Phucmg phdp gidi Todii Hinh hoc theo chuyen de - Nguyen Phu Khdnh, Nguyen Tat Thu 1 xTir; 2) Taco A'N = -^A'B'=^N B'Q = -B'Br:>Q 4 4 4 ^a\f3 a 3aV2 , AT = -A'C'=>P 3 2 2 4 va M 0;a; 0;^;aV^^ Suy ra AN = f aVs AQ = 4 4 aS a 3aV2' ;^;a^/^ , AM = 0;a; — ,AP = 2 2 4 AP,AQ AP,AQ 0; .AN = -^a3; 24 AP,AQ AM = - Dodo VA.MI,Q=7[AP,AQ' .AN AI^AQ AM = aV6 IF w w w w 13a^N/6 VAMPNQ - VA.MI'Q + VA.MPQ - -•—— • £Q BAI TAP Bdi 1.1.3, Trong khong gian Oxyz cho cac vec to: a = (2;3;l), b = (-3;-2;0), c = (x;2;-3) 1) Tim X de a A b vuong goc voi c 2) Tim X de goc giua hai vec to a A b va c bang 120". Jiuang dan gidi Ta CO aAb = (2;-3;5) 1) aAblco(aAb).c = 0c>2x-3.2 + 5.(-3) = 0«x = y (aAb).c - 1 ^ ' • = cosl20°= 2) Yeu cau bai toan o a AB 2x-21 1 = — <=> Vx^+13.V38 2 21 ^21 x< — 2 2(2x-21)2 =19(x^+13) x< 2 o x = llx2+168x-635 = 0 -84 ± V14041 11 Cty TNHHMTV nVVII Khang Vict P^i 2.1 •3- Trong khong gian vai he tpa dp Oxyz cho diem A (3;-2; 4) f im tpa dp cac hinh chieu ciia A len cac true tpa dp va cac mat phSng toa dp. 2j Tir" ^ ^ ^ *a"^ 8'^*^ AMN vuong can tai A 3^ Tim tpa dp diem E thupc mat phang (Oyz) sao cho tam giac AEB can tai E CO dien tich bang 3^y29 voi B(-1;4;-4) . ,^ Jiucang dan gidi \ Gpi Ai,A2,A3 Ian lupt la hinh chieu cua A len cac true Oa, Oy, Oz. ^'"^ chieu cua A len cac mat phang tpa dp (Oxy),(Oyz),(Ozx). Taco: Aj (3;0;0),A2(0;-2;0),A3(0;0;4) 1; " va Bi(3;-2;0),B2(3;0;4),B3(0;-2;4) .y. 2) Do M€Ox=>M(m;0;0), N€Oy=i>N(0;n;0) F' ' Suyra AM = (m-3;2;-4), AN = (-3;n + 2;-4) Tam giac AMN vuong can tai A nen ta c6 • -3(m-3) + 2(n + 2) + 16 = 0 2(n + 2) + 16 AM.AN = 0 AM^ = AN^ (m - 3)^ + 2^ + (-4)2 = (-3)2 + (n + 2)^ + (-4)^ m-3 = 2(n + 2) + 16 3 n2 (1) = (n + 2)2+5 (2) Ta c6: (2) o 4(n + if + 64(n + 2) + 256 = 9(n + if + 45 o5(n + 2)2 -64(n + 2)-211 = 0 n + 2 = n + 2 = 32 + 3V23T 5 32-37231 <=> n = • n = • 22 + 3N/23T 5 22-3^/23T m = - m = 189 + 67231 5 189 - 67231 ' l%y CO hai bp thoa yeu cau bai toan: loSc M, M^ '189 + 67231 15 f 189 - 67231 15 ;0;0 ;0;0 ^^^22 + 37231^^1 0; 22-37231 ;0 Phuortig phap giai Todn Hinh hgc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu 3) Vi Ee(Oyz) nen E(0;x;y) Suyra AE = (-3;y+ 2;z-4), BE = (l;y-4;z + 4) = (8y + 6z-8;4z + 8;10-4y) AE,BE Nen tir gia thiet bai toan ta c6: AE^ = BE^ 12 AE,BE = 3sl29 <=> AE^ = BE^ AE,BE AE^ = BE^ o9 + (y + if + (z-4)^ = 1 + (y-4)^ +{z + if oy = = 1044 4z + l AE,BE '50z-16 = 1044 « (8y + 6z - 8)^ + (4z + 8)^ + (lO - 4yf = W44 2 •(4z + 8)' ^26-16zf_,044 = 0«z.2,z ^ 25 • z = 2::^y = 3 nen E(0;3;2) 34 37 . ^ • z = =>y = nen E 25 ^ 25 V 25 25; Bdi 3.1.3. Trong khong gian Oxyz cho bon diem A(l; 2; 1), B(2; -1; -3), C(-2; 0; 3), D(0; 3; 4) 1) Chung mlnh rkg A, B, C, D khong dong phang. Tinh the tich ciia tu dif n ABCD 2) Tinh chieu cao ve tu B ciia tam giac BCD va chieu cao ciia tii dien ABCD ve tu A. 3) G(?i M, N Ian lugt la trung diem ciia AB va CD. Tinh c6 sin ciia goc giiia hai duong thang CM va BN. 4) Tim E tren duong thSng AB sao cho tam giac ECD c6 di^n tich nho nhat. Jiuang dan gidi l)Tac6 BC = (-4;1;6), BD = (-2;4;7), BA = (-1;3;4) Suyra BCABD = (-17;16;-14), (BCABD)BA = 99^0 Vay A,B,C,D khong d6ng phSng va the tich khoi tu di?n ABCD la: 6 (BCABD)BA =1 (dvtt). 2) Taco: CD = (2;3;l):r>CD = 7l4,S^BCD =: Suy ra duong cao BB' = = ^. BCABD V74T • 214 Cty TNHH MTV DWHKhangVift 3V Chieu cao ciia hinh chop ve tir A la: h = ^ABCD 3) Ta CO M ^•1-1 2'2' 3 7] ~''2'2j - - - ^7 1 Suy ra CM= -;-;-4 V 2 2 ,BN: Nen cos(CM,BN)- CM.BN • 5 13^ ^'2' 2 141 • CM.BN = V24T' 141 4) Ta CO phuong trinh AB: CM.BN 276555 x = l + t y = 2-3t, suy ra E(l + t;2-3t;l-4t) z = l-4t CE = (t + 3; -3t + 2; -4t - 2), CE A CD = (9t + 8; -9t - 7; 9t + 5) Suy ra S^CDE^^ CEACD = i\/243? + 360t + 138 2 1 2^ 243 t + 20f 14 ^ >/42 + — >• 3 6 Vay S^( Q£ nho nhat khi va chi khi E 27, f 7 ,38,107' 27' 9 ' 27 , Bai 4.1.3. Trong khong gian Oxyz, cho ba diem A(l;l;l), B(5;l;-2), C(7;9;l). 