^ Hi^dng din giai ' - , , •
Nh§n xet: Ta can xac djnh kich thirdc cua hlnh chiy nhat. Kich thirdc hint, ,/ chff nhat b^ng khoang each tff mot diem thuoc mot canh cua hlnh chO nhai
den canh do'i di$n.
(h.7.5) Gpi phiTdng trinh canh AB (di qua diem M(4; 5)) a(x - 4) + b(y - 5) = 0(a^ + b^ ^ 0).
Khi d6 phiTdng trlnh diTdng th^ng BC (di qua N(6;
5) vuong g6c vdi AB) la b(x - 6) - a(y - 5) = 0.
Di?n tich hlnh chff nhat la ABCD ^
a(5 -4) + b(2 - 5)| [b(2 -6) - a(l -5)| ^ 4[(a -3b)(a - h) Theo bai ra ta c6
d(P; AB).d(Q; BC) =
4(a-3b)(a-b) = 1 6 o (a-3b)(a-b) =4 (a^+b^): b = l,a = - l b = l;a = - j Vay PTcanh AB m -x + y - 1 = 0hoSc -x + 3y - II = 0
B&i 3. Cho hlnh thoi ABCD c6 phiTdng trlnh hai canh AB. AD theo thuT i\i 1^' X + 2y - 2 = 0, 2x + y + 1 = 0. Canh BD chffa diem M( 1; 2). Tim tpa dp cdc dinh
HiTdng din giai
Nh§n xet: Tim tpa dp diem A de dang. Theo tinh chS't hlnh thoi, diTdng cht cung 1^ diTdng phan giic, nen ta c6 the vie't phiTdng trlnh duTdng ch6o kc dinh A. Mat khdc, dffdng ch6o BD qua M vuong g6c vdi AC nen cung >>
dinh diTdc phffdng trlnh.
Tpa dO diem A nghi^m h$ PT [x + 2 y - 2 = 0 1 2x + y +1 = 0 '
4 5
•3*3
Gpi N(x; y) thuOc tia phSn giic AC (cua BAD).
f^hi d6 N cdch d e u hai diTcJng t h i n g AB. AD. Mat k h a c N. M ciing phia doi
y,ji dff^ng t h i n g AB va cilng phia doi vdi diTdng t h i n g AD. Ttfc la
x + 2 y - 2 2x + y + l (x + 2y-2)(l + 4 - 2 ) > 0 (2x + y + l)(2 + 2 - 2 ) > 0
\. ^
x - y + 3 = 0. '
vay PT diTdng t h i n g chffa c a n h AC la x - y + 3 = 0.
PT diTdng c h e o BD la l(x - 1) + l(y - 2) = 0 o x + y - 3 = 0
TiTd6 B (4 ;-l). D ( ^ ; 7).
TSm c u a hlnh thoi la d i e m 1(0; 3). tiT tpa dO
r4
dilm A, suy ra tpa dp d i e m C - ; — . w 3 y
Chu y : C6 the viet phiTdng trlnh canh BD b^ng cdeh: ^ Wi^t PT dffdng t h i n g qua M tao vdi hai dffdng t h i n g AB. AD hai gdc b^ng nhau.
Ta cung c6 the tlm tpa dp d i e m B, D b l n g cdch suf dung cong thffc tinh dien U'ch hlnh thoi. Gpi c a n h hlnh thoi la a. Khi dd dipn tich hlnh thoi la
8a
SABCD = 2SABD = 2(SAMB + SAMD) = a(d(M; AB) + d(M; AD)) =
1.2+2.1 4 3
r- r- =—=>sina=-
S.S 5 5
2 6 475
= a . - => a =
M4t khac SABCD =2SABD =2AB.ADsinBAD = 2a^sina , trong d6 a \h gdc
giffa hai diTdng thing AB, AD diTdc xic dinh bdi cosa=
Vly ta c6
75 5 3
TiT dd ta cung tlm diTdc tpa dp cAc diem B. D (thupc hai diTdng thing da cho each A mot khoang bkng a).
