Cẩm nang luyện thi đại học hình học Sách dành cho học sinh luyện thi đại học, cao đẳng

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 HAJOL JION Th.S NGUYEN TAT THU 1^ (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 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 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 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 .^BC = -j_ ^^-^C = • - Do do, the tich khoi chop S.ABC la: V = ^SH.S > .RC = ll^.^^^ = ^ . ^ 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 MH. Cach 2: Chuyen tinh khoang each tu M ve tinh khoang each tu diem khac bang each dua \o tinh chat Neu duong thSng MN 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: HM//AC=^HMlAB,HM = -!-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: ' 1 __L^ 1 _ 4 20 aV39 HK^ SH^ HM^ 3a^ a^ 3a^ 26 Ta co: d(C,(SAB)) = ^.d(H,(SAB)) = 2.HK = Cach 2: Ta co d(C,(SAB)) = 3V< S.ABC 'ASAB Ma VsABC=:[;^ Ta CO SMlABi^ >S.^AB = 2SM.AB, 3N/T3 16 SM = Vsrf+HM^ = ^ SuvraS _ ^ aVl3 aVs a^yf39 buy ra b^^g 4 3a Vay d(C,(SAB)) = 4=. 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;-2x-5). Do do, de tim tpa dp diem C ta chi can tim 1 an nen can thie't lap mpt, phuong trinh. Goi I la giao diem ciia AC va BC, ta co tpa dp diem J X-4,-2X + 3'[ B A c M Trong tarn giac vuong BND, ta co IN = IB = lA tu day ta tim dupe x hay tim dupe diem C va I. Khi do ta lap dupe phuong trinh duong thang BN (di qua N va vuong goc voi AC). Dua vao tinh chat IB = lA ta tim dupe tpa dp diem B. Vay ta co loi giai nhu sau: 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^ .<» X = l 2 r-2x+ii^ 2 rx+4] 2 r2x + 13^ <=> + + <=> I 2 ) I 2 J I 2 J I 2 J 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 Ma IB = IA = -AC=> 2 r6b + 37^ 2 1 r2b-i^ I 2 , T I 2 J 250 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. 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. hangVt^t Tu gia thiet ta c6 M(0;a),a > 0 . Trong tam giac vuong MAI, ta c6: 1 1 1 —- + 1 1 1 AM' AM^ 8 10 40 • AM = 2^/To . AH^ Ar => MI = VMA^ + lA^ = 5V2 => MH = MI - HI = 4x/2 . a 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 Hay 3x-2y-z-14 = 0. , ^ Ta CO Me A=>M(6-3t;-l-2t;-2 + t) Ta CO AM = 2N/30 <=> AM^ = 120 o(5-3tf+(-8-2tf+(-5 + tf =120<»t = l hoac t = Suy ra eo hai diem thoa man la M(3;-3;-1), M 51 "7 17 7 Lum m»s mym m vn mnn IH)L - isguyen nn-nm- Cau 8b. Trong khong gian voi he tpa dp Oxyz, cho mat phang: (P): 2x + 3y + z -11 = 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 IH 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 : x-1 y + 2 ^ z-i 2 ~ 3 " 1 Vi Hed=>H(l + 2t;3t-2;l + t). Do H€(P) nen 2(l + 2t) +3(3t-2) +1 + t-11 = 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 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 AH 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=> SHI (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 1HK Dodo HKl(SCD). 