1) Chung minh rang cac diem A, B, C khong thang hang. 2) Tim toa do diem D sao cho ABCD la hinh binh hanh. 3) Tinh cos A, sin B, tan C ciia tam giac ABC. 4) Tinh do dai duong cao tu dinh A, ban kinh duong tron ngoai tiep, ban kinh duong tron noi tiep ciia tam giac ABC. 5) Tim toa dg giao diem ciia phan giac trong, phan giac ngoai goc A v6i duong thSngBC. Jiuang dan giai 1) Ta CO AB(4;0;-3),AC(6;8;0) nen cac diem A, B, C thang hang khi va chi khi [4 = 6k ton tai so thyc k sao cho AB = kAC, tiic la V%y A, B, C khong thang hang. 0 = 8k (v6 li). -3 = O.k 215 Phumigphdpgidi Todit llhih hoc llico clim/eit ile- Ngni/On Phil Khi'iiih, Njjuylit Tat Thu 2) Vi A, B, C khong thing hang nen ABCD la hinh binh hanh khi va chi khi DC = AB, hay 9-yD=0 « 1-ZD=-3 XD=3 yD=9=>D(3;9;4). ZD=4 r • .'1. ^ 3)Tac6 AB = + 0^ + (-3)2 =5, AC =10, BC =777. Vay cosA = cos(AB,AC) = AB.AC .24_12 • 50 ~ 25• AB AC Tuongtu cosB = ^,cosC = ^^^. Vi B,C€(0;7r) nen 385 385 +) sinB>0=>sinB 2^ 2 481 = Vl-cos B =-, 5 V 77 +) tanC,cosC cung dau tanC = .|—I 1 = kos^C 38 4) Dien tich tam giac ABC la S =-BA.BC.sinB =-5.N/77 j—= VisT. 2 2 5 V 77 Do do .) h, = 2S^2S a BC V 77 +) R = = 2 AC '481 2sinB 2sinB = 25. 77 r = ^ = 2S 481 2V48T p AB + BC + CA 15 + V77* Chii y: Co the tinh dipn tich bang cong thuc S = AB,AC 5) Goi E, F Ian lugt la giao diem ciia phan giac trong, phan giac ngoai goc A voi duong thSng BC. Theo h'nh cha't phan giac, ta c6 — = — = ^ ^ EC FC AC +) Vi E nam trong doan BC nen EC = -2EB Giasu E{x^;y^;z^) thi EC(7 - XE;9 - yj:;l - ),EB(5- XE;1 - y^;-2-z^) Nen EC = 2EB tuong duong voi 01 <; Cty TNIIH MTV DVVll Khang Vi?t 7,XE =-2(5-XE) 9_y£=-2(l-yE) « 1 ZE=-2(-2-ZE) 17 Xc =• 11 ^fl7 11 , ZE=-1 Tuong tu, do F nam ngoai doan BC nen FC = 2FB, tu do ta tim dugc tga ^^diem F la F(3;-7;3). gai 5' 1 Trong khong gian Oxyz, cho tam giac ABC c6 A(2;3;l),B(-l;2;0),C(l;l;-2). 1) Tim toa do chan duong vuong goc ke txx A xuong BC 2) Tim tga do H la true tam cua tam giac ABC. 3) Tim tga do I la tam duong tron ngoai tie'p cua tam giac ABC. 4) Gpi G la trgng tam cua tam giac ABC. Chung minh rang cac diem G, H, I nam tren mot duong thing. Jiumigdangiai '.r j; • 1) Ggi K la chan duong vuong goc ke tu A xuong BC. x = -l + 2t Ta CO BC = (2; -1; -2), phuong trinh BC: < y = 2-t z = -2t Suy ra K(2t -1; 2 -1; - 2t), AK(2t - 3; -1 -1; -1 - 2t) Vi AK1 BC AK.BC = 0 o (2t - 3).2 + (-1 - t).(-l) + (-1 - 2t).(-2) = 0 « t = i. Tga do diem K can tim la K M 5 2^ 3 3 3) 2) Ggi H(x;y;z) la tryc tam tam giac ABC. Ta c6 AH(x-2;y-3;z-l),BH(x + l;y-2;z) AB(-3; -1; -1), AC(-1; - 2; - 3), BC(2; -1; - 2) Tich CO huong cua hai vec to AB, AC la r n -1 -1 -1 -3 -3 -1 \ AB,AC -1 -1 • -1 -2 AB,AC \ -2 -3 -3 -1 -1 -2 / = (l;-8;5). Vi H la true tam tam giac ABC nen '"it' VSi 217 Phucntgphapgiai Todn Hinli iiQC theo chuyen de N^iiijcti I'hu Khanh, Nguyen Tat Thu AH±BC BHICA o H 6 (ABC) AH.BC = 0 BH.CA = 0 AB,AC .AH = 0 2x-y-2z = -l x + 2y + 3z = 3 x-8y + 5z = -17 Giai h§ ta dugc H ^ 2 29 1 ^ {15 15 3) 3) Goi I(x;y;z) la tam duong tron ngoai tiep tam giac ABC. Ta c6 AI(x-2;y-3;z-l),BI(x + l;y-2;z),a(x-l;y-l;z + 2). Vi I la tam duong tron ngoai tiep tam giac ABC nen AI^ = BI^ AI^ = CI^ o • AB,AC1.AI = 0 AI = BI AI = CI IG (ABC) 6x + 2y + 2z = 9 x + 2y + 3z = 4 x-8y + 5z = -17 Giai ta dup-c I 61 1^ ^15 30 3, 4) Trpng tam G cua tam giac ABC c6 toa dp thoa man 2-1 + 1 3 + 2 + 1 1 + 0-2^ 2 •2-1 ,3''' 3) Do do HG 115 15 ,GI ^; -1;0 15 30 nen HG = 2GI, tiic la ba diem G, H, I nam tren mot duong thSng. Bai 6.1.3. Cho tam giac deu ABC c6 A(5;3;-l),B(2;3;-4) va diem C nam trong m^t phang (Oxy) c6 tung do nho hon 3. 