?H4N II HIJVHHOC k'': TS, Vu The Huu - Nguygn Vinh Can mm i I tiJHG THIG VA HJT PHIB TRDNE KHONG emii B A I T O A N QUAN H E SONG SONG I KIEN THLfC QUAN TRQNG Dvfcfng t h ^ n g s o n g s o n g vofi diiofng t h a n g a) Dinh nghia : - H a i dudng t h a n g goi l a cheo neu chung k h o n g cung n a m t r o n g mot m a t phang - H a i di/ofng t h a n g goi l a song song neu chung dong phSng va k h o n g c6 diem chung b) Cdc tinh chat : Tinh chat Trong k h o n g gian, qua mot diem ngoai mot diicfng t h a n g c6 m o t va chi mot diiorng t h a n g song song vdi dudng thSng Tinh chat Hai diTofng t h a n g p h a n b i e t cung song song vdfi m o t di/dng t h a n g thuf ba t h i song song v d i Dinh li (ve giao tuyen ciia ba m a t phang) Neu ba m a t phang cSt theo ba giao tuyen p h a n b i e t t h i ba giao tuyen ay hoac dong quy hoac doi mot song song He qua Neu hai m a t phang p h a n b i e t I a n lirgt d i qua h a i du'&ng t h a n g song song t h i giao tuyen ciia chiing (neu c6) song song v d i h a i diTdng t h a n g (hoac t r i j n g vdi mot t r o n g hai diTdng thSng doj Dvfcfng t h ^ n g s o n g s o n g vdfi m a t p h ^ n g a) Dinh nghia : M o t diidng thSng va mot m a t phSng goi l a song song v d i neu chiing k h o n g c6 d i e m chung b) Dieu kien de mot dudng thang song song vdi mot mat phang : Dinh li Neu diidng thang a song song vdi mot dudng thang b nao n a m t r e n mot mat phang (P) khong chufa a t h i a song song vdi m a t phang (P; Dinh li Neu mot dddng t h a n g song song vdi mot m a t phang t h i no song song vdi mot di/dng t h a n g nao n a m t r o n g m a t phang HQC va on luyen theo CTDT mon Toan THPT 193 He qua N e u diTcfng t h a n g a s o n g s o n g v d i m o t m a t p h d n g ( P ) t h i m o i m a t p h a n g ( Q ) chiifa a m a c a t ( P ) t h i cSt ( P ) t h e o g i a o t u y e n s o n g s o n g v d i a He qua N e u h a i m a t p h S n g p h a n b i e t c u n g s o n g s o n g v d i m o t difofng t h a n g t h i g i a o t u y e n c i i a c h i i n g ( n e u c6) c u n g s o n g s o n g v d i diTdng t h a n g H a i m a t p h a n g song song a) Dinh nghia : H a i m a t p h a n g g o i l a s o n g s o n g k h i c h u n g k h o n g c6 d i e m c h u n g b ) Dieu Dinh kien de hai mat phdng song song : li : N e u m o t m a t p h a n g ( P ) chijfa h a i d i i d n g t h a n g a v a b c a t n h a u va cung song song v d i m a t p h a n g (Q) t h i (P) song song v d i (Q) c) Tinh Tinh chat : chat Q u a m o t d i e m n g o a i m o t m a t p h a n g c6 m o t v a c h i m o t m a t p h a n g song song v d i m a t p h a n g He qua N e u d i i d n g t h a n g a s o n g s o n g v d i m a t p h a n g ( Q ) t h i q u a a cd m o t v a chi m o t m a t p h a n g (P) song song v d i m a t p h a n g (Q) He qua H a i m a t p h d n g p h a n b i e t c i i n g s o n g s o n g v d i m o t m a t p h a n g t h d ba t h i song song v d i Tinh chat N e u h a i m a t p h a n g ( P ) v a ( Q ) s o n g s o n g t h i m o i m a t p h a n g ( R ) d a cat (P) t h i p h a i c a t ( Q ) v a cac giao t u y e n ciia c h u n g s o n g song d) Dinh Dinh li Ta-let khong gian : li thuan Ba m a t p h a n g d o i m o t song song chan r a t r e n h a i cat tuyen b a t k i cac d o a n t h a n g ti/cfng t i n g t i l e Dinh li ddo G i a sijf t r e n h a i d i i d n g t h a n g a v a a' I a n l i i d t l a y h a i bo d i e m ( A , B , C) ^, , v a ( A , B , C ) c h o AB = A'B' BC = B'C CA CA' K h i b a d u d n g t h a n g A A ' , B B ' , C C c t i n g s o n g s o n g v d i m o t m a t phang e) Hinh Binh Idng tru va hinh hop : nghla hinh Idng tru H i n h l a n g t r u l a h i n h c6 h a i d a y l a h a i d a g i a c n a m t r o n g h a i m a t j - ' l TS Vu Th6' Hau - NguySn VTnh CJn phSng song song T a t ca cac canh k h o n g thuoc h a i day deu song song vdri H i n h l a n g t r u c6 : - Cac canh ben b&ng - Cac m a t ben l a cac h i n h h i n h h a n h - H a i day l a h a i da giac bang Dinh nghia hinh hop H i n h l a n g t r u c6 day l a h i n h b i n h h a n h diTcfc goi l a h i n h hop H i n h hop c6 sau m a t deu l a h i n h b i n h h a n h , cac m a t d o i dien t h i song song vdri Cac diJcfng cheo ciia h i n h hop cdt t a i t r u n g d i e m ciia m o i di/dfng II CAC DANG BAI TAP VA PHl/dNG PHAP GIAt