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516.0076 C120N Th.S NGUYEN TAT THU (Nhom giao vien chuyen luyen thi Dai hoc) LUYEN TH DAI HOC HINH HOC SACH D1\ CHO HOC SINH LUitN THI DAI HOC, CAO DANG DVL.013508 HA XUAT BAN DAI m QUQJIGJA H A J O L JION 1^ Th.S NGUYEN TAT THU (Nhom giao vien chuyen luyen thi Dai hoc) Mm mum LUYEN THI DAI HOC liiNU nee (:. SACHDANHCHOHOCSINHHJY$NTHIDAIHOC, CAODJNG NHA XUAT BAN DAI HQC QUOC GIA HA Npi L6I NOI D A U 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 luyfn ky nang tir lam bai, t u 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, t u nhien, khoi nguon cam hung khi t u hpc. Kien thuc trpng tam ngan gpn, day dii bao gom ly thuyet, phuong phap giai cac dang bai d i t u 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. V i 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 t u duy, do la mot each hpc toan h i f u 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 v i 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 Jldfi ikiiL f 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 t u 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 v i t r i 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. , ,^, K h o i 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 t u C den (SAB). Phan tich: Yeu cau ciia bai toan c6 2 y: Tinh the tich khoi chop va tinh khoang each t u diem C den mat p h i n g (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 t u 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. Cach 2: Ta co d(C,(SAB)) = 'ASAB Goi H la trung diem canh BC, suy ra SH 1 BC =i> SH 1 (ABC) va SH = Ta co: AB - BC. cos B = suy ra S . ^ B C c = -j_ ^^-^C . , AC = BC. sin B = - , 2 2 = • SM = V s r f + H M ^ = - De' tinh khoang each t u mot diem M den mat phang (a), ta c6 cac each sau = 2SM.AB, ^ S u v ra r a Sb^^g _ ^ aVl3 aVs buy 4 3a Vay d(C,(SAB)) = 4 = . V39 3N/T3 16 a^yf39 , Cau 7a. Trong mat ph^ng voi he tpa do Oxy, cho hinh chi> nhat ABCD co diem Cach 1: Dung hinh chie'u H ciia M len (a) va tinh M H . Cach 2: Chuyen tinh khoang each t u M ve tinh khoang each t u diem khac bang each dua \o tinh chat 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 M D . Tim toa do cac diem B va C, bie't rang N(5;-4). Neu duong thSng M N ck (a) t^i I thi d(M,(a)) = ^ d ( N , ( a ) ) . Diem I t a | Phan tich: Day la bai toan xac djnh thuong chon la chan duong cao ciia hinh chop. 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 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 HK^ __L^ SH^ ^ B c A M AB 1 (SHM) => AB 1 HK . Trong tarn giac SHN ta c6: 1 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 J X - 4 , - 2 X + 3'[ W Mat khac HK 1 S M , do do H K 1 (SAB). ' Ma VsABC=:[;^ Ta CO S M l A B i ^ >S.^AB 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): . 3V 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^ 2 r - 2 x + i i ^ 2 r x + 4 ] 2 r2x + 13^ <=> I 2 ) + I 2 J I 2 J + I 2 J .<» X =l Suy ra I 3 1^ , C(1;-7)=^AC = (5;-15) . Vi AC 1 BN nen phuong trinh BN : x - 3y -17 = 0 => B(3b + 17;b) . 2 ^2 Ma IB = IA = -AC=> r6b + 37^ T r 2 b - i ^ 250 2 I 2 , I 2 J 40b^ + 440b +1120 = 0 « b =-4,b =-7 +) Voi b - - 4 ^ B ( 5 ; - 4 ) = N (loai) +) Voi b = - 7 = > B ( - 4 ; - 7 ) . Vay C(l;-7), B(-4;-7). 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. 