1) Tim tpa dp diem D biet ABCD la tu di?n deu. 2) Tim tpa dp diem S biet SA,SB,SC doi mpt vuong goc. Jiuong dan giai Vi Ce(Oxy) nen C(x;y;0). Ta CO AB(-3; 0; - 3), AC(x - 5;y - 3; 1),BC(x - 2;y - 3;4) Tam giac ABC la tam giac deu nen AB = AC = BC, do do (x-5)2+(y-3)2+1^=18 (x-5)2 +(y-3)2 +1^ =(x-2)2 +(y-3)2 +4^ X = 1; y = 4 x = l |x = l;y=2' AC = AB |AC=BC' |(y-3)2=1 218 Cty TNHl! MTV nVVH Kluuix Viet Vi C CO tung dp nho hon 3 nen C(l;2;0). l)Gpi D('<;y;z)- ' ' Khi do AD(x - 5;y - 3;z + l),BD(x -2;y -3;z + 4),CD(x -1;y - 2;z) Tam giac ABC la tam giac deu nen ABCD la tu difn deu khi va chi khi = BD = CD = AB = 3N/2 . Ta c6 h§ phuong trinh [(X - 5f + (y - 3)2 + (z +1)2 = (X - 2)2 + (y - 3)2 + (z + 4)2 .^J' (x - 5)2 + (y - 3)2 + (z +1)2 = (X -1)2 + (y - 2)2 + z2 • 1 :ir:).'f:> ivu 'ij I ' ,f .(/. (x-5)2+(y-3)2+(z + l)2=18 z = l-x y = 16-5x (x-5)2+(13-5x)2+(2-x)2=18 z = l-x y-16-5x 3x2-16x +20 = 0 V' ' • x = 2 x = —• 10 _2 _7 3' 3' 3 Giai phuong trinh 3x2 - 16x + 20 = 0 ^j^^^^ Vay tpa dp cac diem D la D(2; 6; -1) hoac D 2)Gpi S(x;y;z). Taco: AS(x-5;y-3;z + l),BS(x-2;y-3;z + 4),CS(x-l;y-2;z) SA, SB, SC doi mpt vuong goc khi va chi khi .•;'A .fc' ) it AS.BS = 0 (X • BS.CS = 0 <=> J (X CS.AS - 0 (X ^x2+y2+z2- 7x o • x2+y2+z2- 3x x2+y2+z2- 6x y = 5z + ll o < x = l-z 3z2+10z + 8 = = 0 x + y-4z = 12 -3x - 3z = -3 x2 + y2 + z2 - 6x - 5y + z = -11 Giai phuong trinh 3z2 + lOz + 8 = 0 ta dupe z = -2;z = ' ''^^' 219 Phucmgphapgidi Todn limit UQC theo chtiySn de- Nguyen Phi'i Khanh, Nguyen Tat Thu ^7 13 4~ Suy ra hai diem S thoa man la S(3;1;-2),S 3' 3 Bdi 7.1.3. Trong khong gian Oxyz, cho hinh hpp chu nhat ABCD.A'B'C'D' CO A = 0,B€0x,D€0y,A'G0z va AB = 1, AD = 2, AA' = 3. 1) Tim toa dp cac dinh ciia hinh hop. 2) Tim diem E tren duong thang DD' sao cho B'E 1 A'C 3) Tim diem M thuoc A'C, N thupc BD sao cho MN 1BD,MN 1 A'C. Tu do tinh khoang each giua hai duong thang cheo nhau A'C va BD . Jiucang ddn gidi 1) Taco A(0;0;0),B(1;0;0), D(0;2;0), A'(0;0;3). Hinh chieu ciia C len (Oxy) la C, hinh chieu cua C len Oz la A nen C(l;2;0). Hinh chieu cua B',C',D' len mp(Oxy)va true Oz Ian lupt la cac diem B,C,D va A' nen B'(l;0;3), C'(l;2;3), D'(0;2;3). 2) Vi E thupc duong thang DD' nen E(0;2;Z) , suy ra B'E = (-l;2;z-3) Ma A^ = (l;2;-3) nen B'ElA'C«B^.A^-0 »-l + 4-3(z-3) = 0oz = 4. Vay E(0;2;4). 3) Dat A/M = x.A^; BN = y.BD Ta CO AM = AA^ + A?M = AA; + X.A^ = (x;2x;3 -3x), suy ra M(X;2X;3-3X) AN = AB + BN = AB + y.BD = (l - y;2y;0) => N(1 - y;2y;0) Theo gia thiet cua de bai, ta c6: H. MN.A'C = 0 MN.BD = 0 Ma MN = (l-x-y;2y-2x;3x-3),A'C = (l;2;-3), BD = (-1;2;0) Nen (*) <r> fl - x - y + 4y - 4x - 9x + 9 = 0 [-14x + 3y = -10 -l + x + y + 4y-4x = 0 -3x + 5y = l X = y = 53 61 44 61 Do do M f53.106.24^ 61' 61 '61, Ml 17 88 ^ Vi MN la duong vuong goc chung ciia hai duong thang A'C,BD nen 220 Cty TNHH MTV DWH Khang Viet d(A'C, BD) = MN = ^{1-x-yf + (2y-2xf + {3x - sf = 6761 61 gai 8.1.3. Trong khong gian v6i he true toa dp Oxyz cho hinh chop S.ABCD CO day ABCD la hinh thang vuong tai A, B voi AB = BC = a; AD = 2a ; A = 0,B thupc tia Ox, D thupc tia Oy va S thupc tia Oz. Duong thang SC va BD tao vai nhau mot goc a thoa cosa = - ^ So' V 1) Xacdjnhtpa dp cac dinh cua hinh chop (;••;(• 2) Chung minh rang ASCD vuong, tinh dien tich tam giac SCD va tinh c6 sin cua goc hpp bai hai mat phang (SAB) va (SCD). 