C a c d a n g b a i t^p a) Cac bai t a p t r o n g muc nay, chii yeu l a chijfng m i n h h a i dudng t h a n g cheo nhau, h a i di/dng t h a n g song song, diicJng t h a n g va m a t phSng song song, h a i m a t phang song song De g i a i cac b a i t a p n a y t a thifdng suf dung cac t i n h chat ciia quan he song song va t r o n g n h i e u t r i i d n g hgfp t a suf dung phiCang phdp chiing minh phdn chiJcng b) Ngoai r a cac t i n h chat ciia h i n h l a n g t r u , h i n h hop cung giup t a g i a i cac b a i t a p ve quan he song song (cac canh ciia h i n h l a n g t r u , h i n h hop, cac m a t ben ciia h i n h hop v.v ) ^ Bait^ip Cho bon d i e m k h o n g dong phang A , B, C, D Chufng m i n h r a n g cac du'cfng t h a n g sau l a cac cap dudng t h a n g cheo : A B va C D A D va BC AC va B D Chi dan : Suf dung phi/cfng phap p h a n chufng Cho h i n h chop tuf giac S.ABCD Goi M , N , P theo thuf t\i l a t r u n g diem ciia cac canh SA, SB, SC a) Chufng m i n h mp ( M N P ) // m p (ABCD) b) Chufng m i n h giao d i e m ciia m p ( M N P ) v d i SD l a t r u n g d i e m Q cua doan t h a n g SD c) Goi O l a giao diem ciia A C va B D , O' l a giao d i e m cua MQ va N P Chufng m i n h ba d i e m S, O, O' thSng hang Chi dan : Doc gia tir g i a i Cho h i n h chop tuf giac S.ABCD; day A B C D l a h i n h vuong G o i M , N theo thuf tir la t r u n g d i e m ciia h a i canh SB, SD Hpc va on luyen theo CTDT mon Toan THPT 195 ^^^^^^^^^ a) Chirng m i n h M N // (ABCD) b) Neu each xac d i n h giao d i e m P cua m p ( A M N ) vdfi canh SC e) Xac d i n h giao t u y e n eiia m p (ABCD) \6i mp ( A M N ) C h i dSn : Doc gia tii g i a i Cho h i n h hop ABCD.A'B'C'D'; O la giao diem eiia AC va B D ; O' la giao d i e m eiia A ' C va B D ' a) Chufng m i n h m p (AB'D') // mp(C'BD) b) Chijfng m i n h A O / / C O ' C h i dan : Doc gia tiT g i a i Cho h i n h hop ABCD.A'B'C'D': goi O, O' theo thiif tiT la giao diem cua AC v d i B D va ciia A ' C v d i B'D' a) Chufng m i n h mp(AB'D') // mp(C'BD) b) Chijfng m i n h A'O // CO' C h i dan : Doc gia t\i g i a i BAITOAN QUAN HE VUONG GOC NH0NG BAI TOAN VE KHOANG CACH K h o a n g e a c h tijf m p t d i e m d e n mpt m a t p h a n g Cho m a t p h a n g (P) va d i e m O; H la h i n h chieu vuong goc ciia O t r e n (P) K h o a n g each tii d i e m O den m a t p h a n g (P) l a dp d a i doan t h a n g O H /p\ d(d, (P)) = d ( e d, (P)) 196 tJ.; TS, VQ The Huu - Nguygn VTnh Can K h o a n g e a c h givia h a i dxXdng t h a n g c h e o n h a u Khoang each giCifa h a i difcfng thSng cheo l a d a i doan vuong goe ehung eiia h a i di/ofng t h ^ n g Tinh chat : K h o a n g each giijfa hai diidng t h a n g cheo a; b t h i bSng khoang each tCr m o t d i e m A e a den mot m a t p h a n g (P) qua b va song song vdri difcfng t h a n g a Chuy : a) Ngoai ba t r U c f n g hcrp t r e n day, t a c6 k h o a n g each giiJa h a i diTcrng t h a n g song song, k h o a n g each giiJa h a i m a t phang song song Cac kien thufc n a y dofn g i a n va eung i t gap t r o n g cac b a i t o a n n e n t a k h o n g nhae l a i of day b) De dung mat phang (P) qua b va song song vdfi a, ta l a m n h i i sau : Lay mot diem M e b Q u a M diing difofng t h a n g a' // a M a t phang xae dinh bofi hai dUcfng t h i n g b va a ' chinh la mat phang (P) II CAC BAI TAP VA PHl/dNG PHAP GlAi Lj C a c dang bai tap a) Bai tap ve tim khoang each tit mot diem den mot mat phang De t i m khoang each ttr diem O den mat phang (P), triTdrc het ta t i m h i n h chieu H eiia O t r e n (P), sau t i n h dai doan thang O H Bai toan lien quan den v a n de difofng t h a n g vuong goe vdfi m a t phang va k h i t i n h O H , thiTcfng t a phai sijf dung den cac k i e n thiife ve he thufc Itfofng t a m giac, n h a t la he thijfe lUgfng t a m giae vuong b) Bai tap ve khoang each til mot diCang thang den mot mat phang song song Ta t i n h khoang each gi-iifa ducfng t h a n g A den m o t m a t phang (P) song song vdi A bang each t i m k h o a n g each tiT m o t diem M G A d e n m a t phang (P) Dieu quan t r o n g t r o n g viee g i a i b a i toan l a of cho chon diem M e A m o t each t h i c h hgp, giup t a suf dung het cac gia t h i e t ciia bai toan t i n h duoe d ( M , (P)) e) Bai toan ve khoang each giita hai duang thdng cheo