + 2t)

Vi (S) tiep xiic voi hai mat phing (a) va (p) non d(I,(a)) = d(I,(P))

5t + l l 7t + l

3 3

+) Voi t = - l ^ I ( l ; l ; l ) , R = 2,

5t + n = 7t + l o t = 5,t = - l

Cty TNHH MTV DWHKhang Vi^t

Phuong trinh mat cau (S): (x -1)^ + (y -1)^ + (z -1)^ = 4 I-) Voi t = 5=^l(-5;7;13), R = 12

Phuong trinh mat cau (S): (x + 5)^ + (y - jf + (z -13)^ = 144

Cau 8b Trong khong gian Oxyz cho hai duong thing duong thing

d : ^ = ^ = | , ^ • ^ = Z^ = ^^ va diem A(2;3;3) Viet phuong

trinh mat cau (S) di qua A , c6 tam nam tren duong thing A va tiep xiic voi duong thing d

Lai giii '

Gpi I la tam cua mat cau (S), ta c6 l(2 + t;-t;3 + 2t)

Suy ra AI = (t;-t-3;2t)=> lA = V6t^+6t+ 9 Duong thing d di qua B(-1;-1;0) va c6 u = (-2;l;2) la VTCP

Cau 5 Cho hinh lang try dung A B C A ' B ' C c6 tam giac ABC vuong t^i A,

AB ^- a, BC = 2a va A A ' = 2a G(?i M la trung diem ciia canh BB' Tinh the tich khoi chop BMCA' va c6 sin cua goc giua hai duong thing A'M va BC

Lai giii

Ve duong cao A'H ciia tam giac A ' B ' C

Ta CO A'H 1 (BMC) va A'H = ^'B'.A'C

::—»

B ' C

Ma S^MBC = ^BC.BM = 1 2a.a =

The tich khoi tu dien BMCA' la:

a 2 = ^

V - - A H S , M B C - 3 — ^

Goi N la trung diem CC, ta c6

MN//BC nen goc giija hai duong

thang A ' M va BC bang goc giira hai

duong thang A ' M va M N A

Ta C O A'M^ = A'B'2 + B'M^ = 2a^

A ' N ^ = C'N^ + A'C'^ = a^ + 3a^ = 4a^

/ N / N

Vay CO sin cua goc giiia hai duong thSng A ' M va BC bang

Cau 7a Trong mat phang toa dp Oxy, cho hinh binh hanh ABCD c6 D ( - 6 ; - 6 ) ,

duong trung true cua doan DC c6 phuong trinh la Aj : 2x + 3y +17 = 0 va

duong phan giac goc BAC c6 phuong trinh la : 5x + y - 3 = 0 Xac dinh toa

dp cac dinh con lai cua hinh binh hanh

Lai gidi

Gpi I la trung diem ciia CD, I € Aj nen I (3a - 1 ; -2a - 5)

Taco U j D i = 0 trong do Di(3a + 5;-2a +1) va U j ( - 3 ; 2 ) la vecto chi phuong

ciia Aj suy ra a = - 1

Vay l(-4;-3)=:>C(-2;0)

Vi A e A2 nen tpa dp diem A c6 dang A (a; 3 - 5a)

Mat khac ABCD la hinh binh hanh tuong duong voi DA, DC khong cimg

DA, DC khong cung phuong khi va chi khi * ~~6~~ ^ ^

Duong th^ng A2 la phan giac goc BAC nhan vecto U 2 = (-1;5) lam vec to chi

phuong nen

Cty TNHH MTV DWH Khang Vir 1

cos I AB; u 2 ) = C O S ^ AC; U 2 ) <=> AB.u,

ACu-ABl AC

- <=> a = 1 (thoa man) .'.'j i h ;^ f ->i ,> ij,

" v ^ " ^ ( 2 a f ( 5 a - 3 f Vay tpa dp diem A ( 1 ; - 2 ) , B(5;4)

Cau 7b Trong mat phang Oxy cho ba duong thang dj : 4x + y - 9 = 0,

d 2 : 2 x - y + 6 = 0, d 3 : x - y + 2 = 0 Tim tpa dp cac dinh ciia hinh thoi

A B C D , biet hinh thoi A B C D c6 di^n tich bMng 15, cac dinh A , C thupc

Vay tpa dp cac dinh ciia hinh thoi la:

A(3; 5), B(2; 1),C(-2;0), D(-l;4) hoac A(-2;0), B(2; 1),C(3;5),D(-l;4) i n / i-^

Cau 8a Trong khong gian Oxyz cho duong th^ng d : - — - = = , mat

2 1 3 Ph5ng (a):2x + y - 2 z + 2 = 0 va hai diem A ( 0 ; - 1 ; - 2 ) , B(2;3;1) Viet phuong trinh mat ph^ng (p) di qua hai diem A , B va cSt duong thSng d tai

C sao cho C each mat ph^ng (a) mot khoang bang 2 , ^

Cam nang luyftt thi DH Hinh hgc - N^yen Tat Thu

Cau 8b Trong khong gian Oxyz cho duang thang A : — = = — ^ va

hai diem A(0;1;2), B(1;-1;2) Viet phuang trinh mat phSng (a) di qua A , B

+) Vai t = 1 => AB A AC = (-6;-3;-3) => n = (2;l;l) la VTPT cua (a)

Phuang trinh (a): 2x + y + z - 3 = 0

+) Vai t - -2 AB A AC = (6;3;-3) =^n^ ( 2 ; l ; - l ) la VTPT cua (a)

Phuang trinh (a): 2x + y - z +1 = 0

• Hai vec to u(xj,yj); v ( x 2 ; y 2 ) cung phuang voi nhau <=>

• Goc giiia hai vec to u(xpyj); v ( x 2 ; y 2 ) : cos(u,v) =

• Cho A ( x A ; y A ) ; B ( x B ; y B ) K h i d 6 : 1) AB = ( x B - X A ; y B - y A ) 2) AB= AB = 7(XB - + (yg - y A ) '

3)

X - ^ A + X B

' 2 trong do I la trung diem cua AB

• AB 1 CD <=> AB.CD = 0

• Cho tam giac ABC voi A ( x ^ ; y ^ ) , B(xB;yB)/ C(xc;yc) • Khi do trpng tam

G (x^; y(2 ) ciia tam giac ABC la :

II Phuang trinh.dutmg thang ' !

I Phuang trinh duoiig thing

I I Vec to chi phuang (VTCP), vec to phap tuyen (VTPT) ciia duong thing:

Cho duong thSng d

• n = (a; b) 0 gpi la vec to phap tuyen ciia d neu gia ciia no vuong vai d

• u = ( U j ; u 2 ) ^ 0 goi la vec to chi phuang ciia d neu gia ciia no trung hoac

song song voi duong thSng d s'

M o t d u a n g thSng c6 v6 so VTPT va v6 so VTCP (Cac vec to nay luon ciing

p h u o n g v o i nhau)

• M o i quan h§ giua VTPT va VTCP: n.u = 0

• N e u n = (a; b) la m o t VTPT ciia duong thSng d thi u = (b; -a) la m o t VTCP

cua d u o n g thang d

• D u o n g t h i n g A B c6 AB la VTCP

1.2 Phuang trinh duong thing

1.2.1 Phuang trinh tong quat ciia duong thang:

Cho d u o n g t h i n g d d i qua diem A(xo;yo) va c6 n = (a;b) la VTPT, k h i do

p h u a n g trinh tong quat ciia d c6 dang: a(x - XQ) + b(y - yg) = 0

1.2.2 Phuang trinh tham so cua duong thang:

Cho d u a n g thang d d i qua diem A(xQ;yo) va c6 u = (a;b) la VTCP, k h i do

i , fx = XQ +at

p h u a n g trinh tham so' cua d u o n g thang d la: " , t € R

^ l y = y o + b t

2 V i tri tuong doi giua hai duong thang

Cho hai d u o n g t h i n g d^ : a^x + b j y + c^ = 0; d j : a j X + bjY + C j = 0 K h i do v i

tri t u o n g doi giira chiing p h u thuoc vao so n g h i ^ m cua h ^ :

ajx + b j y + c, = 0

a2X + b 2 y + C 2= 0

• Neu (I) v6 n g h i ^ m thi d j / / d 2

• Neu (I) v6 so n g h i ^ m thi d j = d2

• N e u (I) CO n g h i ^ m d u y nhat thi d j va d2 cat nhau va n g h i ^ m cua h? la tpa

do giao diem

3 Goc giira hai duong thang •'i^^'

Cho hai d u o n g t h i n g d j : a^x + b^y + Cj = 0; d2 : a2X + b2y + C j = 0 Gpi a la

goc nhpn tao bai hai d u o n g t h i n g d j va d2

a^ao + bib^

Taco : c o s a = ' '

^a^ + b ^ ^ a ^ + b ^

4 Khoang each tu mpt diem den duong thang

Cho d u a n g thang A : ax + by + c = 0 va diem M(xo; y g ) IGii do khoang each

a x Q + b y o + c

t u M den A duoc tinh boi cong thiic: d(M,(A)) =

V a ^ + b l

5, Phuong trinh duong phan giac cua goc tao boi hai duong thang

Cho hai d u o n g thang d j : a^x + b j y + C j = 0 va d j : a j X + b j y + C2 = 0

Phuong trinh phan giac ciia goc tao boi hai d u o n g thang la:

a i x + b i y + ci _ ^ a 2 X + b2y + C2 s H i i

-yja^ + b^ ^jal + hi -An^fH rfsU 0 j

, , , -' :rfi8 mfc'i «

1 Phuong trinh duong tron :

Cho d u o n g tron (C) tam I(a; b ) , ban kinh R, khi do phuang trinh ciia (C) la:

( x - a ) 2 + ( y - b ) 2 = R 2

Ngoai ra p h u a n g trinh: x^ + y^ - 2 a x - 2 b y + c = 0 v o i a^ + b^ - c > 0 cung

la phuang trinh ciia d u a n g tron c6 tam I(a; b ) , ban kinh R = +b^ - c

2 Phuong trinh tiep tuyen :

Cho d u o n g tron (C) : ( x - a ) ^ + ( y - b ) 2 = R ^

• Tiep tuyen A ciia (C) tai diem M la d u a n g thang d i qua M va vuong goc voi I M

• D u o n g thang A : A x + By + C = 0 la tiep tuyen ciia (C) <=> d(I, A) = R

• D u o n g tron ( C ) : (x - a)^ + (y - b)^ ^ R^ c6 hai tiep tuyen ciing p h u a n g voi

Oy la X = a ± R Ngoai hai tiep tuyen nay cac tiep tuyen con lai deu c6

dang : y = kx + m

IV E lip ^it •

1 Dinh nghia: Trong mat p h i n g cho hai diem co dinh Fj,F2 c6 F1F2 = 2c Tap

hop cac diem M ciia mat p h i n g sao cho MFj + M F j 2a (2a khong doi va

a > c > 0) la m o t d u o n g elip

• Fj,F2 : la hai tieu diem va 2c la tieu cu ciia elip

• MFj,MF2 : la cac ban k i n h qua tieu

2 Phuong trinh chinh tic ciia elip:

m m mng mygfi im vii iiinii nyiivauym i « t m w

-• Dinh: Ai(-a;0), A2(a;0), Bi(0;-b) va 62(0; b ) A^Aj =2agpi la dp dai

tryc Ion, BiBj = 2b gpi la dp dai tryc be

• Tieudiem: ¥^i-c;0), F2(c;0)