3) Gpi E la trung diem canh AD. Tim tpa dp tam va tinh ban kinh mat cau ngoai tiep hinh chop S.BCE . 4) Tren cac canh SA, SB, BC, CD Ian lupt lay cac diem M, N, P, Q thoa SM = MA, SN = 2NB, BP = 3PC,CQ-4QD . Chung minh rang M, N, P, Q khong dong phang va tinh the tich kho'i chop MNPQ. ' ' ' ' Jiuong ddn gidi 1) Taco A(0;0;0),B(a;0;0),D(0;2a;0),C(a;a;0).Dat SA = x =>S(0;0;x) ^ BD = (-a; 2a; 0), SC - (a; a; -x) :r> DB = a V5, SC = Vx^ + 2a2; BD.SC - a^ a 1 Nen cosa = COS(SC,BD) SC.BD SCBD J5(x2+2a2) N/30 o x^ + 2a^ = 6a^ o X - 2a S(0;0;2a). 2) Ta CO CS = (-a;-a;2a),CD = (-a;a;0) CS.CD = 0 => ASCD vuong tai C. Do do: S.crn = ^CS.CD - i.aV6.aV2 = a^v/s 'ASCD Gpi p = ((SCD),(SAB)). Ta CO hinh chieu cua tam giac SCD len mat phSng (SAB) la tam giac SAB nen ta suy ra 1 S ^^-23 1 cosP = |^=2^= 1 . N ^ASCD a yl3 V3 3) Taco E(0;a;0). Gpi l(x;y;z) la tam mat cau ^ ngo^i tiep hinh chop SBCE ' x 77i Phuang phapgiai Toa,i Hinh hoc theo chuyen dj- NguySn Phii Khdnh, Nguyen Tat Thu = - a)2 + y2 + ^ ^2 ^ y2 ^ _ 2a)2 Khi do J IC2 = « (X - a)2 + (y - a)2 + z2 = x2 + y2 4- (2 - laf . , , I lE^ =IS' -2x + 4z = 3a -x-y + 2z = a<=> -2y + 4z = 3a x^+(y-a)2+z2=x2+y2+(z_2a)2 a X = — 2 z=:a ^a a ^ Ban kinh R = IE =, 4)Tac6 M(0;0;a) Do SN = -SB=>N 3 v2y a a — V 2y + a 2 a iV6 CQ = JCD=^Q r2a I 3 '°' 3 5' 5'" 1 3 — , BP = -BC=>P a; ) 4 V 4 J Suy ra MN = 3 ••"•'-i \ MP = ( 3a ^ a;—-;-a 4 ,MQ = a 9a a^ a^^ MN A MP = • 4 ' 3 ' 2 nen M, N, P, Q khong dong phang"! (MNAMP).MQ = —.^0 VMNPQ =^|(MNAMP).MQ 40 Bdi 9.1.3. Trong khong gian v6i h^ tpa dp Oxyz cho hinh hpp chu nhat ABCD.A'B'C'D' c6 A triing voi goc tpa dp, B(a;0;0),D(0;a;0) ,A'(0;0;b) voi (a > 0,b > O). Gpi M la trung diem cua CC. 1) Tinh the tich cua khoi tu dien BDA' M . 2) Cho a + b = 4. Tim max V^.gPi^ . Jiu&ng ddn gidi 1) Taco: C(a;a;0), B'(a;0;b), C'(a;a;b), D'(0;a;b) z:>M(a;a;|) Suyra ]VB = (a;0;-b), ]VD = (0;a;-b), A'M= a;a; Cty TNHH MTV DVVII Khang Vigt pgn A'B,A'D =(ab;ab;a2) =>A'M.rA'B,A'D Vay V^'MBD = —• 2) Do a,b > 0 nen ap dung BDT Co si ta c6: 4 = a + b = ia + ia + b>33/iX:^a2b<~ 2 2 V4 27 Sa^b DSng thiic xay ra <=> ^ = b 2 <=> a + b = 4 8 a- — 3 64 4 • Vay maxV^.BDM = 27 ^~3 Bai 10.1.3. Trong khong gian Oxyz cho tu di^n deu ABCD c6 A(3;-l;2) va 7' G(l;l;l) la trpng tarn tam giac ABC. Duong thang BC di qua M djnh tpa dp cac dinh con lai va tinh the tich khoi tu dien do. Jiuang ddn gidi 4;4;- Taco: AG = (-2;2;-l), AM = (l;5;y) Suy ra nABC = AG A 2AM = (-12; -24; -24) Phuong trinh (ABC) :x + 2y + 2z-5 = 0. Ug(; = AG A nABC = (6; 3; -6) Phuong trinh BC: x = 4 + 2t y = 4 + t z = -^-2t 2 Gpi a la canh ciia tu di?n ABCD, ta c6 AG = — = 3 => a = 3N/3 2) Xac 4 + 2t;4 + t; 2t 2 Ta c6: B Nen ta dupe phuong trinh: AB = 2t + l;t + 5;-2t- — 2 (2t + l)2+(t + 5)2 + 2t + Buy ra B 2) 4-S I + 2S] -27 ot2+4t + —= 0<»t = -2± , C 7i; 4 + >/3 Phuoiigphdp gidi Todn Hinh hoc theo chuyen di-Nguyen Phii Khdnh, Nguyen Tat Thu X = l + t Do DG 1 (ABC) nen phuong trinh GD J y = 1 + 2t j z = l + 2t Suy ra D(l + t;l + 2t;l + 2t) ' A' Ma GD = VDA2-AG2 = —= 372=^3 t =3V2=>t = ±^y2 3 Dodo S(l + V2;l + 2V2;l + 272) hoac S(1-^;1-2V2;1-2V2). Thetichcua tudien: V ^lDGS.r,r =-—.