De t i m k h o a n g each giOfa h a i diTcfng t h a n g cheo a; b , t a thiidfng l a m nhif sau : Hoc va on luyen theo CTDT mon Toan THPT , T i m mot diem M thich hop thupc a va t i m khoang each tir M den (P) Diem M can chon la diem c6 Hen he vcfi cac gia thiet ciia bai toan de giup t a van dung dirge cac c o n g thijfc, he thijfc liTpfng cac tam giac - T i m m o t m a t p h a n g (P) chufa a va song song song v d i a) M a t p h a n g (P) thiicfng dirofng t h a n g cAt a va song song v i b - song diTgrc v d i b (hoac chijfa b va xac d i n h b o i a va mot Cho h i n h chop t a m giac S.ABC, h a i m a t phang (SAB) va (SAC) vuong C a c b a i tap goc vdfi m a t phang day (ABC) a) N e u each xac d i n h h i n h chieu H cua dinh A t r e n mat phang (SBC) b) B i e t A B = 13cm, B C = 14em, CA = 15cm va SA = 16em T i n h khoang each tiT d i n h A den m a t phang (SBC) CHI DAN (SAB) (ABC) a) Ta CO : (SAC) (ABC) ^ SA ± (ABC) (SAB) n (SAC) = SA T r o n g t a m giac A B C , k e diiomg cao A M A M ± BC Tir SA (ABC) AM ±BC S M BC ( D i n h li diTotng v u o n g g o c ) T r o n g t a m g i a c S A M ke A H S M BC S M • BC (SAM) Tir (1) BC ± A M BC ± ( S A M ) ' A H cz (SAM)J TCr (1) va (2) t a diroc • A H ± BC (2) AHXBC A H (SBC) AH I S M => H l a h i n h chieu cua A t r e n m a t phang (SBC), b) Trirdtc h e t , t a t i n h d i e n t i c h t a m giac A B C theo cong thuTc H e r o n , t a c6 : 2p = 42 => p = suy r a p - a = 8; p - b = 7; p - c = SABC = V21.8.7.6 = 84 Tir day t a eo A M = 2S ^ABC BC _ 19 198 ei TS Vu The Hi;u - Nguyen VTnh CJn Tarn giac S A M vuong t a i A v a A H l a di/dng cao thuoc canh huyen Theo he thiic li/gfng t r o n g tarn giac vuong t a c6 : 1 AH^ SA^ => A H ^ = M A ^ ^ 400 1 A H ^ 16^ A H = 9,6 + 1 12^ 256 TT = + 144 (cm) Chiiy : Ta CO the t i n h A H theo each khac Trirdrc het, t a t i n h S M tiT t a m giac vuong S A M S M ' = S A ' + A M ' = > S M = 20 T i n h A H t\X hai t a m giac vuong dong dang : c^TTA A T A S H A CO A S A M = > AH A H SA = A M S M ^^ =^AH = AM.SA S M =1^=9,6 20 T i n h A H tiT dien t i c h t a m giac S A M 2SsAM = A H S M 2SsAM = S A A M => A H S M = S A A M =^ A H = A M OA SM = 9,6 Ta cung c6 t h e dirng d i e m H theo each l i luan sau : Do SA (ABC) =^ SA BC Dirng qua SA m o t m a t p h a n g (P) vuong goc vdri BC; mSt p h a n g cat BC t a i M T r o n g m a t p h a n g (P), t a ke AH SM t h i v i (SAM) ± (SBC) va (SAM) n (SBC) = SM ma A H A M nen A H (SBC) hay H l a h i n h chieu ciia A t r e n m a t p h a n g (SBC) Cho h i n h chop S.ABC; h a i m a t p h a n g (SAC) va (SAB) vuong goc v6i m a t day (ABC) va SA = a V T i n h k h o a n g each tCr d i n h A den m a t phang (SBC) t r o n g cac t r u d n g hop : a) Day ABC l a t a m giac deu canh a b) Day ABC l a t a m giac can d i n h A, goc A - ° va A B = A C = a c) Day ABC l a t a m giac vuong t a i B; A C = 5a, BC = a CHI D A N a) T a CO SA (ABC) Goi M l a t r u n g d i e m ciia canh BC T r o n g t a m giac S A M ke A H ± S M t h i A H (SBC) T r o n g t a m giac S A M t h i AH' AS' A M ' Hgc va on luyen theo CTDT mon Toan THPT 199 AH'' AH = (aV2)' • + iV66 11 b) Goi M la t r u n g d i e m ciia BC T r o n g t a m giac S A M ke A H ± S M =^ A H (SBC) T r o n g t a m giac S A M t h i AH^ 1 AS' AM' vdi AS - aV2 va A M = AB.coseO" = CO l^f2 ta t i n h duoc A H = c) Ta : SA ± (ABC) BC ± (SAB) T r o n g t a m giac SAB, ke A H SB ^ A H ± (SBC) va t a cung c6 AH' 1 AS' + AB' De t h a y A B ' = a ' AH = 6aVl3 13 Cho h i n h chop S.ABCD, chieu cao bSng a\/3, hai m a t phang (SAC) va (SBD) vuong goc v d i m a t p h a n g day (ABCD); day A B C D la h i n h thoi, canh a, goc n h o n A = 60" T i n h khoang each giOfa h a i dUcfng t h a n g A D va SB CHI DAN Ta CO : Goi O la giao diem ciia hai diTdng cheo AC va BC ciia h i n h thoi A B C D : 200 ,.' TS Vu The Hi/u - Nguy§n VTnh Can (SAC) (ABCD) (SBD) ± (ABCD) (SAC) n (SBD) = SO o SO (ABCD) =:> SO = aVs BC // A D => m a t p h a n g (SBC) la m a t p h a n g chijfa (SB) va song song vdfi A D ^ ^ Qua SO ta difng mot mat phang vuong goc vofi canh BC, mat phang cat AD d P va cat BC d Q Ttr P ke P K SQ t h i P K (SBC) d6d(AD, SB) = d(AD, (SBC)) = d(P, (SBC)) = P K TCr O ke O H SQ =o O H (SBC) De t h a y PQ = AB.sin60° = IN/3 Trong t a m giac vuong SOQ t h i