• Npi tiep trong hinh chif nhat co so PQRS

CO kich thuoc 2a va 2b voi b = a - c^

1 Dinh nghia: Trong mat phMng voi h# tpa dp Oxy cho hai diem FpFj c6

FjF2 =2c Tap hpp cac diem M cua mat phSng sao cho JMF^ - M F 2 | = 2a

(2a khong doi va c> a > 0) la mot Hypebol

• Fj, F2 : la 2 tieu diem va FjFj = 2c la tieu cu

• MFj, MF2 : la cac ban kinh qua tieu

2 2

2 Phuang trinh chinh tac ciia hypebol: —- - ^ = 1 voi b = c - a

a b

3 Tinh chat va hinh dang ciia hypebol (H):

• True doi xiing Ox (true thue), Oy (true ao) Tam doi xiing O

• Dinh: Aj(-a;0), A2 (a;0) Dp dai tryc thue: 2a va dp dai tryc ao: 2b

• Tieu diem Fj(-c; 0), F2( e; O)

VI Parabol

1 Dinh nghia: Parabol la tap hpp cac diem M ciia mat phang each deu mot duong thJing A co' dinh va mot diem F co djnh khong thupc A

A : duong chuan; F : tieu diem va d(F, A) = p > 0 la tham so tieu

2 Phuong trinh chinh tac ciia Parabol: y^ = 2px '

3 Hinh dang ciia ParaboI(P):

• Tryc Ox la tryc doi xiing, dinh O Tieu diem F(-^;0)

• Duong chuan A : x = — j - •

• M ( X ; y ) 6 (P): MF = X + ^ voi x > 0

§1 VIET PHl/ONG TRlNH O I / O N G T H A N G

1 Phuong phap chung

De lap phuong trinh duong thSng A ta thuong diing cac each sau

• Tim diem M ( X o ; y Q ) ma A di qua va mpt VTPT n = (a; b) Khi do phuong

trinh duong thang can lap la: a(x - XQ) + b(y - y^,) = 0

• Gia sii duong thang can lap A: ax + by + e = 0 Dya vao dieu ki^n bai toan ta tim dupe a = mb, e = nb Khi do phuong trinh A : m x + y + n = 0 Phucmg phap nay ta thuong ap dung doi voi bai toan lien quan den khoang each va goc

• Phuong phap quy tich: M(XQ;yg) e A : ax + by + c = 0 <=> axg + byg + c = 0

Chii y: ]) Cho duong thing A : ax + by + c = 0

• Ne'u Aj//A thi Aj : ax + by + Cj = 0, c, 5* c

Phuong phap giai

* Ne'u duong thing di qua hai diem A, B thi AB la VTCP

Hai duong thing song song thi chung co cung VTCP va ciing VTPT

Cam nang luy?n thl Hinh HQC - Nguyen lat iRu

V i dy 1.1.1 Trong mat p h i n g Oxy, choba diem A ( 2 ; 2 ) , B ( 1 ; - 5 ) , C ( 8 ; - 4 )

1) C h u n g m i n h rang A , B , C la ba d i n h cua mpt tam giac

2) Viet p h u a n g trinh d u a n g t r u n g tuyen A M cua tam giac ABC

3) Viet p h u o n g trinh d u o n g t r u n g tr^c canh AB

4) Viet p h u a n g trinh d u o n g phan giac trong va phan giac ngoai goc A

Ta CO A B = ( - l ; - 7 ) la VTPT cua d u o n g t r u n g tr^c c?inh A B nen p h u o n g

trinh d u o n g t r u n g true canh A B la: x + 7y +16 = 0

4) Goi A D la d u o n g phan giac trong goc A De lap p h u o n g trinh A D ta c6 cac

Cty TNHHMIV DWH Khang Vi$t

K h i d o ej +62 = 12 la VTCP cua A D , nen n = ( 3 ; l ) la VTPT cua A D

Dang toan 2: Viet phuang trinh ducmg th4ng di qua m6t diem

va lien quan den khoang each va goc

Phuong phap giai

D u o n g thang d i qua diem A(xQ;yQ) nen p h u o n g trinh c6 dang

a ( x - X o ) + b ( y - y Q ) = 0 v d i a^ + b ^ > 0

V i dv 1.1.2 Trong mat phang Oxy cho hai d i e m A ( l ; 2 ) , B ( 3 ; - l ) Viet

p h u o n g trinh d u o n g thJing A biet:

2) D u o n g thang A d i qua A va each B mpt khoang bang

V2

3) D u o n g t h i n g A d i qua C(1;0) va khoang each iu A den A gap 2 Ian

khoang each t u B den A

Lot gidi

1) Ta CO AB = ( 2 ; - 3 ) la VTPT cua d u o n g th^ng A

Suy ra p h u o n g t r i n h d u o n g thSng A la:

2 ( x - l ) - 3 ( y - 2 ) = 0 » 2 x - 3 y + 4 = 0 2) V i A d i qua A nen p h u o n g trinh A ed dang

Za^ - 24ab + IZb^ = 0 <=> (a - b)(7a - 17b) = 0 »

V i dy 1.1.3 Trong mat phang Oxy cho diem E(2;2) Viet p h u o n g trinh

d u o n g thSng d d i qua E va cat hai tia Ox, Oy tai A , B sao cho:

1) 0 A = 2 0 B

2) Tarn giac A O B c6 di^n tich bang 8

3) Tam giac A O B c6 chu v i nho nhat 7

4) Khoang each t u O den d Ion nhat

b b Phuong trinh d u o n g t h i n g d: - + — = l<=>x + 2 y - 6 = 0

6 3

2) Ta CO dien tich tam giac OAB: S = - O A O B = - a b - 8 ab = 16 hay a = —

^ 2 2 ^ b Thay vao (1) ta dugc: - + - = ! « b ^ - 8 b +16 = 0 < ^ b = 4 = > a = 4

Dang thuc xay ra k h i a = b = 4

1 Vie't p h u o n g trinh d u o n g t h i n g A d i qua P va cat d ^ d j tai hai diem A,B

sao cho tam giac A I B can tai I v o i I = d j n d 2

2 Viet p h u o n g trinh d u o n g t h i n g d d i qua M ( - 3 ; 4 ) cat d p d j Ian l u o t tai E,F sao cho M F = 2 M E

3 Viet p h u o n g trinh d u o n g t h i n g Aj d i qua* P va tao v o i d j m o t goc a thoa cos a =

C a w ttang luyftt thi Oil Uhih hm \'i;iiiu-n \U Jill:

Vi AAIB can tai I nen A ± d ' Do do, p h u o n g trinh A la:

• Vol a - — b , suy ra u =

4 ^ - l b ; b 4 = - f ( 3 ; - 4 )

Phuong trinh A^: 3x - 4y + 25 = 0

V i du 1.1.5 Trong mat phang voi he toa do Oxy, cho diefn P ( - 7 ; 8 ) va hai

d u o n g thang d , : 2x + 5y + 3 = 0, d2 : 5x - 2y - 7 = 0 cat nhau tai A Viet

p h u o n g trinh d u o n g thang d3 d i qua P tao v o i d p d j thanh tam giac can

tai A va c6 dien tich bang 14,5 " "

Toa do diem A la nghiem cua he

29 yjss

M a t k h a c S = — nen d ( A , d 3 ) = - ^ Tir do, ta t i m duoc d3 : 7x + 3y + 25 = 0

V i dy 1.1.6 Tam giac can ABC c6 day BC nam tren duong thang: 2x - 5y + 1 = 0,

canh ben A B nam tren d u o n g thang : 12x - y - 23 = 0 Viet p h u o n g trinh

d u o n g thang A C bie't rang no d i qua diem (3;1)

Vay con l a i : 9a = 8b hay a = 8 va b = 9

^ ^ h u w i g j r i r ^ ^ t i m la: 8x + 9y - 33 = 0 '

du 1.1.7 Viet p h u o n g trinh tie'p tuyen chung ciia hai d u o n g tron:

- — _ J C i ) 2 ( x - 5)2 + (y + 12)2 = 2 2 5 va ( Q ) (x - 1)^ + ( y - 2)^ = 25

Lai gidi

D u o n g tron (Ci) co tam Ii(5 ; -12) ban kinh Ri = 15, D u o n g tron (C2) c6 tam

H^; 2) ban kinh Ri = 5 Neu d u o n g th^ng Ax + By + C = 0 (A^ + ;^ 0) la tie'p

Cam nang luyfn Iht DH Hinh ho: \xuifen Tat Thu

tuyen chung ciia (Ci) va (C2) thi khoang each t u I i va h den d u o n g t h i n g do

Ian lugt bang Ri va R2, tuc la:

Vay CO hai tiep tuyen: (-14 lloV? )x + 21y -203 ± 10%/? = 0

TH2 : 5A - 12B + C = -3(A + 2B + C) C = thay vao (2) ta dugc:

96A? + 28AB + 51B^ ^ 0 Phuang trinh nay v6 nghigm

V i 1.1.8 Trong mat phang vol h ^ toa dp Oxy cho hinh chCr nhat ABCD

CO diem 1(6; 2) la giao diem ciia 2 duang cheo A C va BD D i e m M ( 1 ; 5)

thuoc d u o n g thang AB va trung diem E ciia canh CD thuQC d u a n g thang

d : x + y - 5 = 0 Viet phuong trinh duong thMng AB

Lai gidi

Vi E e d : ^ E ( a ; 5 - a ) = : > i E = ( a - 6 ; 3 - a )

Gpi N la trung diem cua A B , suy ra

I la trung diem ciia EN nen :

• Voi a = 6 => M N = (5;0), suy ra phuong trinh A B : y - 5 = 0

* V o i a = 7 => M N = (4; 1), suy ra phuang trinh A B : x - 4y +19 = 0

Cty TNHHMTVDWH Khang Vift

V i d v 1.1.9 Viet phuang trinh cgnh AB cua hinh chij nhat ABCD, biet AB,

BC, CD, D A Ian lugt d i qua cac diem M(-3;4), N(6;-9), P(8;2), Q(-2;-3)

• ^ = tachpn a = 11, b = - 2 Phuong trinh AB: l l x - 2 y + 41 = 0

V i 1.1.10 Trong mat phang Oxy cho hinh thoi ABCD ngoai tiep duang tron ( C ) : ( x - 1 ) ^ + ( y + l ) 2 = 2 0 Diem B nam tren d u o n g thang

d : 2 x - y - 5 = 0 va X B > 0 Viet phuang trinh canh AB cua h i n h thoi, biet

Cam nang luyftt thi DH Hinh hqc- Nguyen Tat Thu

Phuong trinh AB c6 dang: a(x - 4) + b(y - 3) = 0

3a + 4b Taco: d(I, AB) = IH = 2%/5 <=>

o l l a ^ _24ab + 4b^ = 0 o

a = 2b

11

= 2V5 » ( 3 a + 4b)2 = 20(3^ + b^)