^-^ = (<lvtt). § 2. LAP PHl/ONG TRlNH MAT PHANG iDe lap phuong trinh mat phang (a), ta c6 cac each sau: Cdch l:Tim mot diem M(X(,;yg;zo) ma mat phang (a) di qua va mpt VTPT n = (a;b;c). Khi do phuong trinh ciia (a) c6 dang: a(x-Xo) + b(y-yo) + c(z-Zo) = 0. Mot so' luu y khi tim VTPT cua mat phang (a): • Neu hai vec to a,b khong ciing phuong va c6 gia song song hoac nam tren (a) thi a A b = n la VTPT cua (a). • Neu mat phang (a) di qua ba diem phan biet khong thang hang A, B, C thi AB A AC = n la VTPT cua (a). • Neu (a)//(P) thi 1^ = 1^. • Neu A1 (a) thi n,^ = u^ . • Neu (a)l(P) thi n,^//(a). • Neu A(a; 0; 0), B(0; b; 0), C(0; 0; c) voi abc * 0 thi phuong trinh (ABC): abc Cdch 2: Gia su phuong trinh (a) c6 dang: ax + by + cz + d = 0 Dxfa vao gia thiet cua de bai ta tim dugc ba trong bo'n an a, h, c, d theo 3'^ con lai. Chang han a = mb, c = nb, d = pb. Khi do phuong trinh (a) mx + y + nz + p — 0. Cty TNHH MTV DWII Khang Viet Chii I/: Neu mat phang (a) di qua M(xo;yo;Z(j) thi phuong trinh cua (a) c6 aang: a(x-Xo) + b(y-yo) + c(z-Zo) = 0. i fidu 1.2.3. Lap phuong trinh mat phang (a), biet: 1) (a) di qua ba diem A(1;1;1),B(2;-1;3),C(-1;2;-1), 2) (a) di qua hai diem A, B va song song voi OC 3) (a) di qua M(l;l;l), vuong goc voi ((3): 2x - y + z -1 = 0 va song song vaiA—-j-^, 4) (a) vuong goc voi hai mat phang (P):x+y + z-l = 0, (Q):2x—y+3z-4 = 0 va khoang each tir O den (a) bang ^26 . JCffigidi. 1) Taco AB = (l;-2;2), AC = (-2;l;-2), suy ra ABA AC = (2;-2;-3) Phuong trinh (a): 2x - 2y - 3z + 3 = 0 . 2) Taco OC-(-l;2;-l),suyra ABAOC-(-2;-1;0) Vi (a) di qua A, B va song song voi CD nen (a) nhan n = - AB A OC = (2;1;0) lam VTPT. Suy ra phuong trinh (a): 2x + y — 3 = 0 . 3) Taco: i^-(2;-l;l), S^ = (2;l;-3) Do (a)l(P) .(a)//A =>-a=n,An,=(2;8A). Phuong trinh (a): x + 4y + 2z - 7 = 0. 4) Taco iv,' = (l;l;l), n^ = (2;-l;3) Ian lugt la VTPT cua (P) va (Q). Vi (a) vuong goc voi hai mat phang (P) va (Q) nen (a) nhan vec to n = A n^ = (4; -1; -3) lam VTPT. Suy ra phuong trinh (a) c6 dang :4x-y-3z + d=:0. ^Mat khac: d(0,(a)) = V26 nen ta c6: i = N/26 ^ d = ±26. 726 I Vay phuong trinh (a): 4x - y - 3z ± 26 = 0. ^^idu 2.2.3. Lap phuong trinh mat phang (P), biet: 1) (P) di qua giao tuyen ciia hai mat phiing (a): x-3z-2=0; ([3):y-2z+l=0_ 225 Phuang phdpgiai ToAn Hinh hoc theo chuyen de - Nguyen Phu Khdnh, Nguyen Ta't Thu va khoang each tir M den (P) bang 6^3 ' 2) (P) di qua hai diem A(];2;1),B(-2;1;3) sao cho khoang each tu C(2;-l;i) den (P) bang khoang each tu D (O; 3; 1) den (P). Lai giai. 1) Gia sir (P): ax + by + cz + d = 0. :h Taco A(2;-1;0),B(5;1;1) la diem chung eua (a) va (3) Vi (P) di qua giao tuyen ciia hai mat phlng (a) va (|3) nen A,B e (P) Suy ra 2a-b + d = 0 5a + b + c + d = 0 b = 2a + d c 7a-2d' 1 . — e + d 2 c + 2d| = Va^ + b^ + e^ « 27(e + 2d)^ = 49{a^ + +c^) 3V3 27.493^ = 49 a2+(2a + d)2+(7a + 2d)2 a = -d o27a2+32ad + 5d2 =0o a = -—d 27 • d = -a=>b = a;c = -5a. Suy ra phuang trinh (P) la: ax + ay - 5az - a = 0ox + y-5z-l = 0. 27 17 36 • d = a => b = a;c = a. Suy ra phuang trinh (P) la: 5 5 5 5x-17y-36z-27-0. 2) Gia sir (P): ax + by + cz + d 0 Vi A,Be{P)^- a+2b+c+d=0 -2a + b + 3c + d = 0 <=>s a = • d = - -b + 2c 3 5b + 5c Mat khac: d (C, (P)) = d (D, (P))«|2a - b + c + d| = |3b + c + d <»|5b-c| = |2b-c|oc = |b,c = 0 • Voi c-|b=>a = 2b;d^-yb^(P):4x + 2y + 7z-15 = 0 226 Cty TNHH MTV DWH Khang Viet IVai b = 0=>a = -c;d = c=>(P):2x + 3z-5 = 0. 3 3 fidu 2.2.3. Trong khong gian Oxyz cho diem A(l;2;3) va hai duong , x-1 y + 1 z + 2 , x-2 y + 2 z-1 L"!"'-] d 1 : — / cl 9 : = , . '2 1 -1 1 2-4 'I rU-n.^ u-:-i .•• j) Viet phuong trinh mat phclng (P) di qua A va dj. I 2) Chung minh rang di va di cat nhau. Viet phuong trinh mat phang (Q) chua di va d2. JCffigidi- Taco: Duong th^ng di di qua M(l;-l;-2), VTCP u^ = (2;1;-1) Du6ngthangd2diqua N(2;-2;l), VTCP u^ = (l;2;-4) 1) Taco: AM = (0;-3;-5) Do (P) di qua A va dj nen n^ = AM A u^ = (-8;10;-6) Suy ra phuong trinh (P): 4x - 5y + 3z - 3 = 0. 2) Xet he phuong trinh l + 2t = 2 + t' -l + t = -2 + 2t'o -2-t = l-4t' Suy ra di va d2 cat nhau tai E{3;0;-3). 