1 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. Tu gia thiet ta c6 M(0;a),a > 0 . Trong tam giac vuong MAI, ta c6: 1 1 • AM = 2^/To 1 —1- + 1 1 AM^ 8 10 40 AH^ Ar AM' => M I = V M A ^ + lA^ = 5V2 => M H = M I - HI = 4x/2 . Hay d(M,A) = 4 V 2 o - a^ = 4^y2=>a-8(do a>0) hangVt^t . Do do, phuong trinh IH la: x + y - 8 = 0 . Taco I e M H = > 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 + l _ z +^2 va diem A (l; 7; 3). Viet phuong trinh mat phSng (P) A: x_2- 6 _ y _2 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 Hay 3 x - 2 y - z - 1 4 = 0. , ^ Ta CO M e A=>M(6-3t;-l-2t;-2 + t) Ta CO A M = 2N/30 <=> A M ^ = 120 o ( 5 - 3 t f + ( - 8 - 2 t f + ( - 5 + t f =120<»t = l hoac t = - - . 17 51 Suy ra eo hai diem thoa man la M(3;-3;-1), M "7 7 Lum m»s C t y TNHH MTV DVVH Khang Vi^t mym m vn mnn IH)L - isguyen n n - n m - Cau 8b. Trong khong gian voi he tpa dp Oxyz, cho mat phang: (P): 2x + 3y + z - 1 1 = 0 va mat cau (S): + + - 2x + 4y - 2z - 8 = 0. Chung minh (P) tiep xiic voi (S). Tim tpa dp tiep diem cua (P) va (S). Phan tich: +) De chung minh mat phSng (P) tiep xiic voi m|it cau (S) c6 tam I , ban kinh R • ta Chung minh d(I,(P)) = R. * +) 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 I H voi (P). Ta CO loi giai nhu sau: Lai giai. Mat cau (S) c6 tam la l ( l ; - 2 ; l ) , R = >yi4 . T a c o : d(I,(P)) = 2(1)+ 3(-2) +1-11 = 714 =R. Vay (P) tiep xuc voi (S). Phuong trinh duong thSng d d i qua I va vuong goc voi (P) la : x-1 y+2 ^ z-i * 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 A D = 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, t u day ta tim dupe diem I va viet dupe phuong trinh AC va BD, khi do tpa dp ciia C va D chi nen ta c6 IC = I H = IB, tu do ta tim dupe tpa dp C, B. Dua vao ID = -316 ta nen 2 ( l + 2t) + 3 ( 3 t - 2 ) +1 + t - 1 1 = 0 « t = 1 . Vay P ( 3 ; 2 ; l ) . KhoiB-2013 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 t u diem A den m|t p h i n g (SCD). Phan tich: +) De tinh the tich khoi chop, trudc het ta phai d i xac djnh chan duong cao ciia hinh chop. V i 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 S H 1 A B = > S H I (ABCD) va SH = a^ - CO 1 an. Lai c6, tam giac IBC vuong can tai I va tam giac CBH vuong tai B 2 ~ 3 " 1 Vi H e d = > H ( l + 2 t ; 3 t - 2 ; l + t ) . Do H € ( P ) The tich khoi chop S.ABCD la: V = I s H . S . o r n = ^ . 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 A I . Phuong trinh H I : 2x - y + 8 = 0 . Tpa dp diem I la nghiem ciia h^ < ^ ^~^o • ^~ [2x-y + 8 = 0 [y = 4 => l f - 2 - 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 IC = IB = I H . I la trung diem CH, do do Vi C € l H = * C ( c ; 2 c + 8)=^IC = I H « ( c + 2)^+(2e + 4 f = 5 o c = - l , c = - 3 . Suy ra C ( - l ; 6 ) . Tuong t u B(6 - 2b; b) ^ IB = I H o (8 - 2b) + (4 - b) = 5 o b = 5,b = 3. +) Voi b ^- 3 ^ B(0;3). Ta CO =^ = 1 ^ ID = -316 , suy ra D ( - 8 ; 7 ) +) Voi b = 5=> B(-4;5), tuong tu ta c6 D(4;1) . •-ty iiMtitiTvii Cau 7b. Trong mat phMng vdi tpa dp Oxy, cho tarn giac ABC c6 chan duong cao ha tu dinh A la H 17 , chan duong phan giac trong cvia goc A la 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 V 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), .... B (-1; 2; 3) va duong thSng A: ^ _ y ^ _ z - 3 -2 1 3 . Viet phuong trinh 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 A € A H ^ A ( 3 - 2 a ; a ) ^ B ( 2 a - 3 ; 2 - a ) . Vi d vuong gdc vdi A va AB nen a = AB A u = (6; 2; 4) la VTCP ciia d . 