• a = 2b ta CO phuong trinh AB: 2x + y -11 = 0

• a = —b ta CO phuong trinh AB: 2x + l l y - 41 = 0

Vi du 1.1.11 Trong mat phSng tpa do Oxy cho duong thing (d) c6 phuong

trinh : x - y = 0 va diem M(2;l) Tim phuong trinh duong thang A cat

true hoanh tai A cat duong thing (d) tai B sao cho tarn giac AMB vuong

Cty TNHH MTV DWH Khang Vi(t

Vi dv 1.1.12 Trong mat phing tpa dp Oxy cho duong tron (C) c6 phuang trinh: (x-4)^+y^=25 va diem M(l;-1) Tim phuong trinh duong thing A di qua diem M va cat duong tron (C) tai 2 diem A, B sao cho

MA = 3MB •

Loi gidi

Duong tron (C) c6 tarn 1(4; 0) va c6 ban kinh R = 5; M(l;-1)

MI = VlO < 5 = R nen M nam ben trong duong tron (C)

toan: Aj : 2x - y + 3 = 0 va A2 : x + 2y +1 = 0

Vi dvi 1.1.13 Trong mat phing voi h# tpa dp Oxy, cho duong tron (C):

(x -1)2 + (y + if = 16 tam I va diem A(l + Chung minh rang mpi

duong thing di qua A deu cat duong tron (C) tai hai diem phan bi?t Viet phuong trinh duong thing d di qua A va cat (C) tai hai 3x + y - 4V1O -1 = 0

diem B, C sao cho tam giac IBC nhpn va c6 di^n tich bJng A\l3

Lai gidi

Ta c6: Duong tron (C) tam 1(1; -1), ban kinh R = 2

Ta c6: IA = V3 + 9 = 2V3 < 4 , suy ra diem A nam trong (C) ,

^lAB = |lA.IB.sin BIC = 4N/3 o i.4.4.sin BIC = 4^3 sin BIG =

Duong thing d di qua A, nhan n(a;b) (a^+b2 ^^0) c6 phuong trinh

a ( x - l - V 3 ) + b(y-2) = 0 =>d(I;BC) = 273c:>(V3a-b)2 = 0 ^ V 3 a - b = 0

Chpn a = 1, b = V s

Tu do phuong trinh duong thing d: >/3x + 3y>/39 = 0

-Camnang

V i d y 1.1.14 Trong mat p h i n g v a i tQa d p O x y , cho d u a n g tr6n

( C ) : ( x - 1 ) ^ + ( y- l ) ^ = 1 0 D u o n g tron ( C ) tam r ( - 2 ; - 5 ) c i t (C) tai hai

diem A, B sao cho AB = 2Vs Viet p h u o n g trinh d u a n g t h i n g AB

nen p h u o n g trinh AB la: x + 2y + 2 = 0

T H 2: H khong nam trong doan I F ,

V i dM 1.1.15 T r o n g mat phSng O x y cho d u o n g tron ( C ) : (x - 1 ) ^ + (y +1)^ = 9

CO tam I Viet p h u o n g trinh d u o n g t h i n g A d i qua M ( - 6 ; 3 ) va cSt d u a n g

tron (C) tai hai diem phan bi?t A, B sao cho t a m giac l A B c6 di?n rich

bang 2N/2 va A B > 2

Lot gidi

D u o n g tron (C) c6 tam 1(1;-1), ban kinh R = 3

Gpi H la t r u n g diem cua A B

V i d\f 1.1.16 Trong m a t p h i n g v o i h ^ tpa dp O x y cho hai d u o n g tron

( C i ) : x^ + y2 = 13 va ( C j ) : (x - 6)^ + y^ = 25 G p i A la giao d i e m ciia (Ci)

va (C2) v a i y ^ < 0 • Viet p h u o n g trinh d u o n g t h i n g d i qua A va cat (C^),

(C ) theo 2 day cung c6 d p dai bang nhau

Xet h f : x 2 + y 2 = 1 3

( x - 6 ) ^ + y 2 = 2 5

<=>

Gpi A la d u a n g t h i n g can lap

A = A B thoa yeu cau bai toan

A * A B gia s u A cat hai d u o n g tron (Ci), (C2) Ian lupt tai M , N

Phep d o i x u n g t a m A bieh M

t h a n h N va ( C j ) thanh ( C 3 )

V i M 6 ( C i ) = ^ N 6 ( C 3 ) = > N € ( C 2 ) n ( C 3 )

Phuong trinh (C3): (x - 4)^ + (y + 6f = 13

m nang luy$n thi BH Hinh hqc - Nguyen Tat Thu

3 Bai tap van d\ing

Bai 1.1.1 Trong mat phMng Oxy, cho tam giac ABC c6 A ( l ; l ) , B(0;-6), C(7;-5)

1 Viet phuong trinh duong thang AB

2 Viet phuong trinh duong trung true canh BC va AB Tir do, hay tim tam

duong tron ngoai tiep tam giac ABC

3 Viet phuong trinh duong phan giac trong goc A ciia tam giac ABC

Huong dan giai

1 Ta CO AB = ( - l ; - 7 ) la VTCP ciia duong thSng AB nen n = ( 7 ; - l ) la VTPT

cua duong thang AB Vay phuong trinh duong thang AB la:

7 ( x - l ) - l ( y - l ) = 0 c ^ 7 x - y - 6 = 0

2 GQ'I M la trung diem canh BC, suy ra M 7 11

Ta c6: B C = (7;l) va B C vuong goc voi duong trung tryc cua canh B C nen

phuong trinh trung true canh B C la:

Cach 1: Ta c6 phuong trinh canh AC: x + y - 2 = 0

Gpi N ( x ; y ) e d , k h i do d(N,AB) = d ( N , A C ) hay la

Cty TNHH MTV DWH Khang Vi$t

7 x - y - 6 [ |x + y - 2

' 7 x - y - 6 = 5(x + y - 2 ) 2 x - 7 y + 4 = 0

7 x - y - 6 = -5(x + y - 2 ) L3x + y - 4 = 0 , Voi diem N (x; y ) , ta dat f(N) = 2x - 7y + 4

Ta CO- f(B) = 46 > 0, f(C) = 53 > 0 nen phuong trinh ciia d la: 3x + y - 4 = 0

each 2: Dat e i = ^ A B = Khi do: u = ej + ej =

Phuong trinh ciia d la: 3x + y - 4 = 0

Cach 3 Gpi N ( x ; y ) thuoc d, khi do:

Bai 1.1.2 Trong mat phang voi hf tpa dp Oxy cho duong tron

( C ) : ( x - l ) 2 + ( y - 2 ) 2 = 2 5

1) Viet phuong trinh tiep tuyen ciia (C) tai diem M(4;6) 2) Viet phuong trinh tiep tuyen ciia (C) xuat phat tu diem N(-6;l) 3) Tu E(-6;3) ve hai Hep hiyeh EA,EB (A, B la Hep diem) den (C) Viet phuong trinh duong thSng AB

Hudng dan giai

Duong tron (C) c6 tam 1(1; 2), ban kinh R = 5

1) Tiep tuyen di qua M va vuong goc voi I M nen nhan I M = (3;4) lam VTPT nen phuong trinh Hep tuyen la: 3(x - 4) + 4(y - 6) = 0 <=> 3x + 4y - 36 = 0 2) Gpi A la tiep tuyen can rim

Do A di qua N nen phuong trinh c6 dang A:a(x + 6) + b ( y - l ) = 0<=>ax + by + 6 a - b = 0, a^+b^^O (*)

Ta c6:

d(I,A) = R <=> • 7a+ b

= 5 «|7a + b| = 5Va2+b2 ^ (7a + b)^ = 15{a^ + b^)

, tit i^i> :., - / "

"Cam nang luycu Thi DTT Uinh hoc - Nguyen I at iTuT

Tuong tv ta cung c6 dupe B € A = > A B = A = > A B : 7 x - y + 20 = 0

Bai 1.1.3 Trong mat phang Oxy cho diem M ( 2 ; l ) Lap phuong trinh duong

thSng A di qua M va cki Ox, Oy tai hai diem AB sao cho:

1 Tarn giac OAB can

2 Tam giac OAB c6 dien tich bang 4

3 Khoang each tu O den A Ion nhat

Huong dan giai

Gpi A(a;0), B(0;b) Phuong trinh duong thing A: - + ^ = 1

a b

2 1

Do A di quaMnen: —+ —= 1 (1)

a b

1 Tam giac OAB can khi va chi khi OA = OB <=> b <::>a = +b

• Voi a = b , thay vao (1) ta c6: - = 1 a = 3 Phuang trinh A : x + y - 3 = 0

Huong dan giai

Gpi G la trpng tam cua tam giac, suy ra tpa dp ciia G la nghi^m cua hf J3x + 4 y - 3 = 0

3 x - 1 0 y - 1 7 = 0'

7

^^ = 3 = G

y = - l Gpi E la trung diem ciia BC, suy ra EA = - G A E ( 2 ; - - ) Gia sir B(a; b), suy ra C(4 - a; -5 - b) Tu do ta c6 h?:

3a+Ab-3 = 0 3a + 4 b - 3 = 0 Ja = 5

3 ( 4 - a ) - 1 0 ( - 5 - b ) - 1 7 = 0 ^ [-3a + 10b + 45 = 0 ^ | b = - 3 ' Vay B(5;-3),C(-l;-2)

Bai 1.1.5. Trong mat phing tpa dp Oxy cho tam giac ABC c6 A(-3;0) va phuang trinh hai duong phan giac trong B D : x - y - l = 0, CE:x + 2y + 17 = 0 Tinh tpa dp cac diem B, C ' "

Huong dan giai

Gpi A j doi xung voi A qua BD, suy ra A j e BC va A j ( l ; - 4 ) A2 doi xung voi A qua CE, suy ra A2 e BC va A j 43 56

L « m nang luy?n thi tlH HinH Hoc - Nguyen Tat Thu

Suy ra phuong trinh BC: 3x - 4y -19 = 0

' x - y - l = 0 Toa do B la nghi^m ciia h^:

Tpa dp C la nghi^m ciia h$:

Bai 1.1.6. Trong mat phang tpa dp Oxy cho tam giac ABC c6 C(5;-3) va phuong

trinh duong cao A A ' : x - y + 2 = 0, duong trung tuyen BM: 2x + 5y -13 = 0

Tinh tpa dp cac diem A, B

Huong dan giii

Ta CO phuong trinh BC: x + y - 2 = 0

Suy ra tpa dp ciia B la nghi^m ciia he: x + y - 2 = 0 x = - l

2x + 5y-13 = 0 [y = 3 Gpi A(a;a + 2), suy ra tpa dp ciia trung diem AC la M

B(-l;3)

^a + 5 a - 1 ^

Ma M e B M nen 2 ^ y ^ + 5 ^ - 1 3 = 0 o a = 3:^ A(3;5)

Vay A(3;5),B(-1;3)