OH' ^aV3^' => O H // P K (aVS)^ OQ- OH^ OQ*-^ OH^ = 3a^ 17 + • OS' OH = 17 Trong t a m giac P Q K t h i O H l a diTcfng t r u n g b i n h nen PK = H = aVsi 17 Chii y : Co t h e t i n h O H t i ^ h a i t a m giac vuong dong d a n g SOQ va SHO OH SO SO.OQ ASOQ ASHO OH = OQ SQ SQ vdi SQ-" = SO' + O Q l Cho tuf dien A B C D H a i m a t b e n A B C va DBC n a m t r o n g h a i m a t phang hop \6i m o t goc 60° M a t ben A B C l a m o t t a m giac deu m a t ben DBC l a m o t t a m giac vuong can, d i n h D B i e t D B = a Goi M l a t r u n g d i e m cua canh BC T i n h canh A D T i n h khoang each tCr d i n h A den m a t phang (DBC) va k h o a n g each tCf d i n h D den m a t phang (ABC) T i n h khoang each giufa difdng t h a n g A D va diTcfng t h a n g BC, k h o a n g each giuTa h a i diidng t h a n g AC va D M H Q C va on luyen theo CTDT mon Toan THPT.'.'; 201 T i m d i e m d o i xufng c i i a m p t d i e m q u a m p t m a t p h a n g De t i m d i e m d o i xufng ciia d i e m M q u a m a t p h a n g ( P ) , t a l a m n h i i sau: - T i m t o a d o h i n h chie'u H ciia d i e m M t r e n m a t p h S n g (P) - H S u d u n g h e thuTc vectcf M M ' ^ M H hoac c o n g thufc t o a t r u n g d i e m de s u y r a t o a d i e m M ' , d o i xiifng vdi d i e m M q u a M' m a t p h a n g (P) 11; D O I V l D U d W G THi T i m h i n h c h i e u c i i a d i e m t r e n dii'ofng t h a n g De t i m hinh chieu ciia diem M t r e n diTofng t h a n g A, t a l a m nhii Viet phirong t r i n h sau: m a t p h ^ n g (P) d i qua M v a v u o n g goc v d i A Tim t o a giao d i e m H ciia di/dng t h a n g A va m a t phSng (P) T i m d i e m d o i x i i n g c i i a m p t d i e m q u a m p t ditofng thang De t i m d i e m M ' d o i xufng ciia d i e m M qua difofng t h i n g A, t a l a m n h i f sau: T i m t o a h i n h c h i e u H ciia d i e m M t r e n diTorng t h S n g A S L T d u n g he thOfc vecto M M ' ^ M H hoac c o n g thufc t o a t r u n g d i e m d e s u y r a t o a d i e m M T i m hinh chie'u song song vdi phu'dng ciia dvfcfng thang m p t dvfoTng thang A tren m p t mat phang ( P ) De t i m hinh chieu song song theo p h u o n g ciia diTofng t h a n g d , ciia d i / d n g t h a n g A t r e n m a t phSng (P), t a l a m nhiT sau: V i e t phifcfng t r i n h m a t p h a n g ( Q ) d i qua A v a s o n g s o n g vdfi diTofng t h a n g d V i e t phifcfng t r i n h difofng t h a n g A', giao t u y e n ciia h a i m a t p h a n g (Q) v a ( P ) Hoc va on luyen theo CTDT mon Toan THPT : /; 283 Tim hinh chieu (vuong goc) cua diiifng thang tren mat phang De t i m h i n h chieu cua diidng t h a n g A t r e n m a t p h a n g (P), t a l a m n h i / sau: V i e t phiTcfng t r i n h - V i e t phiicfng t r i n h m a t phSng (Q) d i qua A va vuong goc vdri m a t phang (P) - diicfng thSng A', giao t u y e n ciia h a i m a t phSng (P) va (Q) Chii y: Co t h e coi triiofng hgfp la m o t t r i / d n g h o p dac biet cua trU'dng h o p t r e n day Ili.CAC KHOANG CACH K h o a n g each gii?a h a i diem K h o a n g each giaa h a i d i e m A ( X A ; yA; ZA), B ( X B ; ys; ZB): A B = IABI = ^ix^-x^f+{y^-yj'+{z^-zj' K h o a n g each tijf mpt diem d e n mpt mat phang Khoang each tii diem Mo(xo; yo; Zo) den mp (P): Ax + By + Cz + D = la + By^ + CZQ + D AXQ d ( M , (P)) = VA^TB^TC^ K h o a n g each tijf mpt diem den mpt di^cfng thang K h o a n g each tii m o t d i e m M den diiofng t h a n g A, d i qua diem Mo va CO vectcf chi phifong u l a : d ( M , A) = ll^o-^' t r o n g [ M ( , M , u] la vecto t i c h c6 hLfdrng ciia h a i vectcf MQM va u K h o a n g each giffa h a i dvfcfng thang cheo n h a u Cho h a i dirofng t h a n g A, A' cheo A' d i qua d i e m M ' va c6 vectof chi phiTofng u' - A d i qua M va c6 vectcf chi phuong u - K h o a n g each giaa h a i dudng thSng A, A' l a : d(A, A') = [u, u ' ] M M ' [u, u'] T r o n g do: [ u , u'] l a vectcf t i c h c6 hiTdng ciia u, u' [ u , u ] M M ' la t i c h v6 hi/dfng ciia h a i vectcf [u, u'l va MM' [u, u'] la d a i eiia vector [u, u'] 284 TS Vu The' Huu - Nguyln VTnh Can [u, u ' ] M M ' l a gia t r i tuyet doi cua t i c l i v6 hifdng cua h a i vecto [u, u'] va M M ' Chii y: Ngoai r a t a can nh6 m o t so k i e n thufc ve k h o a n g each t r o n g m o n H i n h K h o n g gian Idp 11: Khoang each giOfa