2t-t' = l t-2t' = -l <=>t-t' = l -t + 4t' = 3 Ta CO ng = Uj A U2 = (-2; 7; 3) Phuong trinh (Q): 2x - 7y - 3z -15 = 0 . ViduL 4.2.3. Trong khong gian Oxyz cho ba duong thang , x-1 y+1 z-1 . x+1 y-1 z , , dj : = ^ = , dj : = - = — va d^ : 1 2 -1^2 3-1 ^ x = -2t y = -l-4t.^,,, . z = -l + 2t " 1) Viet phuong trinh mat phSng (a) di qua d2 va cat di, ds Ian lugt tai A, B sao cho AB = N/IS . 2) Goi (P) la mat phang chua di va di. Lap phuang trinh mat phang (Q) chua d2 va tao voi mat ph^ng (P) mot goc (p thoa cosip = ^) Ta CO A 6 dj =^ A(l + a; -1 + 2a; 1 - a), B e da => B(-2b; -1 - 4b; -1 + 2b) Suy ra AB = (-a-2b-l;-2(a + 2b);a + 2b-2), dat x = a + 2b STU AB = >/13=>(X + 1)2+4X^ +(X-2)^=13OX = -1,X = - Phuong phapgiai Todn Hinh hqc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu • Voi x = -l=> AB = (0;2;-3), ta CO u = (2; 3; -1) la VTCP cua d2 va A(-l; 1; 0) e => A e (a) Suy ra n = [AB, il] = (7; -6; -4) la VTPT cua (a). Phuong trinh (a): 7x - 6y - 4z +13 = 0. 4 f 7 8 2^1 • Voix = —=>AB= —;—;— . 3 V 3 3 3J Suy ra n = [-3AB,u] = (-14;11;5) la VTPT cua (a). Phuong trinh (a): 14x -1 ly - 5z - 25 = 0. 2)Du6ngthang dj di qua M(l;-l;l) c6 VTCP u^ = (l;2;-l) Duong thang dj di qua E(-1;1;0), c6 VTCP u^ = (2;3;-l) Duong thing dj di qua N(0;-1;-1) vacoVTCP u^ = (-2;-4;2) Taco u^ = -2u7=>di//d3. Dodo II7 = U^AMN = (-4;3;2) Vi mat phang (Q) di qua 6.2 nen phuong trinh (Q) c6 dang: ax + by + cz + a - b = 0 (1) voi a^ + b^ + c^ > 0 va 2a + 3b - c = 0 c = 2a + 3b. |4a-3b-2c| 9 b Mat khac cos 9 = Nen ta c6: np.ng —•—. np HQ 9b ^2%a^ + +c^) 729(5a2+12ab + 10b2) 6 V29(5a2+12ab + 10b2) ^ o 3^/5 |b| = 2V5a2+12ab + 10b2 o IQa^ + 48ab - 5b^ = 0 <=> a = -^b 10 a = -Sb 2 • a = —b ta chpn b = 10 => a = l,c = 32 . Phuong trinh (Q) la: x + lOy + 32z - 9 = 0. • a = b tachon b = -2=>a = 5,c=:4. 2 Phuong trinh (Q) la: 5x - 2y + 4z + 7 = 0 . Vi du 5.2.3. Trong khong gian Oxyz cho duong thang A c6 phuong trinh x-l_y+l z+3 -2 va diem M(l;2;0). Cty TNHH MTV DWH Khang Vi?t Viet phuong trinh mat phang (a) di qua M, song song voi A va (a) tao voi ba tia Ox, Oy, Oz mot tu dien c6 the tich bang 8. 2) Viet phuong trinh mat phang (P) di qua M, vuong goc voi (P):x+y+z-3=0 [31 va t^o voi A mot goc (p thoa cos (p = /— . ,,,1 V 34 Xffi giai. I) Gia su (a) cat ba true Ox, Oy, Oz Ian lugt tai A voi a,b,c>0 Khi do, phuong trinh (a): ax + by + cz = 1 Vi (a) di qua M va song song voi A nen ta c6: fa = -2b + l -;0;0 f 1 • ,B 0;-;0 V b ,C 0;0;- c fa + 2b = 1 i <=> <, -2a + 2b + 3c = 0 c = -2b + - 3 Mat khac VQABC = 8 <=> -OA.OB.OC = 8 <=> abc = — o b(2b - l)(2b - -) = J- « 192b3 - 160b^ + 32b -1 = 0 3' 48 ,1 11 b = -=>a = -,c = - 4 2 6 1 1 1' I ,j o (4b - l)(48b2 - 28b +1) = 0 ^ 7 + V37 5-V37 1-N/37 b = => a = ,c = 24 12 12 ^ 7-V37 5 + V37 1 + V37 b = =>a = ,c = 24 12 12 Vay phuong trinh mat can lap la: (aj): 6x + 3y + 2z -12 = 0 ; (a2): 2(5 - v/37)x + (7 + V37)y + 2(1 - V37)z - 24 = 0 Va (a3):2(5+>/37)x + (7-V37)y + 2(l + V37)z-24 = 0. ' 2) Vi mat phang (3) di qua M nen phuong trinh ciia (3) c6 dang: / ' * a(x-l) + b(y-2) + cz = 0^ax + by + cz-a-2b = 0 (*) . (^)'.; ' ^ voi a^ + b^+c^ >0. • I Mat khac, (P) 1 (P) nen rfp.rip = 0oa + b + c = 0c=>c = -a-b. Taco: sin(p= sin(u^,np) u^.np |5a + b| . n, |-2a + 2b + 3c| Vr7.Va2+b2+c2 " 734(77^b2) Phuontg phap giai Toan Hinh hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu . f: — [3" - . ' \5a + b [3" Ma sin (D = vl - cos cp = J— nen ta co: , =, — , V34 V34(a2+ab + b2) V34 (5a + hf = 3{a^ + ab + ) « 22a^ + 7ab - Zb^ = 0 » • a = —b, tachon b^ll=>a =2,c = -13 nen phuong trinh cua (p) la: 2x + lly - 13z- 24 = 0. • a = -^b, tachon b = -2=>a =],c = l nen phuang trinh cua (3) la: x - 2y + z + 3 = 0 = 0. 