Bai 1.1.7. Trong mat phing tpa dp Oxy cho tam giac ABC c6 B(l;-3) va phuong

trinh duong cao AD : 2 x - y +1 = 0, duong phan giac CE:x + y - 2 = 0 Tinh

tpa dp cac diem A, C ,

Huong dan giai

Ta CO phuong trinh BC: x + 2y + 5 = 0

Tpa dp diem C la nghiem ciia he x + y - 2 = 0 fx = 9 <=><

x + 2y + 5 = 0 [y = -7

Gpi B' la diem doi xiing voi B qua CE, suy ra B'(5;l) va B' € AC

Do do, ta CO phuong trinh AC :2x + y - n = 0

^ • [ 6 x - y - 4 = 0 ^ '

Vi B doi xung voi A qua M nen suy ra B = (3; - 2)

Duong thang BC di qua B va vuong goc voi duong thang: 6x - y - 4 = 0 nen suy ra phuong trinh BC : x + 6y + 9 = 0

Toa do trung diem N ciia BC thoa man he: • ^ ^ => N 0; - —

Suy ra AC = 2.MN = ( - 4 ; - 3 ) Phuong trinh duong thang AC : 3x - 4y + 5 = 0 •

Bai 1.1.9. Trong mat phang Oxy cho hai duong thang

d i : 7 x + y + l = 0, d 2 : x - y + 7 = 0

1 Viet phuong trinh duong thMng dj do! xung voi dj qua dj

2 Viet phuong trinh duong phan giac ciia goc tao boi hai duong thSng d ^ d j •

3 Viet phuong trinh duong thSng di qua M(l;2) va dt d ^ d j tai A, B sao cho

M la trung diem AB

Huong dan giai

Xet he phuong trinh 7x + y + l==0 <=> x = - l

y = 6

x - y + 7 = 0 suy ra d•^,d2 catnhau tai I ( - l ; 6 ) 1- Lay N(0;-1) e dj va N ' la diem doi xiing voi N qua d j Phuong trinh N N ' : x + y +1 = 0

Cam nang luyen thi DIl llinh hoc - Nguyen Tat Thu

Phuong trinh can lap la: 13x - y -11 = 0

Bai 1.1.10 Viei phuong trinh canh AB ( A B c6 h^ so goc duong), A D a i a hinh

vuong A B C D biet A (2; - 1 ) va duong cheo B D : x + 2y - 5 = 0

Huong dan giai

Vi A B CO h^ so goc duong nen B(5;0), D(l;2)

=^ A B : x - 3 y - 5 = 0, A D : 3 x + y - 5 = 0

Bai 1.1.11 Trong mat phSng voi h^ toa dp Oxy cho hinh vuong A B C D bie't

M ( 2 ; 1 ) , N ( 4 ; - 2 ) ; P ( 2 ; 0 ) ; Q ( 1 ; 2 ) Ian lugt thupc canh A B , B C , C D ,

AD-Hay lap phuong trinh cac canh ciia hinh vuong

H u o n g dan giai Gia s u duong thang A B qua M va c6 vec to phap tuyen la n(a;b)

{a^ + ^0) suy ra vec to phap tuyen ciia B C la: i i i ( - b ; a )

cry i N H H m i v uvvH Knang

Phuong trinh A B c6 dang: ax + by - 2a - b = 0

A B : x - 2 y = 0 ; C D : x - 2 y - 2 = 0

B C : 2 x + y - 6 = 0 ; A D : 2 x + y - 4 = 0 , b = - a K h i do ,

A B : - x + y + l = 0 ; B C : - x - y + 2 = 0

A D : - X - y + 3 = 0; C D : - x + y + 2 = 0 Bai 1.1.12 Trong mat phang Oxy cho bon diem M(4;5), N(6;5), P(5;2), Q ( 2 ; l ) Viet phuong trinh canh A B ciia hinh c h u nhat A B C D biet cac duong thang A B , B C , C D , D A Ian lupt di qua M , N , P, Q v a di?n tich hinh chii nhat bang 16

Huong dan giai Phuong trinh A B c6 dang: a(x - 4) + b(y - 5) = 0 voi a^ + b^ > 0 Phuong trinh B C : b(x - 6) - a(y - 5) = 0

4 ( a - 3 b ) ( a - b ) Di?n tich cua hinh chir nhat: S - d(P, AB).d(Q, BC) = •

Bai 1.1.13 Trong mat phSng voi h§ toa dp Oxy cho hinh vuong A B C D biet

M ( 2 ; l ) , N ( 4 ; - 2 ) ; P(2;0); Q ( 1 ; 2 ) Ian lupt thupc canh A B , B C , C D , A D Hay lap phuong trinh cac canh ciia hinh vuong

Huong dan giai Truoc het ta chung minh tinh chat sau day:

"Cho hinh vuong A B C D , cac diem M, N, P, Q Ian lupt nkm tren cac duong

thang AB, B C , C D , D A Khi do MP = N Q o MP 1 N Q " •' Chiing minh:

Ve M E l C D , E e C D ; N F l A D , F € A D

Cum tiaii}; lin/etttnt tJtl HinH Hoc - Nguyen lat lHu

Hai tarn giac vuong M E P va N F Q c6 NF = M E

Do do MP = N Q <=> AMEP = ANFQ

« EPM = FQN <=> QIM = 90" « MP 1 NQ

Tro lai bai toan:

Ta c6: MP = (0;-l) => MP = 1

Goi d la duong thang di qua N va

vuong goc vol MP

Suy ra phuang trinh d: x - 4 = 0

GQI E la giao diem cua d voi duong thang

AD, ap dung tinh chat tren ta suy ra NE = MP

Phuong trinh AB: 3x - y - 5 = 0, BC: x + 3y + 2 = 0, CD: 3x - y - 6 == 0

Bai 1.1,14 Trong mat phang toa do Oxy, cho duong tron (C): (x -4) + y = 25

va M ( ] ; - l ) Viet phuong trinh duong thang d di qua M cat (C) tai hai diem

A, B sao cho MA = 3MB

Huang dan giai

Duong tron (C ) c6 tarn 1(4; 0), ban kinh R = 5

Do I M < 5 nen M nam trong duong tron (C)

Goi H la hinh chieu cua I tren AB, H la trung diem cua AB

Do MA= 3MB nen M la trung diem cua HB

Xet hai tarn giac vuong I H M va IHB ta c6

IH^ + HM^ = IM^ I IH^ + HM^ -10

IH^ + HB^ = IB^ IH^ + 4HM^ = 25

„ Voi b = 2a Phuang trinh d: x + 2y +1 = 0

, Voi a = -2b Phuang trinh d: 2x - y - 3 = 0 y , Vay phuang trinh duong thJing (d) la x + 2y +1 =0 hoac 2x - y -3 = 0

Bai 1.1.15 Trong mat phang tpa do Oxy, cho duong tron (C):x^+y2-6x-2y+l=0

Viet phuong trinh duong thing (d) di qua M(0;2) va cat (C) theo day cung

Huong dan giai i> :o ; <

Duong tron (C) c6 tam 1(3; 1), ban kinh R = 3

< » 2 a 2 - 3 a b - 2 b ^ =0<^a = 2b,a = - i b

2

Tudo, ta CO phuang trinh duong thang dla: x - 2 y + 4 = 0; 2x + y - 2 = 0

Bai 1.1.16 Trong mat phSng voi h$ tpa dp Oxy, cho duong tron (C):

( x - 1 ) ^ + ( y + 1)^ =16 tam I va diem A(1 + N/3;2) Chung minh rang mpi

duong thang di qua A deu cSt duong tron (C) tai hai diem phan biet Viet phuong trinh duong thing d di qua A va cat (C) tai hai diem B, C sao cho

tam giac IBC nhpn va c6 di^n tich bang 4\f3

Huong dan giai

Ta c6: Duong tron (C) tam 1(1; -1), ban kinh R = 2

• lA = V3 + 9 = 2^3 < 4 , suy ra diem A nam trong (C) => dpcm

• Sj^g =iiAJB.sinBIC = 4V3c>i.4.4.sinBIC = 4V3r^sinBIC = — '

2 2 BIC = 60° r r>d(I;BC) = 2>/3

•

BIC = 120°(loai) " ' '

Duong thing d di qua A, nhan n{a;h) (a^ + b^ ^ 0) c6 phuong trinh

a ( x - l - V 3 ) + b ( y - 2 ) = 0:^d(I;BC) = 2 V 3 « ( V 3 a - b ) 2 = 0 o V 3 a - b = 0 Chpn a = 1, b = Vs

Tir do phuang trinh duong thing d: yjsx + 3y-\/3-9==0

Bai 1.1.17 Trong mat phing vai h# tpa do Oxy cho duong tron:

(C): + y^ - 2x + 6y -15 = 0 Viet phuong trinh duong thing A vuong goc

voi duong thing: 4x - 3y + 2 = 0 va cSt duong tron (C) tai A, B sao cho

AB = 6

Huong dan giai

Cqi H la trung diem AB thi A H = 3 va I H 1 AB, suy ra I H = 4

Mat khac I H = d(I; A )

Vi A // d: 4x - 3y + 2 = 0 nen phuong trihh cua A c6 dang: 3x + 4y + c = 0

c - 9

= 4 » c = 29,c = - l l d(I,A) = 4 o

Vay CO 2 duong thang thoa man bai toan: 3x + 4y + 29 = 0 va 3x + 4y - 11 = 0

Bai 1.1.18 Trong mat phang voi he tpa dp Oxy, cho duong tron

(C): x^ + y^ = 1 Duong tron (C) tam I (2; 2) cat (C) tai hai diem A, B sao

cho AB = \/2 Viet phuong trinh duong thang AB

Huong dan giai

' Ta CO OA^ + OB^ = AB^ = 2 => AOAB vuong tai O Mat khac OI la duong

trung true cua doan thang AB nen A, B thupc cac true toa dp Vay:

• A(1;0);B(0;1), phuong trinh duong thang AB:x + y - l = 0

• A ( - l ; 0); B(0; -1), phuong trinh duong thing AB:x + y + l = 0

Bai 1.1.19 Trong mat phang voi h^ tpa dp Oxy, cho duong tron (C) c6 phuong

trinh: x ^ + y ^ - 2 x - 6 y + 6 = 0 va diem M ( - 3 ; l ) Goi Tj,T2 la cac tiep diem

cua cac tiep tuyen ke tu M den (C) Viet phuong trinh duong thing di qua

Huong dan giai

Duong tron (C) c6 tam 1(1; 3) va ban kinh R = 2

Do I M - lyfs > R nen diem M 6 ngoai duong tron (C)

Gpi T(xQ;y(j) la tiep diem cua tiep tuyen ke tu M

Taco: T 6 ( C )

M T I I T

Te(C) MT.IT = 0

Cty T3VHH MTV DWH Khang Vift

Tpa dp cac tiep diem T^Tj thoa man ding thuc (1)

Vay phuong trinh duong thang di qua Tj,T2 la: 2x + y - 3 = 0

Bai 1.1.20 Trong mat phing Oxy cho hai duong tron

(Ci):(x + 5)2 +(y-7f =73, (C2):(x-1)2 + ( y - l ) 2 =13

Gpi A la giao diem cua (Cj),(C2) voi x^ < 0 Viet phuong trinh duong thing A di qua A, cat (Cj), (Cj) Ian lupt tai M , N (khac A) sao cho A la

trung diem doan M N

Huong dan giai

A = AB thoa yeu cau bai toan

A 5^ AB Khi do, phep doi xung tam A bien M thanh N va (Cj) thanh (C3)