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: 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" Loi giai. Ta cd n = (2;3;-l) la VTPT ciia (P). Suy ra duong thSng d di qua A => AABC deu, suy ra AM 1 BC va AM = va vuong gdc vdi (P) nhan n lam VTCP. x = 3 + 2t Phuong trinh d : y = 5 + 3t, t e R . z = -t Tarn giac SAM vuong tai M va SMA = 45° Gpi H la giao diem ciia d vdi (P) va A' la diem doi xung vdi A qua (P). • SA = AM = aVi • , D Ma S A B C D = 2 S ^ B C = 2 . - Cant nang luyen thi DH Hinh Hoc - Nguyen latThti 1 1 a V s a^S The tich k h o i chop S.ABCD la: V = - S A . S A B C D " 2 • ~ 2 ~ ~ 2 V i AD//BC ^ AD//(SBC) Ta c6: IC^ = l A ^ _3a^ ~8~' ri Do BC 1 S A . • BC1(SAM)=>BC1 AH T u d o suy ra A H 1 (SBC) => d(A,(SBC)) = A H . Ma 1 1 AH^ AM^ _8_ 1 - +• SA^ AM^ +5=0 c = l,c = 5 Voi a = -5=> A ( - 5 ; - 2 ) , B ( - 4 ; 5 ) ^ B H = ( 2 ; - l ) Phuongtrinh A C : 2 x - y + 8 = 0=>C(c;2c + 8). ' " T a c o : I C ' = I B 2 =>(c + l ) c =-l,c +(2c + 7) = 25 « c^ + 6c + 5 = 0 ' ' =-5 Suy ra C ( - l ; 6 ) . >AH = aV6 Vay C ( 4 ; l ) hoac C ( - l ; 6 ) . 3a' j,,, Cau 7b. T r o n g m a t phang v o i he tpa dp Oxy, cho d u o n g t r o n (C): (x -1)^ + (y - 1)'^ = 4 : r < va d u o n g thang A : y - 3 = 0 . T a m giac M N P c6 true tam t r u n g v o i tam cua (C), cac d i n h N va P thupc A , d i n h M va t r u n g d i e m Vay d(D,(SBC)) = ciia canh M N thupc (C). T i m tpa dp d i e m P. C a u 7a. T r o n g m a t p h 5 n g O x y cho tam giac A B C c6 M 9 3 '2'2 la t r u n g d i e m Phan t i c h : Ta thay (C) «- va A tiep xuc v o i nhau tai T, ma tam I la true tam nen M la giao cua T I v o i (C). A B , d i e m H ( - 2 ; 4 ) , I ( - 1 ; 1 ) Ian l u o t la chan d u o n g cao ve t u B va t a m ducmg Goi J la t r u n g d i e m M N , suy ra IJ la d u o n g t r u n g b i n h nen IJ song song v o i t r o n ngoai tiep t a m giac ABC. T i m toa dp d i n h C. A va J thupc (C) nen ta t i m d u p e tpa dp d i e m J, lay d o i x u n g ta c6 d i e m N . Phan tich: V i M va I la t r u n g d i e m canh A B va t a m d u o n g t r o n ngoai tiep V i P thupc A nen tpa d p cau P chi c6 1 an, dua vao N I v u o n g goc v o i M P ta t a m giac A B C nen I M v u o n g goc v o i AB, t u day ta c6 p h u o n g t r i n h cua A B . t i m dupe P. Vay ta c6 l o i giai n h u sau: Ta goi toa d o cua A (chi c6 1 an) lay d o i x u n g qua M ta c6 toa d o ciia B . L a i giai. D u o n g tron (C) c6 tam 1(1; 1), R = 2. D u a vao A H va B H v u o n g goc ta t i m d u p e tpa d p ciia A va B. Ta C O d ( l , A ) = K Co A , H nen ta c6 p h u o n g t r i n h A C . D u a vao l A = IC ta t i m d u p e tpa dp d i e m C. 7 1 2'2 nen suy ra A tiep xuc (C) tai T. Do L o i giai. Ta c6 I M = 1 la true tam tam giac P M N nen M I v u o n g goc . V i I M 1 A B nen p h u o n g t r i n h A B : A A, suy ra x^^ - x, =^1 . 7x - y + 33 = 0 . M a M thupc (C) nen M ( l ; -1) Suy ra A ( a ; 7a + 33) H A = (a + 2; 7a + 29) Gpi J la t r u n g d i e m M N suy ra B ( - a - 9;-7a - 30) ^ B H = (a + 7;7a + 34). IJ la d u o n g t r u n g b i n h cua tam Do BH 1 A H o giac M T N , suy ra y j = y , = 1 BH.HA = 0 M a J thupc (C) nen J(3; 1) hay J(-l; 1). (a + 2)(a + 7) + (7a + 29)(7a + 34) = 0 <=> a^ + 9a + 20 = 0 <^ a = -4,a = - 5 . . c^-6c SuyraC(4;l). d(D,(SBC)) = d(A,(SBC)) Ve A H 1 S M . BCl AM (7 - 2c)^ + (c - i f = 25 « Voi a = -4 A ( - 4 ; 5), B ( - 5 ; -2) ^ P h u o n g t r i n h A C : x + 2y - 6 = 0 BH B = (3; 6 C ( 6 - 2c;c). C +) V o i J(3; ] ) t h i N(5; 3). Gpi P(t; 3) thupc A . Ta c6 M 