diibng thang A den m a t phang (P) song song vdfi A: (A // (P)) t h i bang khoang each tU mot diem M e A den m a t phang (P) Khoang each giOa hai mat phang song song (P) // (Q) t h i bkng khoang each t i i mot diem M thupc mat phang den mat phang Khoang each giOfa h a i diidng t h a n g song song t h i bang k h o a n g each tif m o t d i e m thuoc diTcrng t h a n g den diicfng t h a n g k i a BAI TAP 130 T i m h i n h ehieu cua d i e m M ( l ; - ; 2) t r e n m a t phang (P): 2x - y + 3z + = CHi DAN M a t phang (P) c6 vectof phap tuyen n = (2; - ; 3) Diiofng t h a n g A d i qua M ( l ; - ; 2) va vuong goe vdi (P) n h a n vecto n l a m m o t vectcf chi (x = l + 2t phifong, do, phifong t r i n h ciia A l a : A: d(M, A) = MHI = |z = V(2 + D ' + (-3 + 3)2 + (3 - 2f d ( M , A) = / l O Cdch A CO vector chi phiromg u = ( ; 2; - ) M a t phSng (P) d i qua M ( - l ; - ; 2) va vuong goc vdri A n h a n u l a m m o t vecto phap tuyen Phifcfng t r i n h m a t phSng (P) l a : l ( x + 1) + 2(y + 3) - 3(z - 2) = X + 2y - 3z + 13 = T h a m so t ufng vdri giao diem ciia A va (P) l a nghiem ciia phi/ong t r i n h (3 + t ) + 2(-l + t ) - ( - t ) + 13 = ^ t = - TCr day t a c6 H ( ; - ; 3) va M H = VlO 286 T S Vu The Huu - Nguyin Vinh CSn Chu y: De t i n h M H , ta c6 the suf dung cong thufc M H = d ( M , A) = [MoM, u] Vdi Mo(3; - ; 0), M ( - l ; - ; 2) ^ M ^ M = (-4; - ; 2) Ta t i n h di/oc [M ^M , u] = (2; - ; 6) |[MoM, ul! = Vlio , iui = V l =^ M H = d ( M , A) = Vl40 Vl4 = Vio b) H l a t r u n g d i e m cua M M ' cho ta M M ' = M H M ( - l ; - ; j , H ( ; - ; 3)=^ M H = (3; 0; 1) M'(x; y; z), M ( - l ; - ; 2) => M M ' = (x + 1; y + 3; z - 2) x + = 2.3 M M ' = M H o y + = 2.0 z - = 2.1 X = o y - - => M'(5; - ; 4) z = Chu y: Co the siTf dung cong thijfc cho toa dp trung diem doan thang x - 4t Cho diTong t h a n g A: y = 7t z = + 2t va m a t phSng (P): x - y + z + = 133 T i m phurong t r i n h h i n h chieu vuong goc ciia A tren mat phang (P) CHI DAN H i n h chieu A' ciia A t r e n m a t phang (P) l a giao t u y e n cua h a i m a t phang: m p (P) va m p (Q) d i qua A va vuong goc v(Ji (P) Difcfng t h a n g A c6 vectcf chi phiTcfng u = (4; 7; 2) M a t phang (P) c6 vecto phap tuyen n = ( ; - ; 1) M a t phang (Q) chufa A, d i qua d i e m Mo(0; 0; 3) va c6 vecto phap t u y e n q = [u,H] = ( 1 ; - ; - ) Phirong t r i n h m a t p h a n g (Q): l l x - 2y - 15z + 45 = Ducfng thang A' la giao tuyen ciia (P) va (Q) nen vector chi phuomg ciia A' l a u' = [u,q] Ta t i n h diiOc u' = (-28; 4; 20) = 4(-7; 1; 5) Ta chpn vectof (-7; 1; 5) l a m vecto chi phifcfng ciia A' Hoc va on luyen theo CTDT mon Toan THPT ' 11 287 De t h a y d i e m M 0;i5;r X = -7t thuQC A' Phiiorng trinh A': y = — + t ^ z = - + 5t Chiiy: B a i toan gom h a i b a i toan cor ban: - V i e t phi/dng t r i n h m a t phSng d i qua m o t difofng t h a n g va vuong goc vdri m o t m a t phang - V i e t phi/ong t r i n h giao t u y e n cua h a i m a t phang B a i toan v i e t phuong t r i n h h i n h chieu A' ciia diiorng thSng A t r e n mat p h a n g (P) theo phi/Ong ciia m o t difcfng thSng d cho triTdfc cung gom hai b a i toan cO b a n : - V i e t phiiong t r i n h m a t p h a n g (Q) chufa m o t di/ofng t h a n g A va song song vdfi dirofng t h i n g d ( A va d cheo nhau) - V i e t phuong t r i n h giao t u y e n A ciia (P) va (Q) CAC CAU HOI TlJ ON TAP VE CACH VIET PHlJCfNG TRINH MAT PHANG VA DlJClNG THANG H a y neu cac phifong phap g i a i cac dang bai toan sau : Cho ba d i e m A , B, C a) Chufng m i n h ba d i e m A, B, C k h o n g t h a n g hang b) V i e t phuong t r i n h m a t p h a n g (ABC) Cho d i e m A va dudrng t h a n g A a) Chufng m i n h A g A b) V i e t phiiong t r i n h m a t p h a n g (A, A), xac d i n h hdi A va A Cho h a i duc^ng t h a n g A, A' a) Chufng m i n h A, A' cat T i m toa giao diem b) V i e t phircfng t r i n h m a t p h a n g xac d i n h bdfi A, A' Chufng m i n h h a i difcfng t h a n g song song v d i va viet phiicfng t r i n h m a t p h a n g chufa h a i difcfng t h a n g song song Chufng m i n h h a i di/6ng t h a n g cimg thuoc m o t m a t phang H D : Co the chufng m i n h chung cat hoac song song vdri 288 H TS VD The Hyu - Nguyin Vinh CSn V i e t phLTorng t r i n h m a t p h d n g d i qua m o t d i e m va song