Vi M e ( C i ) ^ N e ( C 3 ) ^ N e ( C 2 ) n ( C 3 ) Phuong trinh (C3): (x -1)^ + (y + 9)^ = 73

Suy ra N : < ( X - l ) 2 + ( y _ l ) 2 ^ 1 3

( x - l ) 2 +(y + 9)2 =73 N(4; -1) Phuong trinh A : y +1 = 0

Bai 1.1.21 Trong mat phing tpa dp Oxy, cho tam giac ABC vuong tai A , c6

dinh C(-4;l), phan giac trong goc A c6 phuong trinh x + y - 5 = 0 Viet phuong trinh duong thing BC, bie't di|n tich tam giac ABC bang 24 va

dinh A CO hoanh dp duong

Huong dan giai

Gpi D la diem doi xung voi C qua duong thing d:x + y - 5 = 0, tatimdupc D(4;9)

Vi A thupc duong tron duong kinh CD nen A la giao diem cua duong thang d va duong tron duong kinh CD, suy ra tpa dp cua A la nghi^m aia h#:

Vi AB va A D ciing huang nen ta c6 B(4;7)

V|y phuong tri'nh BC: 3x - 4y +16 = 0 ^

LAm HdHg lU^^h Ihi UH Hmh hqc - Nguyen lat IfnT

Bai 1.1.22. Trong mat phiing voi h | toa dp Oxy, cho tarn giac ABC c6 goc

A = 120" , dinh A c6 tung dp duong va A thupc duong thang x + y - 1 = 0 ,

Biet cac canh AB, AC ciing tiep xiic voi duong tron X -s + ( y - i ) = 4

Hay lap phuong trinh cua cac canh AB, AC cua tam giac ABC

Huong dan giai

Gia sir duong tron c6 tam I 'tiep xiic voi AB, AC tai D, E Khi do do

A = 120° nen de dang suy ra : lAE = 60°

Xet AIAE vuong, c6 lAE = 60° nen tinh dupe: l A = R

sin 60° S'

Gia su: A ( a , l - a ) , a < l ta c6 : lA = a

-Vay taco: A(0,1)

Phuong trinh duong thang A di qua A c6 dang:

Suy ra phuong trinh AB, AC la: ±42>\ y - 1 = 0

Bai 1.1.23 Trong mat phang toa dp Oxy cho diem A(3; 1) Lap phuong trinh

duong thang d qua A va di chieu duong cac true tpa dp Ox, Oy thu tu tai

P, Q sao cho dien tich tam giac OPQ nho nhat

Huong dan giai

Tu gia thiet ta c6 P(a;0);Q(0;b),a > 0,b > 0

Cty TNHH MTV DWH Khang Vift

§ 2 PHaONG TRINH Ol/dNG T R 6 N

1 Tom tat ly thuyet

De lap phuong trinh duong tron (C) ta thuang su dung cac each sau

Cach 1: Tim tam I(a;b) va ban kfnh cua duong tron Khi do phuong trinh duong tron c6 dang: (x - a)^ + (y - b)^ = R^

Cach 2: Gia su phuang trinh duong tron c6 dang: x^ + y^ - 2ax - 2by + c = 0 Dua vao gia thiet cua bai toan ta tim dupe a,b,e Cach nay ta thuang ap dung khi yeu eau viet phuang trinh duong tron di qua ba diem

2 Cac vi du minh hpa

Vi du 1.2.1

Trong mat phcing Oxy cho tam giac ABC voi A(2;-2), B(-5;-l), C(l;5) 1) Viet phuang trinh duong tron (C) ngoai tiep tam giac ABC

2) Viet phuang trinh duong tron (C) npi tiep tam giac ABC

3) Viet phuong trinh duong tron (Cj) di qua A,B va eo tam nam tren duong th3ng d : 5x - 2y = 0

Vay phuong trinh (C): x^ + y^ + x - y -17 = 0

Suy ra a = ( l ; 1), b = ( l ; -3) Ian lupt la VTPT cua phan giac trong goc A va B

Do do, phan giac trong goc A va B Ian lupt c6 phuong trinh la

x + y = 0 va x 3 y + 2 = 0 1+ • '^P' I la tam duong tron npi tiep tam giac ABC, ta c6 , •.• (<•; / *

-I : x + y = 0

X - 3y + 2 = 0 <=>

^ 2 ' = 2 ' 2'2

Cam ttang luyfti thi DH Hinh hoc - Nguyen Tat Thu

Va phuong trinh B C : x - y + 4 = 0 nen d ( l , B C ) =

Vay phuong trinh ( C ) : ( 1 '

V i (Cj) d i qua A , B nen ta c6 E A = E B <=> E A ^ = E B ^

« ( 2 x - if + (5x + if = (2x + sf + (5x + if <^x = -l=> E ( - 2 ; - 5 )

Va R = E A = 5

2 2

Phuong trinh duong tron ( C j ) : (x + 2) + (y + 5) = 25

V i d ^ 1.2,2, Viet phuong trinh duong tron (C), biet:

1) ( C ) d i q u a A ( 1 ; - 1 ) , B(0;2) va tiep xiic voi duong thSng d:2x + y - 1 0 = 0

2) (C) d i qua C ( 0 ; - l ) va tiep xiic voi hai duong thJing Aj :x + 2 y - 1 4 = 0,

A 2 : 2 x + y - 9 = 0

3) ( C ) CO tam nam tren duong t h i n g d | : 2x + y + 2 = 0, d i qua D ( - 6 ; 1 ) va

tiep xiic voi duong thSng d j : 4x - y - 7 = 0

Thay (1) vao (2) tadupc

(3y - if + (y - 2)^ = sJlOy^ - lOy + s) = 49y^ - 168y +144

- , ,2 (x + 2 y - 1 4 ) ^

IC = d ( I , A i) « x 2 + ( y + l ) =A L '- n

p , X = y - 5 thay vao (*) ta c6

( 3 y - 1 9 f

<=> y^ + 74y - 231 = 0 <=> y = 3,y = - 7 7 ( y - 5 f ( y l f =

Phuang trinh (C) la: X +-370

V i dv 1.2.3 Tron mat phang Oxy cho hai diem M(0;-1), N ( 2 ; 5 ) va duong

t h i n g A : x + y - 3 = 0 Viet phuang trinh duang tron (C) di qua M , N va

cat A tai hai diem A, B sao cho AB = 6V2

Phuong trinh duong tron (C) la: (x + 2 ) ^ + (y - 3)^ ^ 2 0

V i d\ 1.2.4 Trong mat phSng voi he tpa dp Oxy cho hai diem A(3;0) va

B(0;4) Chung minh rang duong tron npi tiep tam giac OAB tiep xiic voi

duong tron di qua cac trung diem cac canh cua tam giac OAB

Lot giai

Ta c6: OA = 3 ; OB = 4 ; AB = 5

P la nua chu v i tam giac OAB P = 6

S la dien tich tam giac OAB : S = 6

r la ban kinh duong tron noi tiep r = — = 1

P

L r i / i s\n.n wnry rang vter

Tam giac OAB nam a goc phan t u thu nhat c6 2 canh la 2 true toa do nen

tam I cua duong tron noi tiep (C) c6 toa do duong each 2 true toa do 1 khoang b l n g 1 => I ( l ; l ) -

G<?i M , N , P Ian lugt la trung diem cac canh OA; OB; AB

Gia su duong tron ( C ) qua M , N , P c6 phuang trinh:

( C ) : x^ + y^ + 2ax + 2by + c = 0 Ta c6 he phuang trinh

3a + c + - = 0

4

4 b + c + 4 = 0

25 3a + 4b + c + — = 0

Duong tron (C) c6 tam r(3/4; 1), ban kinh !"' = - •

Ta CO U' = — = r ' - r Vay hai duong tron (C) va (C) tiep xuc trong

4

V i dv 1.2.5 Trong mat phSng Oxy cho duong tron ( C i ) : x ^ + / - 2 x - 2 y - 1 8 = 0

va duong tron ( C 2 ) : ( x + 1)^ + ( y - 2 ) ^ = 8 Chung minh rang hai duong

tron (Cj) va ( C j ) cat nhau tai hai diem phan biet A, B Viet phuong trinh duong tron (C) di qua ba diem A, B, M(0; 6)

Lot giai

Duong tron (C,) c6 tam Ij (1; 1), ban kinh Rj = i S •

Duong tron ( C 2 ) c6 tam l 2 ( - l ; 2 ) , ban kinh R 2 = 2^2

Do 2N/5-2V2 = R I - R J < I i l 2 = 7 5 < R I + R 2 = 2 ^ 5 + 2 7 2 nen (Cj) cat nhau tai hai diem phan biet A, B

Tga do giao diem cua (Cj) va ( C 2 ) la nghiem cua he:

, x2 + y2 _ 2 x - 2 y - 1 8 = 0 j x ^ + y^ - 2 x - 2 y - 1 8 = 0 l(x + l ) 2 + ( y _ 2 ) 2 =8 x ^ + y ^ + 2 x - 4 y - 3 - 0

T T i D i uniif; linffii JTil r>TT Tliiih hoc - Nguyen lat Ihu

Goi X j , X 2 la hai nghi^m ciia (*), suy ra A X j ; 2 x j + 15 X 2 ; 2 X 2 + — 15 PBPTH

Cty TNHH MTV DWH Khang Vift

Phuong trinh duong th3ng AB: 4x - 2y +15 = 0 nen

Phuong trinh duang trung tryc A cua doan AB: x + 2y - 3 = 0

Goi I la tam cua duang tron (C), suy ra I e A => I(2a + 3; -a)

Mat khac:

^2n AB^ i*.r2 (10a+ 27)2 - ^ j ^

d''(I,AB) + - ^ = I M ^ 20 +-^ = i^^ + ^) +(a + 6r <=>a = l

Suy ra 1(5;-1), ban kinh R = I M = 5>/2

Vay phuang trinh ciia (C): (x - 5)^ + (y +1)^ = 74

Chii y: Ngoai each giai tren, ta c6 the sir dyng chiim duang tron de giai Cu

the:

Vi (C) di qua cac giao diem cua (Cj) va (Cj) nen phuong trinh ciia (C) c6

dang: m{\^ + y^ - 2x - 2y -18) + n{x^ + y^ + 2x - 4y - 3) = 0

Do (C) d i qua M(0;6) nen ta c6: 2m + 3n = 0, ta chpn m = 3,n = -2

Khi do phuang trinh (C): x^ + y^ - lOx + 2y - 48 = 0

V i 1.2.6. Trong h? tpa dp Oxy, cho duong tron [C):(x-6f +{y-2f=4

Viet phuang trinh duong tron ( C ) tiep xiic voi hai tryc tpa dp Ox,Oy

dong thai tie'p xiic ngoai voi (C)

Lot giai

Duong tron (C) c6 tam I (6; 2), ban kinh R = 2

Goi ( C ' ) : ( x - a ) ^ + ( y - b f - R ' ^ thi ( C ) c6 tam I ' ( a ; b ) , ban kinh R'