4/MP =^ t = -1 =^ P ( - l ; 3). +) V o i J ( - ] ; ] ) t h i N ( - 3 ; 3). Gpi P(t; 3) thupc A. Ta c6 N I 1 M P =^ t = 3 => P(3; 3). C a u 8a. T r o n g k h o n g gian v o i 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 d p 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 . A B C D c6 day A B C D la hinh thoi canh a va = 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 ( S A B ) bSng 60° . Tinh the tich khoi chop S . A B C D va khoang each giua hai duong thang SC va A B . BAD Loi giai. la trong tam tam giac A B C , 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 ( S A B ) va ( A B C D ) , do do SMH = 60". CQ'I H HB 1 . AT.\s ax/s f suy ra SH = MH. tan 60° = | . Mat khac tam giac ABD deu canh a nen S^BCD = ^ S ^ B D = 2 . - j — = - y - The tich khoi chop S.ABCD la V = -3S H . S^^^^ ^ 3 e D3 = 2- - - 2- ^ = Ta CO AB//(SCD) r:>d(AB,SC) = d(AB,(SCD)) = d(B,(SCD)) . ^ = |d(H,(SCD)) ^ Gpi N, K theo thu hx la hinh chieu cua H len CD va SN, khi do d(H,(SCD)) = HK. ViHN=2d(B,CD)=2£V3^aV3 ' ' ^ 3 Vay d(AB,SC) = ^ ^ HK= ^"'^^ VSH^+HN^ 7 • 11 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 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 ) . .6 Cty TNHH MTV DWHKhang Phuong trinh mat cau (S): (x -1)^ + (y -1)^ + (z -1)^ = 4 . « n = -l,n = - l . +) Voi n = - i = ^ X B = 2 X N 5 =^>2 I-) Voi t = 5=^l(-5;7;13), R = 12. (loai). Phuong trinh mat cau (S): (x + 5)^ + (y - jf +) Voi n = - l = > X B = l < 2 , y B = - 3 ^ B ( l ; - 3 ) . d : ^ =^ Phuongtrinh C D : X = 5 = > D ( 5 ; 1 ) . Cau 7b. Trong mat phSng Oxycho tarn giac ABCco true tarn H(-6;7), tarn = | , ^ • ^ = Z^ = ^^ duong thSng BC . Tim tpa do dinh A . Lai giii. Duong thing d di qua B(-1;-1;0) va c6 u = (-2;l;2) la VTCP. BI = (t + 3; -t +1; 2t + 3) => BIA u = (-4t -1; -6t -12; -t + 5) Goi M la trung diem canh B C , ta c6 I M = d(l,BC) = N/S . Kc duong kinh BB', khi dcS AHB'C la hinh binh hanh Dodo d(l,d) = non A H = B ' C - 2 1 M - 2 N / 5 . A ( 8 ~ 2a;a) ^ A H = {2a -14;7 - a). BIAU ^(4t + l)2+(6t + 1 2 f + ( t - 5 f V53t2+142t + 170 Theo de bai, ta c6 d{l,d) = lA <=> 53t^ + 142t +170 = 54t^ + 54t + 81 Suy ra (2a -14^ + ( a - 7 ^ = 2 0 ^ ( a - 7 ^ = 4 => a = 9,a = 5. o t ^ - 8 8 t - 8 9 = 0 o t = - l , t = 89. Vay A (2; 5) hoac A (-10; 9). +) Voi t = - l : ^ l ( l ; l ; l ) , R = IA = 3. ^ y-2 Cau 8a. Trong khong gian Oxyz cho duong thang ^ ' — ^ z- 3 = 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)) Phuong trinh mat cau (S): (x -1)^ + (y -1)^ + (z -1)^ = 9 . +) Voi t = 89=>l(91;-89;18l), R = IA = 748069. Phuong trinh mat cau (5): (x - 9l)^ + (y + 89f + (z - 18l)^ = 48069. * Kh6iD 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 5t + n = 7t + l o t = 5,t = - l +) Voi t = - l ^ I ( l ; l ; l ) , R = 2, ' Suy ra AI = (t;-t-3;2t)=> lA = V6t^+6t+ 9 . suy ra phuong trinh B C : 2 x - y + 4 = 0. Phuong trinh D H : x + 2y - 8 = 0 . 7t + l 3 va diem A(2;3;3). Viet phuong Gpi I la tam cua mat cau (S), ta c6 l(2 + t;-t;3 + 2t). Loi giai. 5t + l l 3 thing trinh mat cau (S) di qua A , c6 tam nam tren duong thing A va tiep xiic voi duong thing d. duong tron ngoai tiep l ( l ; l ) va D(0;4) la hinh chieu vuong goc ciia A len Vi A G D H + (z -13)^ = 144 . Cau 8b. Trong khong gian Oxyz cho hai duong thing duong Phuang trinh BC: y = -3 => C (5; -3). Ta CO HD = (b;-3), Vi^t 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 :—»

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