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 AI vuong goc ' voi BD nen H thupc AI 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 AI 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 IC = IB = IH. Vi C€lH=*C(c;2c + 8)=^IC = IH«(c + 2)^+(2e + 4f =5oc = -l,c = -3. Suy ra C(-l;6). Tuong tu B(6 - 2b; b) ^ IB = IH 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) . 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 AH 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 AH: x + 2y - 3 = 0 Vi A€ AH^ A(3-2a;a)^B(2a-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, teR. 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 iiMtitiTvii 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), ^_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 AM = Tarn giac SAM vuong tai M va SMA = 45° aVi , • SA = AM = • Ma SABCD=2S^BC=2 D M Cant nang luyen thi DH Hinh Hoc - Nguyen latThti ri Do 1 1 aVs a^S _3a^ The tich khoi chop S.ABCD la: V = -SA.SABCD " 2 •~2~~2 ~8~' Vi AD//BC ^ AD//(SBC) d(D,(SBC)) = d(A,(SBC)) Ve AH 1SM . BCl AM BC 1SA Tu do suy ra AH 1 (SBC) => d(A,(SBC)) = AH. • BC1(SAM)=>BC1 AH Ma 1 1 1 - + • AH^ AM^ SA^ AM^ _8_ 3a' > AH = aV6 Vay d(D,(SBC)) = Cau 7a. Trong mat ph5ng Oxy cho tam giac ABC c6 M 9 3 '2'2 la trung diem AB, diem H(-2;4),I(-1;1) Ian luot la chan duong cao ve tu B va tam ducmg tron ngoai tiep tam giac ABC. Tim toa dp dinh C. Phan tich: Vi M va I la trung diem canh AB va tam duong tron ngoai tiep tam giac ABC nen IM vuong goc voi AB, tu day ta c6 phuong trinh cua AB. Ta goi toa do cua A (chi c6 1 an) lay doi xung qua M ta c6 toa do ciia B . Dua vao AH va BH vuong goc ta tim dupe tpa dp ciia A va B. Co A, H nen ta c6 phuong trinh AC. Dua vao lA = IC ta tim dupe tpa dp diem C. 7 1 2'2 Loi giai. Ta c6 IM = 7x - y + 33 = 0. Suy ra A(a; 7a + 33) HA = (a + 2; 7a + 29) B(-a - 9;-7a - 30) ^ BH = (a + 7;7a + 34). Do BH 1 AH BH.HA = 0 o (a + 2)(a + 7) + (7a + 29)(7a + 34) = 0 <=> a^ + 9a + 20 = 0 <^ a = -4,a = -5. . Vi IM 1 AB nen phuong trinh AB: A B . Voi a = -4 A(-4; 5), B(-5; -2) ^ BH = (3; 6 Phuong trinh AC : x + 2y - 6 = 0 C(6 - 2c;c). C Ta c6: IC^ = lA^ (7 - 2c)^ + (c - if = 25 « c^-6c + 5 = 0 c = l,c = 5 SuyraC(4;l). . Voi a = -5=> A(-5;-2),B(-4;5)^BH = (2;-l) Phuongtrinh AC:2x-y + 8 = 0=>C(c;2c + 8). '" ' ' Taco: IC'=IB2 =>(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 voi he tpa dp Oxy, cho duong tron (C): : r < (x -1)^ + (y - 1)'^ =4 va duong thang A : y - 3 = 0 . Tam giac MNP c6 true tam trung voi tam cua (C), cac dinh N va P thupc A , dinh M va trung diem ciia canh MN thupc (C). Tim 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 TI voi (C). Goi J la trung diem MN, suy ra IJ la duong trung binh nen IJ song song voi A va J thupc (C) nen ta tim dupe tpa dp diem J, lay doi xung ta c6 diem N. Vi P thupc A nen tpa dp cau P chi c6 1 an, dua vao NI vuong goc voi MP ta tim dupe P. Vay ta c6 loi giai nhu sau: Lai giai. Duong 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 PMN nen MI vuong goc A, suy ra x^^ - x, =^1 . Ma M