Vi ( C ) tie'p xiic voi Ox,Oy nen suy ra

d ( r , O x ) = d ( r , O y ) < » a = b = R ' «

la = - b Hon nua ( C ) tiep xiic vai Ox,Oy va tiep xiic ngoai v o i (C) nen ( C ) nam

ben phai true O y , do do a > 0 •

V$y truong hpp nay c6 1 duang tron la ^Cj j : (x - 6 ) ' + (y - 6 ) ' = 36

Tom l a i , c6 3 duong tron thoa can t i m la : (x - 2f + (y - 2f = 4, (x - 1 8 ) ' + (y - 1 8 ) ' = 18^ va (x - 6 ) ' + (y - 6 ) ' = 36

a = 2

a = 18

V i 1.2.7 Trong mat phang Oxy, cho duong tron {C):{x-lf +{y-lf =25

va duong thSng d : 2 x - y - l = 0 Lap phuong trinh duang tron ( C ) c6 tam nam tren d va hoanh dp Ion hon 2, dong thai ( C ) cat (C) t ^ i hai diem

A , B sao cho day cung A B c6 dp dai bang 4V5 va tiep xiic v o i duang thSng A : 3 x - y + 15 = 0

Lai giai

Duong tron (C) c6 tam 1(1; 1 ) , ban kinh R = 5

Gpi r la tam ciia duong tron ( C ) , Ved nen suy ra r ( m ; 2 m - l ) , m > 2 va

R' la ban kinh

_ , m + 16 Taco: R' = d ( r , A ) =

^/lO Gpi H la giao diem ciia 11' va A B , suy ra H la trung diem ciia A B nen

Luwi HUHjj luym im un mm ni^ii - ivguyen lai rmr

c = 8 hay c = -8 Vaic = 8: I ( t ; - t + 8)

o 50 - 1072(m^ +32m + 56) + m^ + 32m + 56 = 50m^ - 100m + 50

o 49m^ - 132m - 56 + lo72(m2 + 32m + 56) = 0 (1)

Do m > 2 nen 49m^-132m-56 + 1 0 ^ m +32m + 56) > 32 nen (1) v6 nghiem

Vay phuong trinh (C'):(x-4)2 +{y-7f =40

d (I; A) = t - ( 8 - t ) | = N/2 = I H

I (5; 3)

Vi du 1.2.8 Trong mat ph3ng Oxy, cho d : 2 x - y + 3 = 0 Viet phuong trinh

duong tron tam thuoc d cat true Ox tai A,B, cat true Oy tai C, D sao cho

AB = CD = 2

Lai giai

Goi tam ciia duong tron la I ta c6: I (a; 2a + 3)

Goi M , N Ian lugt la trung diem ciia AB,CD ta c6:

lA = IC, M A = NC, IMA = INC nen AIM A = AINC IM = I N

Do do, d ( I , AB) = d ( I , CD) <=> d (I,Ox) = d(I,Oy) • 2a+ 3 a = -3

Cach 2: Gpi I la tam ciia duong tron can tim, H la trung diem ciia AB, M la giao

diem ciia hai tiep tuyen tai A, B

Tu gia thiet ta c6 M (O; a), a > 0

Trong tam giac vuong MAI: \

+ V6i a = -1 ^ l ( - l ; l ) , I M = 1 ^ R = lA = VlM^ + MA^ = 72

2 2 Phuong trinh duong tron c6 dang: ( x + l) + ( y - l ) = 2

+ Vai a = -3=>l(-3;-3),IM = 3=>R = \/lM^ +MA^ = VlO

2 2 Phuong trinh duong tron CO dang: ( x + 3) + ( y + 3) =10

1 AH^ A I ^ ^ A M ^ AM^ 8 10 40 • A M = 2 v r o

Vi dv 1.2.9 Trong mat phang voi hf tpa dp Oxy, cho duong thang

A :x - y = 0 Duong tron (C) c6 ban kinh R =\/l0 cat A tai hai diem A va

B sao cho AB = A-Jl Tiep tuyen ciia (C) tai A va B cat nhau t^ii mpt diem

thupc tia Oy Viet phuong trinh duong tron (C)

MI = N / M A ^ T L A ^ = 5V2 M H = MI - HI = 4V2 Hay d(M,A) = 4V2 c^-^ = 4V2=>a = 8(do a>0)

v 2

Do do, phuong trinh IH la: x + y - 8 = 0 Taco I e M H r ^ l ( b ; 8 - b )

= 5V2 o V2b^ = 5N/2 « b = ±5 + Voi b = 5,PTciia (C) la ( x - 5 ) ^ + ( y - 3 ) ^ =10 + V6i b = -5,PTcua(C)la (x + 5)^+(y-13)^ =10 ( ! ; ' ' i ' O l : »''->

Cant nang Ivy en thi VH Htnh HQC -isigayen rat inn

Vi dv 1.2.10 Viet phuang trinJi duong tron npi tiep tarn giac ABC v6i cac

Gia su tarn I ciia duong tron npi tiep c6 tung dp la b Khi do hoanh dp la

1 - b va ban kinh ciang bang b Vi khoang each tu I toi AC cung phai bang b

npi tiep A ABC la: ( 1 ' X 2 + y ~

I 2j v 2J 4

V i d\ 1.2.11 Trong mat phSng tpa dp Oxy, cho hai duong thing

dj : Vsx + y = 0 va d j : VSx - y = 0 Gpi (T) la duong tron tiep xuc voi dj

tai A , cat d j tai hai diem B va C sao cho tam giac ABC vuong tai B Viet

phuong trinh ciia (T), biet tam giac ABC c6 di^n tich bang ^ va diem A

CO hoanh dp duong

Cty TNHH M T V D W H Kkang Viet

Lai gidi

Cich 1: Vi AABC vuong tai B nen AC la duong kinh ciia (T)

Gpi ASB = (<Vd7) = t ta CO BAC = ASB = t (gocc6 canh tuong ung vuong goc)

Gia su ban kinh (T) la R ta c6:

l^^t khac cost = N/3.V3 + 1.(-1)

Suyra S ^ B C = l ^ ^ - y t u d 6 c 6 R = l

Do A G d p C e d 2 nen A(a;-aV3),c(c;cV3) Mat khac vector chi phuong

ciia dj la u^ihS) c6 phuong vuong goc voi AC nen:

AC.Uj = 0 o c - a - 3(c + a) = 0 <=> c = -2a

Mat khac AC = 2 R - 2 ^ ( c - a ) ^ +(V3(c + a))^ - 2 c:>2|a|V3 = 2 vi a>0

s

nen a = Tam duong tron la trung diem cua AC la :

Cach 2: Ta c6 dj tiep xiic voi (T) c6 duong kinh la AC nen AC 1 d j

Tu gia thiet ta c6 : AOx = 60^BOx -120° ^ AOB-eo"; ACB = 30°

S^3, = iAB.BC = AB^ AB^ = AB = 1

2 2 1

Vi A e d j A ( X ; - V 3 X ) , X >0;OA = - ^ A B = ^ ^^S''^^'

0 C = 2 0 A =

N/3 I S ;-2

Cam ttang luy$n thi DH Hinh hpc - Nguyen Tat Thu

Duong tron (T) duong kinh AC c6: I

^2 ,

Phuong trinh (T): X + 1 3^

= 1

V i du 1.2.12 Trong mat phang Oxy cho tam giac ABC c6 trong tam G(2;3)

Gpi H la tryc tam ciia tam giac ABC Bie't duong tron di qua ba trung diem

ciia ba doan thing HA, HB, HC c6 phuong trinh: (x-1)^ + ( y - l ) ^ =10

Viet phuong trinh duong tron ngoai tiep tam giac ABC

"Trong tam giac, 9 diem gom trung diem

cua ba canh, chan ba duong cao va ba

trung diem ciia cac doan noi true tam

voi dinh nam tren mpt duong tron c6

tam I, G, H thSng hang va I H = 3IG "

Goi E la tam duong tron ngoai tiep tam

giac ABC va M la trung diem BC Ta c6:

Phep vi t u V(c,_2) : I ^ E, M -> A va M e (C)

nentaco: E(4;7) va EA = 2IM = 2VlO

Vay phuong trinh duong tron ngoai tiep tam giac ABC la:

( x - l) 2 + ( y - 1 0 ) 2 =40

V i du 1.2.13

Cho duong t h c i n g A:x + y + 2 = 0va duong tron (C): x^ + y^ - 4x - 2y = 0

Go! I la tam va M thuQC duong thiing A Qua M ke tiep tuyen MA, MB

Tim M sao cho di^n tich Kr giac MAIB bang 10

Lai gidi

D u o n g tron (C) c6 tam 1(2; 1), ban k i n h R = N/S => A I = >/5

1 MatkhacS^^^_ 2 -

Suy ra IM^ = lA^ + AM^ = 25

la hai diem can tim

Vi dv 1.2.14 Trong mat ph^ng Oxy, cho duong tron (C): (x - 4)^ + y^ = 4 va

diem E(4;1) Tim toa dp diem M tren tryc tung sao cho tu M ke dupe hai

tiep tuyen MA, MB den duong tron (C) voi A,B la hai Hep diem sao cho duong thang AB di qua diem E

Ta c6: x^+y^ - 8 x + 12 = 0

x^ + y^ - 4x - my = 0

=:>4x-my-12 = 0 MT.IT = 0

Do do, phuong trinh duong thSng AB: 4x - my -12 = 0

N e n A B d i q u a E <=> 16 - m -12 = 0 o m = 4

Vay M(0; 4) la diem can tim

Vi dy 1.2.15 Trong mat ph^ng voi h? tpa dp Oxy, cho duong tron

( C ) : x^ + y^ - 2x + 4y = 0 va duong thSng d : x - y = 0 Tim tpa dp cac

diem M tren duong thing d, bie't tu M ke dupe hai tiep tuyen M A , M B

«leh ( C ) ( A , B la cac tiep diem) va khoang each tir diem N(1;-1) den A B

ing

Vi M e d = > M ( m ; m ) Gpi A (X ( , ; y „ ) Khi do, ta c6:

Loi giai

Cam nang luy^n thi BH Hinh HQC - Nguyen Tat Thu

IA.MA = 0 J'^o +yo -(m+l)xo - ( m - 2)yo - m = 0

A e ( C ) [ x 2 + y 2 - 2 x o + 4 y o = 0

Suy ra (m - 1)XQ + (m + 2)yo + m = 0

Do do, ta CO phuang trinh AB la: (m - l)x + (m + 2)y + m = 0

Mat khac: d(N, AB) = - 7 = nen ta c6 phuong trinh:

lf+{m + 2f

58 Giai phirong trinh nay ta tim dirge m = 0,m = - —

Ta loai m = 0, vi khi do M € (C)

Vay CO mpt diem M thoa yeu cau bai toan: M

58 58 13' 13 j

Vi du 1.2.16 Cho duong tron (C):(x-1)^+(y-])^ =13 va duong thSng

d:3x + 2y + 6 = 0 Gpi ( C ) la duong tron c6 ban kinh bang 2%/l3 tiep

xiic ngoai voi (C) tai A va tiep xiic voi d tai B Tinh doan AB

/ Lai giai

Duong tron (C) c6 tam 1(1; 1), ban kinh R Vl3

Gpi r(a;b), R' Ian lupt la tam va ban kinh ciia (C), suy ra R' = 2\/l3 va

ir = R + R' = 3Vl3

Ap dung dinh li c6 sin cho tam giac AI'B ta c6:

AB^ = r A^ + r B^ - 2.r A.rB.cos ATB = 104(1 - cos ATB)

Ma cos AI'B = cos|n^,rij n^.ri 3(a-l) + 2(b-l)

Mat khac: d(r,A) = 27T3:

= 64=i> AB = 8

= — => AB =

3 3

Cty TNHH MTV DWH Khang Viet

y{ dV 1-2.17 Trong mat phlng voi h? toa dp Oxy cho duong thang

^.x + y - 2 = 0 va duong tron (T): x^ + y^ - 2x + 2y - 7 = 0

Chung minh ring A cat (T) tai hai diem phan biet A, B va tim toa dp

diem C tren (T) sao cho tam giac ABC c6 dien tich bang (3 + V2 )^/7

Lbi giai

Du-ong tron (T) c6 tam 1(1;-1), ban kinh R = 3

Xa CO d(I ,A) = ^/2 < R = i > A va ( T ) cat nhau tai hai diem phan biet A , B

Va A B = 2 V R ^ - d 2 ( I , A ) = 2 N / 7 Giasu C(xo;yo)€(T)

=^(xo-i)'+(yo+i)'=9 (1) Di?n tich tam giac A B C :

SAABC = ^ d ( C , A B ) A B = V7.d(C, A) Dodo, S ^ABC =(3 +V2)V7c^d(C,A) = 3 + V2

Ma d (C,A) = -^|x(, +yo - 2| = V2 + 3 <=> [xg + yg - 2| = 2 + 3^2

O X Q +yo - 2 = +(2 + 3V2)

X(, + yg - 2 = 2 + 3V2 = > XQ = 4 + 3^2 - thay vao (1), ta c6 dupe:

(yo - 3V2 - 3)^ + (yg +1)^ = 9 v6 nghiem

XQ + yo - 2 = -2 - 3N/2 => x^ = -3^2 - y^ thay vao (1) ta c6

( y o+ 3 V 2 - l ) 2 + ( y o + l ) 2 = 9 « y o = - l - A ' 3 x „ = l - ^

Vay C

A 't>M

Vi du 1,2.18 Trong mat phang tpa dp Oxy, cho duong thang d : x + y = 0

Gpi (C) la duong tron tam I, (C) ck d tai A va B sao cho OA.OB = 6, dong

thoi tam giac AIB vuong tai I va c6 dien tich bang 2 Viet phuong trinh

- - j u a j C ) , biet O 6 ngoai (C)

Lai giai

Goi R la ban kinh duong tron (C) Vi tam giac lAB vuong va c6 dien tich t'ang 2 nen ta co:

Cam nang luy$n thi DH Hinit hoc X^m/cn TaiThu

ilA.IB = 2<=>R2=4=>R = 2=>AB = Is/l

2

Gia su OB > OA, do O € d nen O, A,B thSng hang va OB = OA + AB

Do do OA.OB = 6 o OA (OA + AB) = 6 o OA^ + 2%/20A - 6 = 0<=>OA = V2

Ma Aed=> A(a;-a)=>OA = (a;-a)=>OA^ = 2a^ =2=>a = ±l

• a - 1 => A ( l ; - l ) Do OB = 30A B(3;-3), trung diem ciia AB la M(2;-2)

Phuong trinh I M : x - y - 4 = 0=> l(m;m - 4) => AI = (m - l ; m - 3)

Ma lA = 2 (m -1)^ + (m - 3)^ = 4 o -4m + 3 = 0 m = l,m = 3

+) Voi m = 1 => l(l;-3), phuong trinh (C): (x -1)^ + (y + 3)^ = 4

+) Voi m = 3 ^ I(3;l), phuong trinh (C): (x - 3)^ + (y -1)^ = 4

• a = - l i ^ A(-l;l)=>B(-3;3)=>M(-2;2) la trung diem AB

Phuong trinh I M : x - y + 4 = 0=> l(m;m + 4) AI = (m +1;m + 3)

Ma IA = 2=>m^ +4m + 3 = 0 <=> m = - l , m =-3

+) Vdim =-!=>!(-1; 3), phuong trinh (C): (x +1)^ + (y - 3)^ = 4

+) Vol m - -3 l(-3;l), phuong trinh (C): (x + 3)^ + (y -1)^ = 4

3 Bai tap v|n dvng

Bai 1.2.1 Viet phuong trinh duong tron (C) di qua hai diem A(2;1),B(4;3) v a

CO tam thuQC duong thang A : x - y + 5 = 0

Huomg dan giii

a - b + 5 = 0 [c = 5 Vay phuong trinh (C): x^ + y^ - lOy + 5 = 0

Bai 1.2.2 Trong m|t phing Oxy, cho hai diem A(0;5),B(2;3) Viet phuong

trirth duong tron di qua hai diem A, B va c6 ban kinh R = -y/lO

Cty TNHH MTV DWH Khang Vift

Huong dan gi^i Gpi la tam ciia du6ng tron (C)

Ta CO phuong trinh (C): (x - a)^ + (y - b)^ = 10

(x + l f + ( y - 2 ) ' = 1 0 va ( x - 3 f + ( y - 6 f =10

Bai 1.2.3 Viet phuong trinh duong tron di qua hai diem A(1;0),B(2;0) vatiep xiic voi duong thang d : x - y = O

Huong dan giAi

Gia su duong tron (C) can tim c6 phuong trinh la:

1.2.4. Trong mat phSng voi tpa dp Oxy, cho duong tron

( C ) : x 2 + y 2 - 2 x - 2 y + l = 0 va duong thing d : x - y + 3 = 0 Viet phuong

Wnh duong tron (C) c6 tam M nSm tren d, ban kinh b ^ g 2 Ian ban kinh duong tron (C) va (C) tiep xuc ngoai voi duong tron (C)

Hu6ng dan giii

^"•ong tron (C) c6 tam 1(1; 1), ban kinh R = 1

Taco Med=>M(x;x + 3)

^» (C) va (C) tiep xiic ngoai nen ta c6 MI = 3R o (x -1)^ + (x + 2)^ = 9

Cam nang luyftt thi DH Hinh hgc - Nguyen Tat Thu

T u ( l ) v a (2) g i a i h # thu dupe a = 0 , b = l,c = 0 hoac a = l , b = 0,c = 0

Vay CO hai d u o n g tron thoa man la : x^ + y^ - 2y = 0 va x^ + y^ - 2x = 0

B a i 1.2.6 Trong mat phang Oxy cho diem M ( 2 ; 3 ) Viet p h u o n g trinh duong

tron (C) d i qua M va tiep xuc v o i hai tryc tpa dp

H u o n g d a n g i a i

Gpi I(a; b) la tarn cua d u o n g tron (C) V i (C) tiep xuc v o i hai tryc tpa dp va

di qua M(2;3) nen ta suy ra dupe I nam trong goc phan t u t h u nhat hay

Viet p h u o n g trinh d u o n g tron ( C ) d i qua cac giao diem cua ( C i ) , (C2) va

tam nam tren d u o n g thang A : x + 6 y - 6 = 0

Cty TNHHMTVUVVH Khang V«?r

Xet he p h u o n g trinh

H u o n g d a n g i a i

x^ + y ^ - 1 0 x = 0

x2 + y 2 + 4 x _ 2 y - 2 0 = 0 Giai he nay ta dupe hai cap nghiem (x;y) = ( l ; - 3 ) , ( 2 ; 4 )

S u y r a (C^) va ( C j ) c S t n h a u t a i A(1;-3),B(2;4)

Gpi I la tam ciia (C), ta c6 1 ( 6 - 6 m ; m )

Vi l A = IB nen ta c6: (5 - 6m)^ + (m + 3)^ = (4 - 6m)^ + (m - 4)^ <^ m = - 1

uy ra 1(12;-1), R = l A = sVs vay p h u o n g trinh (C): (x -12)^ + (y +1)^ = 125

Bai 1.2.8 Lap p h u o n g trinh d u o n g tron (C), biet

1 (C) d i qua M(5;5), N(4;6) va tiep xuc v o i d u o n g th^ng A : 3x + 4y +14 = 0

2 (C) d i qua P(2;l) va tiep xiic v o i hai true toa dp

25(2a^-18a + 4 l j = (7a + 1 8 f -702a + 701 = 0 c> a = l , a = 701

• Voi a = 1 =^ 1(1; 2), R = I M = 5 Phuong trinh ( C ) : (x -1)^ + (y - 2f = 25

• V a i a = 7 0 1 ^ l ( 7 0 1 ; 7 0 2 ) , R = 985

Phuong trinh ( C ) : (x - 7 0 l ) ^ + (y -702)^ - 985^ ^ Gpi I(a; b) la tam eiia d u o n g tron (C)

Do (C) tiep xuc v o i hai tryc toa dp nen a = b = R Phuong trinh ( C ) : ( x - a ) ^ + (y - b)^ = a^

(C) d i qua P(2; 1) nen: (2 - a)^ + ( l - b)^ = a ^ * ) '•' '

T u b ta C O a = ±b

Cam nang luyftt thi DH Hinh hpc - Nguyen Tat Thu

• a = b thay vao (*): (2 - a)^ + (l - a)^ = a^ -» a^ - 6a + 5 = 0 <=> a = l,a = 5

Phuongtrinh ( C ) : ( x - l f + ( y - l f =1 va {x-sf+{y-sf =25

• a = -b thay vao (*): (2 - a)^ + (l + a)^ = a^ <=> a^ - 2a + 5 = 0 phuong trinh

v6 nghi?m

Bai 1.2.9 L^p phuong trinh duong tron (C), biet:

1) (C) di qua A(3;4) va cac hinh chieu cua A len cac tryc tpa dp

2) (C) CO tarn nSm tren duong tron (Ci):(x-2)^ = - va tiep xuc voi hai

5 duong th5ng Aj : x - y = 0 va Aj : x - 7y = 0

2) Gpi I(a;b) la tarn cua duong tron (C), vi I € (Cj) nen: (a - 2)^ + b^ = - (1)

5

Do (C) tiep xuc voi hai duong thSng Aj, Aj nen d(I, Aj) = d(I,A2)

<=> a - b a-7b o b = 72 5>/2 -2a,a = 2b

• b =-2a thay vao (1) ta CO dupe:

(a - 2)^ + 4a^ = — o 5a^ - 4a + — = 0 phuong trinh nay v6 nghif m

• a = 2b thay vao (1) ta c6: (2b-2)^+b^ =l<::>b = | , a = |

5 5 5 Suy ra R = D(I,Ai) = -

Vay phuong trinh (C): f 8'

2 f

X — + y

I 5, y 5J _8_ 25

Cty TPmH MTV DWH Khang Vic I

Bai I'2-IO Trong mat phang Oxy cho hai duong tron:

(Ci):(x + l)2+(y + 3)2=20, (Cj):(x-8)^ + y2 = 50 Viet phuong trinh duong tron (C) di qua hai giao diem cua (Cj), (Cj) va

cit duong thiing A : x + 2y + 7 = 0 tai hai diem A, B sao cho AB = 6^/5

<=> x = l V x = 3

y = - 5 ' Dodo (Cj) va (Cj) cat nhau tai M(l;l), N(3;-5)

Gpi I(a;b) la tam duong tron (C), ta c6: IM = IN

o ( a - l ) ^ + ( b - l ) 2 =ia-3f +{h + 5f

o a - 3 b = 8 ^ a = 3b + 8 (1)

Taco: IM^ = ^ + (1^(1, A)

o ( a - l ) ^ ( b - l ) ^ = 4 5 i i l ^ (2) Thay (1) vao (2), ta dupe:

(3b + 7)2+(b-l)2 =45 + 5(b + 3)^ o b ^ + 2 b - 8 = 0 o b - - 4 , b = 2

• V6ib = 2=^a = 14=>(C):(x-14)2 +(y-2)2 =170

• V6ib = -4=>a = -4^(C):(x + 4)2 +(y + 4)2 =50

1.2.11 Trong mat phang voi h^ true tpa dp Oxy, cho tam giac ABC c6 A(0;2),B(-2;-2), C(4;-2) Gpi H la chan duong cao ke tu B; M,N Ian lupt 'a trung diem ciia AB, AC Viet phuong trinh duong tron di qua cac diem

H, M, N

( mil H / i F i x not' " ^^guyen lui iriu

Hu6ng dan giai

Taco M(-l;0),N(l;-2),AC = (4;-4).GQi H(x,y),tac6:

m i X c ^ [ 4 ( x + 2)-4(y + 2) = 0 ^ r x = l

H ( l ; l )

4x + 4(y-2) = 0 Gia su phuang trinh duong tron: x^ + y^ + ax + by + c = 0

Ba diem M,N,H thuQC duong tron nen ta c6 h$ phuong trinh :

Bai 1.2.12 Trong mat phang voi h$ tpa dp Oxy, cho cho hai diem A(2;0) va

B(6;4) Viet phuong trinh duong tron (C) tiep xiic voi tryc hoanh tai A va

khoang each tu tam cua (C) den diem B bMng 5

Huong dan giai

Goi I(a;b) va R Ian lugt la tam cua va ban kinh ciia (C)

Vi (C) tiep xiic voi Ox tai A nen a = 2 va R = b

Matkhac: IB = 5^4^ + ( b - 4 ^ - 5 ^ b = l,b = 7

2 2

+ Voi b = 1 thi phuong trinh duong tron (C): (x - 2) + (y -1) = 1 •

2 2

+ Voi b = 7 thi phuong trinh duong tron (C): (x - 2) + (y - 7) = 49

Bai 1.2.13 Trong mat phang Oxy cho diem M(6;6) va hai duong thSng

Aj : 4x - 3y - 24 = 0, Aj : 4x + 3y + 8 = 0 Viet phuong trinh duong tron (C)

di qua M va tiep xiic voi hai duong thang Aj, Aj

Huong dan giai

Gpi I(a;b) la tam va R la ban kinh cua duong tron (C)

Vi (C) tiep xiic voi hai duong thang A^ va Aj nen ta c6 d(I, Aj) = d(I, A2)

4a-3b-24 4a + 3b + 8 = R o 4a-3b-24 = 4a + 3b + 8

4a-3b-24 = - 4 a - 3 b - 8 ' b.-Ii 3

a = 2

2 2

• Voi a = 2, phuong trinh (C):(x-2) + ( y - b ) = (3b + 16r 25

CtyTNHHl hang Viet

Do Me(C) nen {6-af + 16 _(4a-8)^ 25 phuong trinh v6 nghiem

Bai 1.2.14 Trong mat phSng voi he toa do Oxy, cho duong tron (C): x^ + y^ - 2x - 2y +1 = 0 va duong thang d : x - y + 3 = 0 Viet phuong trinh duong tron (C) c6 tam M tren d, ban kinh bang 2 Ian ban kinh duong tron (C) va tiep xiic ngoai voi duong tron (C)

Huang dan giai

Duong tron (C) c6 tam 1(1; 1), ban kinh R = 1

Goi r la tam va R' la ban kinh ciia duong tron (C), ta c6 R' = 2R = 2va

red=>r(a;a + 3)

Vi (C) va (C) tiep xiic ngoai voi nhau nen 11' = R + R' = 3

» ( a - l ) 2 +(a + 2)^ = 9 < ^ a 2 + a - 2 = 0«a = l,a = -2

• V 6 i a =l=^r(l;4)^(C'):(x-l)2+(y-4)^=4

• V6ia = -2:^r(-2;l)=^(C'):(x + 2)^+(y-l)2 =4

Bai 1.2.15 Trong mat p h k g Oxy cho duong tron (C): (x - 4)^ + (y -1)^ = 10

1 Viet phuong trinh duong tron (Cj) di qua A(3;8), c6 tam nam tren duong thang 3 x - y - l l = 0 va tiep xiic ngoai voi (C)

2 Viet phuong trinh duong tron (Cj) di qua B(-4;7),C(4;11) va cit (C) tai

hai diem M, N sao cho MN = 2V5

Huang dan giai

Duong tron (C) c6 tam 1(4; 1), ban kinh R = VlO /

1 Goi Ijlatamduongtron (Cj)=>Ij(m;3m-ll), Ban kinh Rj = IjA = ^(m - 3)^ + (3m-19)^

Vi (C) va (C]) tiep xiic ngoai voi nhau nen ta c6:

III =R + Ri o 7(m -4)2 + (3m -12)^ = VlO + V (m -3)^ + (3m -19)^

Cam naiiv luyen thi DH Hinh hoc Nguyen lat fHiT

Vay phuang trinh (C^): (x - 6)' + (y - 7)' = 10

2 Gpi M la trung diem doan B C , suy ra M ( 0 ; 9 ) D O B C = (8; 4) rien phuong

trinh duong trung true doan B C : 2x + y - 9 = 0

G Q I I2 la tarn cua duong tron (Cj) va H la hinh chieu cua I len M N , ta c6

Bai 1.2.16. Trong mat phSng Oxy cho duong tron (C): (x - if + (y _ ^^2 ^ 2 5

1) Lap phuong trinh tiep tuyen cua (C), biet tiep tuyeh di qua A(3._6)

2) Tu diem D(-4;5)ve den (C) hai tiep tuyen D M , D N (M, N tiep diem)

Viet phuong trinh duong thang M N

Huong dan gi^

Duong tron ( C ) c6 tarn 1(2; 1), ban kinh R = 5 1) Gia su A : ax + by + c = 0 la tiep tuyen cua ( C )

Do B 6 A nen 3a - 6b + c = 0 => c = 6b - 3a

2a + b + c

Cty TNHHMTVTyWHKhanx Viet

A la tiep tuyen cua ( C ) nen d(I, A) = R <=>

7 ^ = 5<:> -a + 7b

= 5 a^+b^

o n a ^ +7ab-12b^ = 0 o a l b 4

a = - i b

3 Tir do, ta CO dupic phuong trinh tiep tuyen la:

•2X(j-6yo-23 = 0 , ' ' o + y o - 6 x o + 4 y o = - 3

Vay phuong trinh M N : 2x - 6y - 23 = 0

Bai 1.2.17 Trong mat phing Oxy cho duong tron (C): (x -1)^ + (y - 2)^ = 9 c6 tam I va diemM(5;-3) Chung minh rang tu M , ta c6 the ve den (C) hai tiep tuyeh MA, MB (A,B la tiep diem) Tinh di?n tich ciia t u giac M A I B

Huang dan giai

Duong tron (C) c6 tam 1(1; 2), ban kinh R = 3

Vi M I = V i l > R nen M nam ngoai duong tron (C), do do t u M ta luon ve dugc hai tiep tuyeh toi duong tron (C)

™ S^^AiB = 2SAMAI = IA.MA = R V M I 2 - R 2 = 3.V41-9 = 12^2 (dvdt)

Bai 1.2.18 Trong mat phSng voi hf toa dp Oxy, cho duong tron

(C): (x - 1 ) ' + (y + 2 ) ' = 9 va duong thJing d : 3x - 4y + m = 0 Tim m de

tren d c6 duy nhat mpt diem P ma tu do c6 the ke dupe hai tiep tuyeh

PA, PB toi (C) ( A , B la cac tiep diem ) sao cho tam giac PAB deu

Huong dan giai

Duong tron (C) c6 tam va ban kinh Ian lupt la: 1(1; -2); R = 3

Do tarn giac PAB deu

~ IP = 2IA = 2R = 6

nen API = 30^'

Suy ra P thuQc vao duong tron ( C ) c6

tarn I va ban kinh R' = 6

Ma P e d nen P chinh la giao diem ciia

duong t h i n g d va duong tron ( C )

Suy ra tren d c6 duy nhat diem P thoa

man yeu cau bai toan khi va chi khi

duong t h i n g d tiep xuc voi duong tron

( C ) tai P, hay la

d ( I , d ) = 6 o m = 19,m = - 4 1

Bai 1.2.19 Trong mat phSng voi he toa do Oxy, cho duong tron

(C): x^ + y^ - 2x + 4y = 0 va duong thSng d : x - y = 0 T i m toa dp cac diem

M tren duong th^ng d, biet t u M ke duoc hai tiep tuyen M A , M B den (C) (A,

3

B la cac tiep diem) va duong thang AB tao voi d mpt goc cp v o l coscp =

Hirang dan giai

Duong tron (C) c6 tam 1(1;-2), ban kinh R = Vs

Goi M ( m ; m ) va T(xo;yQ) la tiep diem ve tu M den (C) Khi do, ta c6

3 Mat khac AB tao voi d mpt goc cp voi coscp = - y = nen ta c6:

m - l - m - 2

<=> Ts = V2m^ +2m +5 o m ^ + m = 0<=>m = 0,m = - l

T h u lai ta thay ca hai truong hop nay ta deu I M = R hay M e (C)

Vay khong c6 diem M thoa yeu cau bai toan

Bai 1.2.20 Cho hai duong tron

( q ) : (x - 3)' + (y - 2 ) ' = 9 va (C^): (x - vf + (y + i f = 4 Chung minh (C^) va (C2) tiep xiic ngoai voi nhau tai A Viet phuong trinh tiep tuyen chung ciia (C|) va (C2) tai A Goi d la mot tiep tuyen chung cua (Cj) va (C2) khong di qua A, duong thang d cat duong thing noi hai tam tai B Tim toa do diem B

Huang dan giai

Duong tron (Cj) c6 tam 1(3; 2) va ban kinh R = 3

Duong tron (C2) c6 tam r ( 7 ; - l ) va ban kinh R' = 2 Goi A ( x ; y ) Theo gia thiet ta c6:

Tiep tuyen chung cua (C,) va (C2) tai A

Vec to phap tuyen cua tiep tuyen tai A : n = 11' = (4;-3) Phuong trinh tiep tuyen chung cua (Cj) va (C2) tai A la: 4x - 3y - 21 = 0

BI' R' Goi B( X y ; yQ), theo gia thiet ta c6 BI R

Duong tron (C) c6 tam 1 (-2; -2), ban kinh R = 72