4. Cho tam gidc ABC can tai B. phffdng trlnh canh AB cd dang
>'^x-y-2>/3-0. tarn diTdng tron ngoai ticp tam gidc la 1(0; 2), B e Ox.
Tim tpa dp cdc dinh tam giac.
HiTdng dSn giai
x^t: Tpa dp diem B xac djnh de dang. Diem A cung xac djnh de dang,
™ ta sit dung tinh cha't lA = IB = R.
tim C, ta sur dung tinh cha't doi xffng cua C va A qua dffdng phan giac tam giac ABC can tai B) hoac viel phffdng trlnh dffdng thing chffa canh b^ng cdch xac djnh g6c giffa dffdng thing BC va true hoanh.
Ll^n gidi di^ ln/<1c ky Ihi />// < iniin Bdc, Trun^. Sam IK::: < r , Nguyln VOn TMng
^ Tpa dp B(2; 0). Goi A(&;S&-2y[3)
vdia?t2. Ta c6: IA = IB
•y I r /- \ Fa ==2 (loai) o a ^ + ( N / 3 a - 2 V 3 - l ) =8 o ^
^ ' [a = l + 73
V$y tpa do diem A(I + >/3;3-N/S).
' Do g6c giCfa IB vdi true hoinh \k 45", g6c giffa AB vdi true ho^nh b^ng 60", suy ra iBC = 30"
hay gdc giffa BC vdi true ho^nh \l 30". Vay hp
so g6e eua difcJng thAng B C k = tan30" = -|=. ;( , . Vay phffdng trinh dffdng thing chffa eanh BC y = -U(x - 2).
v3
\ o
^ c
; i-
Gpi tpa dp diem C ( c-2) vdi c ?t 2.
Ta c6 IC = IB o + 1 ( c- 2 ) - 2 \ = 8: e = 2 (loai) . c = ^/3-I
Vay c( V 3 - l ; I - > / 3 ) .
Bai 5. Cho tarn giic ABC c6 phiTdng trlnh canh BC la y = 2. dinh A thupc di/tlnJ thing x + y - 2 = 0va dipn tich tam giac la . Tim tpa dp edc dinh cu|
tam gidc ABC, biet A c6 ho^nh dp dffdng. ' Htfdng din giai
Nh§n xet: Tff gid thiet tam gidc ABC deu va dipn tich da biet, ta xac djnl difdc dp d^i canh cua tam gidc, tff dd tinh dffdc dp d^i dffdng cao AH.
khdc, AH bSng khoang cdch tff dinh A den dffdng thing BC, tff dd ta tfn^
dffdc tpa dp dinh A. Do BC song song vdi true ho^nh va AH vuong go*- true ho^nh nen dffa v^o dp dii canh da biet ta de d^ng xac dinh dffdc c u a B , C .
(h.7.7) Gpi a la dp d^i canh tam gidc deu ABC.
2 41
Ta CO dipn tich tam giac 1^ S^BC ^ * "4"
AH = a. : ^ = x/2.
2 8.
=>a = , - '3 \
po dinh A thupc dffdng thing x + y - 2 = 0 nen A(t; 2 - t).
jChi <16 AH = d(A; BC) = |(2 -1) - 2| = |t| = >^ I = po A cd ho^nh dp dffdng nen A( > ^ ; 2 - > ^ ) .
pifa v^o hinh ve, ta de suy ra tpa dp dinh:
B = B
\
= c
V 4i
3
>/6
;2
t i v i i i w : ; Hinh7.7 '
pal 6. Cho lam giac ABC vdi dffdng cao AH cd phffdng trlnh x = 3V3 , phffdng trinh hai dffdng phan giac Irong gdc ABC v^ ACB Ian Iffdt 1^ y = — x ;
\/3 y = —|=x + 6. Bdn kinh dffdng trdn npi tiep tam gidc b^ng 3. Viet phffdng
v3
trinh cdc canh cua tam gidc, bie't dinh A cd lung dp dffdng.
Hffdng dSn giai
Nhqn xet: Ta thay hai dffdng phan gidc v i dffdng cao dong quy tai mot diem, canh BC song song hoSc trilng vdi true hoanh. Hai dffdng phan giac lao vdi true ho^nh hai gdc b^ng nhau nen tam gidc n^y can tai A.