11 a = -lb 2 Vi du 6.2.3. Trong khong gian Oxyz cho hai mat phang (P):x+2y+2z-3=0, x-4 _y _ z mat phang (Q): 2x - y + 2z - 9 = 0 va duong thang ^' ^ ^ 2 1) Gpi (a) la mat phang phan giac cua goc hop boi hai mat phang (P) va (Q). Tim giao diem cua duong thang A va mat phSng (a). 2) Viet phuong trinh mat phSng (3) di qua giao tuyen ciia (P) va (Q), dong thoicach E(8;-2;-9) mot khoang Ion nhat. 1) Goi M la giao diem cua mat phang (a) va duong thSng A. Taco MGA nen M(4 + t;t;-2t) Vi M e (a) nen ta c6: d(M,(P)) = d(M,(Q)) |4 + t + 2t-4t-3| |2(4 + t)-t-4t-9| , „ , „ —^ = ^— ^o|t-l| = |3t + l|<::>t = 0,t = -l o o Tu do ta CO duoc hai diem M la: Mi(4;0;0) va M2(3;-l;2). 2) Ta CO A(3;-l;l) va B(-4;l;9) la hai diem thuoc giao cua (P) va (Q). Do do (3) di qua giao tuyen cua hai mat phang (P) va (Q) khi va chi khi A,B€(3). fx = 3-7t Taco AB = (-7;2;8) nen phuong trinh AB: Gpi K la hinh chieu cua E len AB, suy ra K(3 - 7t; -1 + 2t; 1 + 8t) => EK = (-5 - 7t; 1 + 2t; 10 + St) y = -l + 2t, t€] z = l + 8t Cty TNHH MTV DWH Khang Viet Vi EK 1 AB => EK.AB = 0 -7(-5 - 7t) + 2(1 + 2t) + 8(10 + 8t) = 0 » t = -1 Suyra K(10;-3;-7), EK = (2;-1;2). Goi H la hinh chieu cua E len mat phang (3), khi do: d(E,(P)) = EH < EK Suy ra d(E,(3)) Ion nhat khi va chi khi H = K hay (3) la mat phang di qua va vuong goc voi EK , ; , Phuong trinh (p): 2x - y + 2z - 9 = 0 . (ta thay (P) = (Q)). Vi du 7.2.3. Trong he toa do Oxyz, cho hai duong thang t, , x-1 y-2 z-4 X y-3 z-2 , , ^ , nx di •. — = ^^ = —^,d2 :- = ^-j- = — vadiem A(-2;l;0) . Chung minh A,di,d2 cung nam trong mot mat phang. Tim toa do cac dinh B,C cua tam giac ABC biet duong cao tir B nam tren dj va duong phan giac trong goc C nam tren 62 . JCgigidi. Duong thang di di qua M(l;2;4) vac c6 VTCP u," = (1;1;1) Duong thang d2 di qua N(0;3;2) vac c6 VTCP u^ = (l;-l;2) * ' Goi I la giao diem ciia di,d2 => I(l;2;4) " ' ' Mat phang (a) chua di, d2 c6 n^ = Ui,U2 =(3;-l;-2) va di qua I nen jphuongtrinh :3x-y-2z + 7 = 0. Ta thay A e (a). Vay A,di,d2 ciing thuoc mp (a) Xac djnh diem C: Goi (3) la mp di qua A va vuong goc voi dj => (p): x + y + z +1 = 0 Co C = (P) n (d2) nen tpa do diem C la nghiem cua h^ phuong trinh : x+y+z+l=0 • ; ' x_y-3_z-2^C(-3;6;-4) 1 -1 2 Goi A' la diem do'i xung cua A qua d2 . Ke AHi.d2 =>H(t;3-t;2 + 2t) => AH = (t + 2;2-t;2 + 2t) AH.U2 -0o(t + 2)-(2-t) + 2(2 + 2t) = 0r:>t •H 2.n.2 '3' 3 '3j A' 2.19.4 3' 3 '3) Co duong thang BC la duong thang BA' di qua C va c6 VTCP = CA' = i(ll;l;16) chon u' = (ll;l;16) 3 [...]... '55.7 va d j 3' 3 '3; 7'7'7 ;B 13 10 16^ 5 1) Xet he phuang trinh: y -3 z -3 2 1 ; \3' 3 '3 x= l y= l ~ 3 5 B- z= 2 Vay d j cat d2 tai giao diem l ( l ; l ; 2 ) d i d i qua diem M i ( 3 ; 3 ; 3 ) c6 ^ = (2;2;1) l a V T C P ; d j d i q u a M2(-5;-2;0) vaco u ^ - ( 6 ; 3 ; 2 ) laVTCP Gpi cp la goc giiia hai duong thang d j va d2 32 ^1 1 U 3 5\0 16^ 3 \7 7 7 1 X- Phuong trinh A : 32 ^ 3 23 13^ V 2 1 ' "21'"... Ian lugt tai A, B 2 ( l - a ) + 3a-2(-a) + 4 - 0 fa = - 2 2 ( - l + 2b) + 3 ( - l - b) - 2(1 + 2b) + 4 = 0 \ = -1 Suy ra A (3; -2;2), AB = (-6; 2; -3) 3) Ta CO y+ 2 la - b + c| 1 \ /3( 77b^^ ~ 3V3 |5a + c = - o 9(5a + c)^ = 13a^ - 8ac + 13c^ Vl3a2-8ac + 13c^ 3 212a2 + 98ac - 4 c 2 = 0 o a = — c, a = - i c 53 2 266 phuang trinh: —^— = ^-^— = z va mat phang (P): x + 2y - 3z + 4 = 0 Viet P (Trich cau 6b... 3 13 ^32 23 13> AB = 21 21 21 ri> 1 y- 21 7 32 5 23 1^ -1 l7'7'7J "2 5 I21'""21' z- 4 -5 r i 4 12^ -V ;B — = > A x+5 _ y+2 _ z 6 3- l7'7'7 Xffi gidi 2 1 1 11] K7'7'7) 13 10 16 7 ay ta c6 4 truong hop la: 2) Viet phuong trinh duong phan giac cua goc tao boi hai duong thang d j 'x -3 ViT S = -.IA.IB.sin(p = a = r:>a = l 2 42 42 aco: A e d j => A (3 +2t ;3 + 2t ;3 +1) => l A = (2t + 2;2t + 2;t +1) Be 6 .3. 