thupc (C) nen M(l; -1) Gpi J la trung diem MN suy ra IJ la duong trung binh cua tam giac MTN, suy ra yj = 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 NI 1 MP =^ t = 3 => P(3; 3). Cau 8a. Trong khong gian voi h^ tpa dp Oxyz, cho cac diem A(-l; -1; -2), B(0; 1; 1) va mat ph^ng (P):x + y + z- l = 0. Tim tpa dp hinh 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 ABCD la hinh thoi canh a va BAD = 60^'. Hinh chieu ciia S len mat phing (ABCD) la trpng tam tam giac ABC. Goc giiia mat phSng (ABCD) 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 (ABCD). Ke MH vuong goc voi AB, M thuoc AB. Ta CO SMH la goc giua hai mat phSng (SAB) va (ABCD), do do SMH = 60". HB 1 . AT.\s ax/s Vi = — nen MH = —d(D,AB) = = . DB 3 3 ^ 3 2 6 f suy ra SH = MH. tan 60° = |. Mat khac tam giac ABD deu canh a nen S^BCD = ^S^BD = 2 j— = -y- The tich khoi chop S.ABCD la V = -SH.S^3eD= ^ = ^ 3 ^^^^ 3 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 HK= ^"'^^ ' ' ^ 3 VSH^+HN^ 7 • Vay d(AB,SC) = ^^ 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 .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, DK:x-2y-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 «n = -l,n = -l. +) Voi n = -i=^XB =2XN =^>2 (loai). 5 +) Voi n = -l=>XB=l<2,yB=-3^B(l;-3). Phuang trinh BC: y = -3 => C (5; -3). Phuongtrinh CD:X = 5=>D(5;1). 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 BC:2x-y+ 4 = 0. Phuong trinh DH : x + 2y - 8 = 0 . Goi M la trung diem canh BC, ta c6 IM = d(l,BC) = N/S . Kc duong kinh BB', khi dcS AHB'C la hinh binh hanh non AH = B'C-21M-2N/5 . Vi A G DH A (8 ~ 2a;a) ^ AH = {2a -14;7 - a). Suy ra (2a -14^ + (a-7^ = 20^ (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), led 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 + ll 7t + l 3 3 +) Voi t = -l^I(l;l;l), R = 2, 5t + n = 7t + l ot = 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. BI = (t + 3; -t +1; 2t + 3) => BIA u = (-4t -1; -6t -12; -t + 5) Dodo d(l,d) = BIAU ^(4t + l)2+(6t + 12f+(t-5f V53t2+142t + 170 Theo de bai, ta c6 d{l,d) = lA <=> 53t^ + 142t +170 = 54t^ + 54t + 81 ot^ -88t-89 = 0ot = -l,t = 89. +) Voi t = -l:^l(l;l;l), R = IA = 3. 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 ABCA'B'C c6 tam giac ABC vuong t^i A, AB ^- a, BC = 2a va AA' = 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 [...]... TNHH MTV DWH Cam nang luyftt thi DH Hinh hqc- Nguyen Tat Thu Phuong trinh AB c6 dang: a(x - 4) + b(y - 3) = 0 Taco: d(I, AB) = I H = 2 / %5 3a + 4b = 2V5 » ( 3 a + 4b)2 = 20(3^ + b^) Khang Vi(t V i 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 a = 2b MA... b ta CO phuong trinh A B : 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 can tai M Vi A nkm tren Ox nenA(a;0), B nam tren duong thing x - y = 0 nen B(b;b) Do M(2;l) =>MA = ( a - 2 ; - l ) , M B = ( b - 2 ; b - l ) Tarn... thang can tim di qua B, M vay c6 hai duong thing thoa man bai 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): 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... Trong