song vdri m o t m a t p h a n g cho triTdfc Cho h a i diTorng thSng A va A' a) Chufng m i n h A va A' l a h a i duomg t h a n g cheo b) V i e t phirong t r i n h m a t p h a n g (P) t r o n g cac trirofng hgrp : - (P) chijfa mot di/cfng thang va song song vdi diromg thang lai - (P) di qua mot diem M cho tnidc va song song vdfi ca hai dUcfng thang Cho di/orng t h a n g A va m o t diem M V i e t phifofng t r i n h m a t phSng (P) qua M va vuong goc v6i A Cho dirdng thSng A va m a t phSng (P) V i e t phuong t r i n h m a t p h a n g (Q) d i qua A va vuong goc vdfi m a t p h a n g (P) 10 Cho h a i m a t phang (P), (Q) va m o t diem M Viet phifong t r i n h m a t phSng (R) di qua M va vuong goc vdi ca hai m a t phSng (P), (Q) H.D : Chia b a i toan t h a n h h a i bai toan cor b a n : + V i e t phiTcfng t r i n h giao tuyen A cua (P) va (Q) + V i e t phuofng t r i n h m a t p h a n g (R) d i qua M va vuong goc v6i A 11 Viet phiiomg t r i n h t h a m so va phiTcfng t r i n h chinh tac ciia diiotng t h a n g A di qua diem Mo(xo; yo; ZQ) va c6 vectcf chi phiforng u = (ai; a2; as) 12 V i e t phirorng t r i n h di/orng t h a n g d i qua m o t d i e m va song song v6i m o t diidng t h d n g cho trLfdc 13 V i e t phiicfn^ t r i n h diidng thSng d i qua m o t d i e m va vuong goc v d i m o t m a t phang «ho triidfc 14 V i e t phifong t r i n h diidng thSng d i qua m o t diem va vuong goc va cSt mot diicfng t h a n g cho trifdic 15 V i e t phirorng t r i n h giao tuyen ciia hai m a t phang 16 Chijfng m i n h diTofng thang va m a t phang cat T i m toa giao diem 17 T i m toa h i n h chieu cua m o t d i e m t r e n m o t m a t phang 18 T i m toa d i e m doi xufng v d i d i e m M qua m a t p h a n g (P) 19 T i m t o a h i n h chieu cua m o t d i e m t r e n m o t difcfng t h a n g T i n h khoang each tiT m o t d i e m den m o t difcfng t h a n g 20 T i m toa diem doi xufng v6i mot diem qua mot difofng t h a n g cho trifdfc Hpc va on luyen theo CTDT mon Toan THPT U: 289 CHyYENflEVI MAT ClVBMHTPHiG BAITOAN18 PHl/dNG TRINH MAT CAU Phifcfng t r i n h m a t c a u a) M a t cau tarn I { a ; b; c), b a n k i n h r c6 phuofng t r i n h (x - a)^ + (y - b ) ' + (z - c)^ = r ' (1) b) Phi/ofng t r i n h bac h a i dang x' + Vdi + + + + 2Ax + 2By + 2Cz + D = (2) - D > la phuorng t r i n h ciia mot m a t cau t a m I ( - A ; - B; - C ) va b a n k i n h r = V A - ^ + B ' + C - D c) V i t r i tiforng doi giiia m a t cau va m a t phang Cho m a t cau t a m I , b a n k i n h r va m a t phang (P) * d ( I , (P)) > r => M a t p h a n g v a m a t cau k h o n g c&t * d ( I , (P)) = r (3) M a t p h a n g va m a t cau t i e p xiic v d i "Dicu khodng kien can vd dii de mat phang (P) tiep xuc vai mat cau (S) Id each tit tam I ciia mat cau den mat phang b&ng ban kinh ciia mat cau" !iiJcAC BAVTOAN c:d B A N ^ C a c d a n g t o a n Ve m a t cau va m a t phang t a thadrng gap cac b a i t o a n sau : a) Cho phuang trinh mat cau, t i m t a m v a b a n k i n h : co sd de g i a i loai t o a n n a y l a cac cong thiifc (1), (2) b) Viet phuang trinh mat cau De v i e t phi/cfng t r i n h m a t cau, t a can xac d i n h t a m I va b a n k i n h r ciia no, sau sii dung cong thufc (1) hoac (2) c) Toan lien quan den mat phang V i e t phiicfng t r i n h m a t cau t i e p xiic v d i m a t p h d n g : Siif dung cong - X e t v i t r i ti/Ong doi cua m a t cau vdfi m a t phdng - thufc (3) - Viet phuorng t r i n h m a t phang tiep xuc vdi mat cau va d i qua mot diem hoac tiep xuc t a i mot diem thuoc mat cau : Sijf dung cong thijfc (3) 290 III TS Vu The Hi^u - NguySn Vinh Can - T i m t a m va ban k i n h dtfcfng t r o n giao tuyen ciia mat phSng vdfi mat cau : + T a m K ciia di/ofng t r o n giao tuyen (C) ciia m a t phSng (P) v d i m a t cau (S) l a h i n h chieu vuong goc ciia t a m I ciia m a t cau t r e n m a t phang (P) + Ban k i n h r' ciia diTcfng tron giao tuyen d diicfc t i n h theo cong thijfc r'^ = r^ - d'^ t r o n g r l a ban k i n h m a t cau, d = d ( I , (P)) = I K Chu y : T r o n g k h i g i a i cac bai toan c6 l i e n quan den v a n de tiep xiic giCira m a t phSng vdii m a t cau hoac giOfa diiong t h i n g v d i m a t cau t a can luon luon n h d : " M a t phang (P) (hoac dircfng thang d) tiep xuc v6i mat cau t a m I t a i diem M t h