(h.7.8) Do dffdng cao AH cd
phffdng trinh \ 3\f3 nen 1 dffdng thing B C song song
hoac tr&ng vdi true ho^nh.
Hai dffdng phan giac tao vdi true hoanh hai gdc b^ng nhau
b k g 30" k = ±
y = -
, nen
Hinh 7.8 tam giac A B C deu.
Tam dffdng iron npi tiep la l(3>/3;3). Khoang each tff I den B C bing 3, nen phffdng trinh dffdng thing B C la y = 0 hoSic y = 6. Neu phffdng trlnh dffdng
* l n g B C m y = 6, thi lung dp cua diem A Ik -3 (loai).
^ay phffdng trinh canh B C 1^ y = 0.
Toa dp cic diem B va C IJl B(0; 0), C(6N^;0) . ' •
dffdng thing A B cd hp so g6ck = yf3,vh dffdng thing C A cd h^ so' gdc P'= - 7 3 , vay phffdng trlnh cua chiing Ian lff(?tm y = V3x; y = -V3x + i8.
Luy?n gi&i 3S IfUOt Jg> IM UH J mten-BiB^TTimgrnam lomnpc- lyguym van i nong
%huyht M IS:
B A D t f d N G C 6N I C ( E U P , H Y P E R B O L , P A R A B O L ) I (Gidi tfafch: t f a e o chiftfng tdnh mdi)
1. T 6 M T A T Li T H U Y E T ^ '
A.EUP ' M / ' ^ -f^:' V. . ^a..,!:,.;
/. Dfnh nghta Elip: . v
- Trong mat p h i n g cho hai diem co dinh F i , F 2 vdi F 1 F 2 = 2 c > 0. Tap hpp diem M cua m5t p h i n g sao cho ta luon c6 M F i + M F 2 = 2a (a > c) (a la h^n so) goi Ik mpt elip.
- Hai diem F,, F 2 gpi Ik tieu diem cua elip. ' '^^^ " ' ' ^ ' - 2c gpi 1^ tieu c\i cua elip.
- Neu M n^m tren elip thi M F i , M F 2 gpi 1^ cdc bdn kinh qua tieu d i l m cua M.
2. Phuang trinh chtnh tdc cua elip.
- Chpn he tpa dp Oxy sao cho Fi vh F 2 c6 c^c tpa dp F i ( - c ; 0), F2(c; 0). Khi d(
2 ..2
elip c6 phi/dng trinh ^ + ^ = 1 (b^ = a^ - c^) (1) a^ b^
( 1) gpi \k phiTcfng trinh chinh t^c cua elip. \ Elip c6 bon dinh A , ( - a ; 0). A 2( a ; 0), B,(0; - b ) , 6 2 ( 0 ; b).
2a gpi Ik dp d^i ciia true Idn A 1 A 2 ; 2b gpi Ik dp d^i cfla true nh6 B 1 B 2 cua elip F|(-c; 0), F2(c; 0) la hai tieu diem cua elip.
Ta CO cong thufc sau de tim cic bdn kinh qua tieu: Ne'u M ( X o ; yo) n^m tren elip thi MF, = a + ^ ; M F j = a - ^0
a a
- Dai liTdng e = — goi la tam sai cua elip. NhiT vay 0 < e < 1.
a
- Hinh chff nhat gidi han bdi ckc difdng t h i n g x = ±a, y = ±b gpi 1^ hinh chff nhat cd sd cua elip.
- Elip c6 hai di/dng chuan:
+ DiTcJng chuan x = - la difcfng chuan tfng vdi tieu diem F2(c; 0) e
+ DiTcfng chuan x = - - la di/dng chuan uTng vdi tieu diem F i ( - c ; 0) a e
3. Dinh It
Neu M(x,); y„) thupc elip. K i hi?u M H : , M H 2 tiTdng tfng Ik ckc khodng <- tir M den hai diTdng chuan x = - - . x = - . K h i d6: i!lS_^J^, vdi ni'''
^ e e M H 2 M H ,
diem M .
piSu ki$n di dudng thdng tigp xiic vdi elip.
pi/^jng t h i n g A x + By + C = 0 la ti^p tiiyen da elip ndi trdn (1) khi vk chi
' ^ h i: A V + B V = C^ w ,
^ f)inhnghia hypebol
Tr6n mat p h i n g cho hai diem co dinh F, va F 2 , vdi F 1 F 2 = 2 c > 0. Tap hdp c^c diem M cua mat p h i n g sao cho |M F , - M F J] = 2a (trong do a la mot so difdng khong ddi nho hdn c) gpi la mpt hypebol.