khong gian Oxyz cho duong th^ng d : - — - = 2 Ph5ng ( a ) : 2 x + y - 2 z + 2 = 0 va hai diem = 1 , mat 3 A ( 0 ; - 1 ; - 2 ) , B(2;3;1) Viet Duong th^ng A2 la phan giac goc BAC nhan vecto U 2 = (-1;5) lam vec to chi phuong trinh mat ph^ng (p) di qua hai diem A , B va cSt duong thSng d tai phuong nen C sao cho C each mat ph^ng (a) mot khoang bang 2 , ^ Cam nang luyftt thi DH Hinh hgc - N^yen... thang = l va Cho d u a n g thang A : ax + by + c = 0 va d i e m M ( x o ; y g ) IGii d o k h o a n g each axQ+byo+c t u M den A d u o c t i n h b o i cong thiic: d ( M , ( A ) ) = 2a (2a a > c > 0 ) la m o t d u o n g e l i p • •'i^^' Cho hai d u o n g t h i n g d j : a^x + b^y + Cj = 0; d2 : a2X + b 2 y + C j = 0 G p i a Taco : cosa= ^it • Va^+bl 3- T i n h chat va h i n h dang ciia elip: Cho ( E )... Tam doi xiing O • 6 - • §1 VIET PHl/ONG TRlNH O I / O N G 1 D i n h nghia: Trong mat phMng voi h# tpa dp Oxy cho hai diem FpFj c6 • ' ^ Phuong phap giai >a * Ne'u duong t h i n g di qua hai diem A, B thi AB la VTCP Hai duong t h i n g song song thi chung co cung VTCP va ciing VTPT ^ Cam nang luy?n thl Hinh HQC - Nguyen lat Cty TNHHMIV iRu V i dy 1.1.1 T r o n g mat p h i n g O x y , c h o b a d i... Chpn a = 1, b = V s Tu do phuong trinh duong thing d: >/3x + 3 y - > / 3 - 9 = 0 - Camnang V i d y 1.1.14 T r o n g m a t p h i n g v a i ( C ) : ( x - 1 ) ^ + ( y - l ) ^ = 1 0 D u o n g tron ( C ) t a m r ( - 2 ; - 5 ) c i t (C) tai h a i d i e m A, B sao cho AB = 2Vs Viet p h u o n g t r i n h d u a n g t h i n g AB Lai L o t gidi tQa d p O x y , cho d u a n g t r 6 n gidi D u o n g t r o n (C)... dp Oxy, cho hai duong thing CO hoanh dp duong Ss/Sa^ Vay phuong trinh ciia (T) la b) b - 3 = -5b=>b = i 2 npi tiep A ABC la: ( X a S, 'a + c = N/3 I AB^ = 2 2 >0;OA = - ^ A B = ^ S ;-2 AB = 1 1 ^^S''^^' a>0 Cam ttang luy$n thi DH Hinh hpc - Nguyen CtyTNHHMTVl Tat Thu I^a M e A Duong tron (T) duong kinh AC c6: I Phuong trinh (T): X + ^2 1 , 3^ suy ra = 1 V i du 1.2.12 Trong mat phang Oxy cho tam giac... tryc ao: 2b • Phuong phap quy tich: M ( X Q ; y g ) e A : ax + by + c = 0 axg + byg + c = 0 • Tieu diem Fj(-c; 0), F2( e; O) Chii y: ]) Cho duong t h i n g A : ax + by + c = 0 • Ne'u Aj//A thi Aj : ax + by + C j = 0, c, 5* c • Haiti^mcan: y = ±—x • Ne'u A 2 1 A thi A j : bx - ay + C j = 0 • Hinh ehu nhat co so PQRS c6 kich thuoc 2a, 2b voi b^ = c^ - a^ 2) Duong thSng di qua hai diem A(a;0), B(0;b);... 0 Tuong t v 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 Gpi A(a;0), B(0;b) Phuong trinh duong thing A : - + ^ = 1 a b 2 1 Do A di q u a M n e n : —+ —= 1 (1) a b 1 Tam giac OAB can khi va chi khi OA . 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 DAU Cac em hpc sinh than. 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. sach dupe hoan thi^ n hon. Nguyen Tat Thu Jldfi ikiiL f * Theo cau tnic de thi Dai hpc - Cao dSng ciia Bp Giao dic thi trong de thi vao cac truong Dai hpc - Cao dSng c6 3 diem