i (P) (hoac d) vuong goc v6i ban k i n h I M ciia mat cau" hoac I M = r 2] B a i tap 134 Trong cac phi/ofng t r i n h sau day, phuang t r i n h nao l a phuorng t r i n h cua mot m a t cau va t r o n g t r i i d n g hop l a phiicfng t r i n h ciia m a t cau t h i hay t i m t a m va b a n k i n h ciia m a t cau ay a) (S) : x ' + y^ + - 4x + 6y - 2z + 10 = b) (S) : x ' + y^ + - 3x - y + 2z + 15 = c) (S) : X - + y^ + z^ + 4x - 2z - 2z - = CHI D A N a) Dua phircfng t r i n h ve dang : (x - 2f + (y + 3)^ + (z - i f = Ta di/tfc m a t cau t a m 1(2; - ; 1), ban k i n h r = Chil y : Co t h e g i a i nhif sau TCf viec ap dung phirorng t r i n h dang (2), ta C O 2A = - => A = - 2B = ^ 2C = - => C = - B= V i A^ + B^ + C^ - D = (-2)^ + 3^ + (-1)^ - (+10) = > => PhiiOng t r i n h x^ + y^ + z^ - 4x + 6y - 2z - 10 l a phi/Ong t r i n h ciia m a t cau t a m I ( - A ; - B ; - C ) = (2; - ; 1) Ban k i n h r = Vi = b) Khong phai phi/Ong t r i n h mat cau v i t a c o A = — ; B = — ; C = - ; D = 15 2 =^ A^ + B^ + C^ - D < c) M a t cau t a m I ( - ; 1; 1) va r = VlO HQC va on luyen theo CTOT mon Toan THPT 291 Viet phiicfng t r i n h mat cau cac trifdng hop : a) Co tam 1(1; 2; - ) , ban kinh r = 2V3 b) Co tam I ( - l ; 2; -3) va di qua diem M(0; 4; 2) c) Di qua hai diem A(2; - ; - ) , B(2; - ; 3) va c6 tam nam tren difcfng X = + 2t thang A : y = - i - t z = 3-2t d) Nhan doan thSng AB la dudng kinh, vdi A ( - l ; 2; 0), B(3; 4; -2) C H I a) D A N (S): (x - 1)^ + (y - 2? + (z + 3)^ = 12 o x'^ + y'"^ + b) (S): (x + if + (y - 2? + (z + f = 30 - 2x - 4y + 6z + = x^ + y^ + z^ - 2x - 4y + 6z - 24 = c) Tam I cua mat cau la giao diem ciia A vdi mat phSng trung trifc ( P ) cua AB Ta C O : ( P ) : 2x + y - z - = TCr day t i m dirge 1(3; - ; 1), ban kinh r = l A = Vs ^ (S): (x - 3)^ + (y + 2f + (z - 1)' = x^ + y^ + z^ - 6x + 4y - 2z + •= d) Tam mat cau I la trung diem ciia doan thSng AB : 1(1; 3; - ) Ban kinh r = l A = V G =^ ( S ) : (x - 1)^ + (y - 3)^ + (z + 1)^ = o x^ + y^ + z^ - 2x - 6y + 2z + = Chu y : Co the giai nhif sau : Goi M(x; y; z) la diem thupc mat cau t h i MA = (x + 1; y - 2; z) MB = (x - 3; y - 4; z + 2) M thupc mat cau dircfng kinh AB nen MA J_ MB MA.MB = (x + l)(x - 3) + (y - 2)(y - 4) + z(z + 2) = x^ + y'^ + z- - 2x - 6y + 2z + = Chufng minh rSng mat phSng ( P ) : 2x + 2y - z - = va mat cau ( S ) : x"^ + y^ + z^ - 2x + 2y + = tiep xuc vdi Tim toa tiep diem C H I - D A N Mat cau (S) c6 tam 1(1; - ; 0) va ban kinh r = (1) Mat khac d(I, (P)) = 292 2+ ( - ) - - = => d(I, ( P ) ) = r => dpcm V2^72^+M7 TS, VO The Huu - Nguygn Vinti Can - Tiep d i e m M cua m a t p h a n g c h i n h l a h i n h chie'u cua d i e m I t r e n (P) ^5 1 ^ Ta t i m difofc M 3'3' 137 Cho m a t cau (S) : + + + 4x - 4y + 6z + 13 = v a diem M ( - ; 2; - ) a) Chufng m i n h d i e m M nam t r e n m a t cau (S) ( b) Viet phiforng t r i n h mat phang (P) tiep xiic vdi mat cau (S) t a i diem M CHi DAN a) The toa cua d i e m M vao ve t r a i ciia phifong t r i n h m a t cau (S) de thay toa d i e m M thoa m a n phiicfng t r i n h m a t cau b) M a t cau (S) c6 tarn I ( - ; 2; - ) va ban k i n h r = Ta c6 I M = (2; 0; 0) M a t phang (P) vuong goc vdi I M , n h a n I M l a m vecto phap t u y e n T a CO : (P) : x + = 138 Cho m a t cau + y^ + + 2x - 2y + 2z + = v a m a t p h a n g (P) : 2x - 3y + 6z + = V i e t phiiOng t r i n h m a t phang (Q) d i qua d i e m M ( l ; - ; - ) song song vo'i m a t p h a n g (P) va tie'p xiic vdri m a t cau (S) CHI D A N M a t cau (S) c6 t a m I ( - l ; 1; - ) v a b a n k i n h r = M a t phang (Q) song song vdi m a t p h a n g (P) c6 phtfOng t r i n h (Q) : 2x - 3y + 6z + D = 0, D ^ Ta CO : d ( I , (Q)) = - - - +D -12+ D V ' + (-3)' + ' (Q)tiep xiicvdri m a t c a u (S): d ( I , (Q))= r D-12 = 1=> D - * V d i D - 12 = => D = 19 ( t m ) va t a di/gc m a t p h a n g ( Q i ) : 2x - 3y + 6z + 19 = * Vdri D - = - = > D = ( t m ) v a t a diTOc m a t phang (Q2) : 2x - 3y + 6z + = 139 Cho diem M ( l ; 1; V ) va m a t p h a n g (P) : 6x + 2y - 3z - = V i e t phuong t r i n h m a t cau (S), t a m M va t i e p xuc v d i (P) Hoc va on luyen theo CTOT mon Toan THPT ij" 293 CHI DAN Ta CO : d(M, (P)) = 3V7 V7 Phi/ong t r i n h mat cau (S) : (x - 1)^ + (y - 1)^ + (z - V?)' = y 140 Cho mat phSng (P) : 2x - y - 2z + 10 = va