3.Trong... 14 ,3. 3 Trong khong gian Oxyz cho hai duong thang ^ 1 - 1 1 1 - 1 ; t) =^ ME = (t - 4; 4 - 1 ; t - 4 ) Suyra E(6;-5;6), F ( 2 ; - l ; 2 ) Tuong tu: I(-4 + 3t'; -5 - 1 ' ; 4 - 1 ' ) => N I = (3t'- 6; - t ' + 2; - t ' + 2) 2 3 _^ _^ • 2 73 kumtg phapgiai Toiitt Hinh hoc theo chuyen de- Nguyen Phil Khanh, Nguyen Tat Thu Do do N I = xn(t'-2)^ =12t' = 2 + ^ ^ b 733 ^ 2V 33 11 ' Suy ra A 2 3' 11 T 6V 33. .. +2t ;3 + 2t ;3 +1) => l A = (2t + 2;2t + 2;t +1) Be 6 .3. 3.Trong khong gian voi h§ toa dp Oxyz cho hai duong thang , CO Nen lA^ = l c ^ 9 ( t + l)^ = l o ' ^ ~ ^ : ^ A , ( ^ ; ^ ; ^ ) , A 2 ( i ; i ; ^ ) t=- l 3 3 3^ 2 ^3 3 3^ " 3 x=t Vidu 20 r — Vii — => sm (p = v l - cos (D = U2 U1.U2 a ~ 5 z - - 23 ^ =- 13 ^ / '1.1.5> r i 4 i2> 4.5.1' ;B =>AB = 3' 3 '3> "2l'2l'21 l7'7'7; \ Khang Viet Cty TNHH MTV DWH... 5 3 t = 3 =^ I (3; -2;6), ban kinh R = - J Phuang trinh (S): (x - 3) ^ + (y + if t = U l^5 3' 3 ' 3 •I' 3 +{z-6f^^ ^'^\un^inhRJ- 5' >2 / 10 64 Phuong trinh (S): ^81 Vidu 3. 4 .3 Lap phuong trinh mat cau S(I; R) bie't ^ 1) Mat cau c6 tam I(-2 ;3; -l) va tiep xiic voi duong thang , x - 1 y+1 z + 2 A: = 1 1 -2 X — 2) Mat cau c6 tam I(-2 ;3; 5) va cSt A': ^^-^ = sao cho AB = 4 = ^—^ tai hai diem A, B X—2 y 3. .. N = 3\ f3 va M N song song voi mp (a): x - 2y + 3z = 0 2) Viet phuong trinh duong thang d d i qua O va cat d j tai A sao cho mat X y z Phuong trinh d : — = — = — 1 5 -4 ^ 233 157 148^ 130 • a= , suy ra A 9 ' 9 ' 9 Phuang trinh d : 1) V i M e d p N e d j nensuy ra M ( l + m ; - l + 2m;-2m), N (3 + 2n; -3 + n ; 2 - n ) Do do M N = ( 2 n - m + 2 ; n - 2 m - 2 ; - n + 2m + 2) 233 157 z -148 Vi du 5 .3. 3 Trong... o - s + 13 = 0 o s = 13 =>C(27;-41;18) JJitang ddn gidi t - l = 2s + l 2t + 3 = s + 5 x-1 i VI A C i i ^ = 0r^2(2s + 2) + 3 ( - 3 s - l ) + 4(s + 3) = 0 3 t - 2 = 2(-3s-2)=^t = - = > B Voi b = - a - a V 3 , chon a = 1 => b = - V s - 1 => c = A/S-1 = T u do ta CO hai duong thMng thoa man c6 phuong trinh la : 3f-? ^ Tpa dp trung diem M cua A B la M t - l ; : ^ ; 2 t + 3 = Voi b - - a + ax /3, chon a^\^h... + 2 2x+y-z-5=0 Ta CO AB = X=— 3 y = - l =>B 3 8 z= — 3 phirong trinh: 3 x = -12 x = 16 M(-12;16;0) z=0 y =t z=0 3x+2y-z+4=0 Diem 1(2; 2; 0) la trung diem cua doan thing AB 3' ^' 3 Duong thang A chi'nh la duong thang AB x - 1 _ y - 2 _ z_+l V | y phuong trinh A : -5 2 -9 3i kh Bdi 10 .3, 3 Trong kliong gian voi hf tryc toa dg Oxyz, cho diem A(l;2 ;3) va x-2 _ y + 2 _ x -3 x - l _ y - l _ z +l ; : 2 -1... =-10a + 7b Suy ra v==(a;b;-10a+7b) la VTPTcua (a) -39 a + 30 b n.v — • v 2].Va^+b^+(7b-10ar -39 a +30 b V^.^a2+b2+(7b-10a)2 23 ^679 I 1 9 4 , -+—+-=1 a b c 9b = 4c a = 4c a = -b,a = — b 85 Qc u 98 49 •a = 98;b = — ; c = : — 9 2 huong trinh mat phang (a) can tim la x + 9y + 4z - 98 = 0 \Cdchl:Tac6: • >/97 |39 a -30 b| = 23^ 3(l01a2+50b2-140ab • 3. 97(l3a -lOb)^ = 23^ (lOla^ - 140ab + 50b^ j AM.BC = 0 BM.CA = 0 a . 2)-211 = 0 n + 2 = n + 2 = 32 + 3V23T 5 32 -37 231 <=> n = • n = • 22 + 3N/23T 5 22 -3^ /23T m = - m = 189 + 67 231 5 189 - 67 231 ' l%y CO hai bp thoa. Nen theo gia thiet ta c6: cos9 = n.v -39 a + 30 b — n • v 2].Va^+b^+(7b-10ar 23 Suy ra coscp = . o -39 a +30 b 23 V679 V^.^a2+b2+(7b-10a)2 ^679 • >/97 |39 a -30 b| = 23^ 3(l01a2+50b2-140ab. ^''" ^ 42 9a-2b-7c 7a2+b2+c2 3 3 30a - 16b = 76(1 03^ - 12ab + 5h^) / o 3( 15a - 8hf = 2(1 03^ - 12ab + 5b^)» 134 03^ - 1428ab + 37 9b2 = 0 37 9, 1, <»a = b, a = —b. 670 2