mat cau (S) : + y^ + - 2x - 4y + 2z - 19 = a) Chufng minh rang mat cau (S) va mat phang (P) cSt b) Tim tam K va ban kinh r' ciia diTcfng tron giao tuyen CHI DAN a) Ta CO mat cau (S) c6 tam 1(1; 2; - ) va b a n kinh r = d(I, (P)) = 4; b) De thay r'^ = d(I, (P)) < r => (P) v a (S) cat n h a u - d^ = 5^ - 4^ = r' = Goi K la tam ciia difofng tron giao tuyen ciia (S) va (P) t h i K la hinh 16 5^ 3' ' j chieu vuong goc ciia I tren (P) Ta tinh K Cho mp (P) : 2x - 2y - z - = va diem 1(1; 2; 3) Viet phircmg trinh mat cau (S) CO tam la diem I , cat mat phang (P) theo mot difofng tron c6 ban kinh r' = Va tim tam ciia difcmg tron giao tuyen CHI DAN Ta CO : d(I, (P)) = 2-4-3-4 72^ + {-2? -9 + {-If Ban kinh mat cau r : r^ = d^ + r'^ -3 r^ = 3^ + 4^ = 25 => r = Phirorng t r i n h mat cau (S) : (S) : (x - if + (y - 2f + (z - 3)^ = 25 Tam K ciia dirofng tron giao tuyen la hinh chieu vuong goc ciia I tren mat phang (P) Ta tinh dirge : K(3; - ; 4) X Cho dtfdng t h i n g A : =t y =1 va diem M(0; 2; 4) Viet phiTcfng z=-l-2t t r i n h mat cau (S) tam M va tiep xuc vdi A CHIDAN Ta CO : r = d(M, A) Ta tinh duac : d(M, A) = S, suy (P) : x^ + (y - 2f + (z - 4)" = l:i T S Vu The HI^J - Nguygn Vinh Can MVCLUC DAI SO VA GIAI TICH « CHUYEN DE I DAI SO TO HOP VA XAC SUAT §1 Hoan vi, chinh hap, to hap §2 Nhi thCfc Niutan §3 Xac suat CHUYEN DE II PHUdNG TRINH VA BAT PHl/dNG TRINH DAI SO §1 Phuong trinh, bat phuong trinh bac nhat mot an §2 Phuong trinh, bat phuong trinh bac hai mot an §3 Bat phuong trinh bac hai mot an §3 Phuong trinh, bat phudng trinh chifa gia trj tuyet doi §4 Phuong trinh, bat phuong trinh chifa cSn thi/c §5 phuong trinh nhieu an CHUYEN DE III PHUdNG TRINH LI/ONG 5 13 17 23 23 28 31 36 39 48 60 GIAC CHUYEN DE IV PHUdNG TRINH, BAT PHUdNG TRINH MU VA LOGARIT §1 Luy thi/a va ham so luy thil/a §2 Logarit, ham so mu, ham so logarit §3 Phuong trinh, he phuong trinh mu va logarit §4 Bat phuong trinh, h? bat phuong trinh mu va logarit CHUYEN OE V Gidl HAN - OAO HAM - KHAO SAT HAM SO VA DAO HAM §1 Gidi ban cua day so §2 Gidi han cua ham so §3 Dao ham va quy tSc tinh dao ham §1 Md dau §2 Mien xac djnh, mien gia trj §3 Su dong bien, nghjch bien cua ham so §4 Ci^c tri cua ham so §5 Gia trj Idn nhat, gia trj nho nhat cua ham so §6 Dudng ti?m can cua thi ham so §7 Su loi Idm va diem ud'n §8 Khao sat su bien thien va ve thj mgt so ham so §9 MOt so bai toan thudng gap ve thj ham so §10 Sir dung djnh |[ lagrang va tinh don dieu cua ham so de chifng minh va giai bat ding thi/c, bat phuong trinh Hoc va on luy?n theo CTDT mon Toan THPT 82 82 83 88 95 100 100 105 107 110 113 117 121 125 129 132 132 146 153 •295 CHUYEN O E VI NGUYEN HAM VA Tl'CH PHAN 160 184 § Qng dung cua tfch phan 170 § Tich phan 160 § Nguyen ham H I N H HOC CHUYEN D E I DUCiNG THANG VA MAT PHANG TRONG KHONG GIAN BAI TOAN Quan he song song 193 193 BAI TOAN Quan hO vuong goc nhOng bai toan ve khoang each BAI TOAN Cac bai toan ve goc 196 206 C H U Y E N O E II T H E TICH CAC KHOI 209 221 BAI TOAN The tich cac khoi tron xoay 209 BAI TOAN Tinh the tich cac khoi da dien C H U Y E N D E III VECTd - VECTd VA TQA 00 (TRONG MAT PHANG) 226 BAI TOAN Xac dinh tpa dp cua diem - Tpa dp cija vecto - Vecto cung phi/ong 2 BAI TOAN Tich v6 hudng cua hai vecto C H U Y E N D E IV Dl/CJNG THANG TRONG MAT PHANG 229 232 254 BAI TOAN 10 Cac van de ve khoang each va goc 241 BAI TOAN : Ducing thing vuong goc 232 BAI TOAN Viet phuong trinh dudng thing C H U Y E N D E V Dl/dNG TRON VA E L I P BAI T O A N 1 Xac 260 dinh phuong trinh dUcJng tron xac dinh tarn va ban kinh ducing tron 260 BAI TOAN 1 Elip va phuong trinh chinh tic C H U Y E N D E VI VECTd VA TQA DQ TRONG KHONG GIAN 266 268 272 BAI TOAN 15 Viet phUOng trinh mat phing 271 BAI TOAN 14 Tpa dp vectO Tich c6 hu6ng cua hai vecto 269 BAI TOAN 13 Bieu thtfc tpa dp cua tich v6 hudng cua hai vecto 268 BAI TOAN Vecto bang - Vecto cung phuong TOAN 17 BAI TOAN 16 BAI Viet phUOng trinh ducing thing Mot so bai toan lien quan den tinh vuong goc va khoang each C H U Y E N OE VII MAT CAU VA MAT PHANG BAI 296 TOAN 18 Phuong trinh m^t cau 276 282 290 290 TS VQ ThS' Hiiu - Nguygn VTnh Can vv w w 1 1 i i ^ ; i c " 1111( ) n ^ i i n > n i v n Email: baolongco 18DNguyen Thj Minh DT: 38246706 (M&i^ N G A N H A N G N G A N - 08083021 ha