F I va F 2 gpi la cac tieu diem cua hypebol. ^ Khoang each F 1 F 2 = 2c gpi la tieu ciT cua hypebol.
N^u M n^m tren hypebol, thi MF|, M F 2 gpi la ckc ban kinh qua tieu diem c u a M . . ^ . j
2. Phuang tiinh chtnh tdc vdcdcyiutdcim hypebol.
Chpn he tpa dp Oxy sao cho F|(c; 0) khi do phi/dng trinh cua hypebol la - ^ - ^ = 1 (2) v d i b ' = c ' - a '
a^ b^
phi/dng trinh tren gpi la phiTdng trinh chinh t i c cua hypebol. i j uv, - Cic diem A i ( - a ; 0) va A 2( a ; 0) gpi la cac dinh ciia hypebol t ^
Ox gpi la true thiTc, Oy gpi la true ao cua hypebol (2) (do no khong c i t true Oy);
Fi(-c; 0) va F2(c; 0) gpi la hai tieu diem cua hypebol, con 2c gpi la tieu ciT hypebol;
•2a gpi la dp dai true thiTc, c6n 2b la dp dai cua true a o ; ' y = — x , y = - — X la hai dirdng tipm can cua hypebol;
a a '^nf'' •"' ' ' Hinh chi? nhat tao bdi cac diTcfng t h i n g x = ±a, y = ±b gpi la hinh chff nhat cd
sdciia hypebol. cjii, .vc , Cac cong thtfc ban kinh qua tieu
Neu x„ > 0 thi MF, = a + ; MF2 = - a + ^
a a , Neu x„ < 0 thi MF, = - a ; MF2 = a - - ^
a a • ••
sai e ciia hypebol diTdc djnh nghia e = - . NhiT vay vdi hypebol ta c6 e > I . a
^u'clng chuan cua hypebol:
EJiTcJng chua'n x gpi la diTdng chuan cua hypebol tiTdng tfng vdi tieu diem
P 2( c ; 0 ) ^ .
Luyen gfM di truOc thi DH 3 miin Bdc, Trung, Nam Todn hoc - Nyuvl^n Van ThOng
+ Di/dng chuan x = — g o i 1^ diTdng chuan c u a hypebol ttfdng uTng vdi ij^^
diemF |(-c ;0)
+ Goi H,, H2 tiTdng lirng 1^ hlnh c h i c u tCf M d e n hai diftlng c h u a n x = , x - ^ c
MH, MH,
K h i d o t a c o : llll-L^ L=e. , rfn!' • • - - t : "nhfin
3. Diiu ki^n di dudng thdng tiSp xOc vdi hypebol
- Dc dirdng thdng Ax + By + C = 0 (A^ + > 0) la ticp tuycn vdi hypcrbol (i) dieu kiOn can va dij la AV - B V =
C.PARABOL im^^-^U'ifi'^i^^^
1. DinhnghJa. ^'
- Parabol la tap hcfp cdc diem cua mat phdng each deu mot dirc:>ng thang A (.6 djnh va mot diem F co djnh khong thupc A .
+ Diem F di/dc goi la tieu diem cua parabol.
+ DiTclng thdng A goi li di/dng chuan cua parabol.
2. Phuang trlnh chtnh tdc cm parabol
Chpn he true toa do: True Ox 1^ difcJng thdng di qua tieu diem F va vuong goc vdi drftfng chuan A, hi/dng difdng tif P den F (P \h giao diem cua Ox vdi
A). True Oy la trung trifc cua PF.
- Trong hp true niy F , P
\2 ) \ )
- Difdng chuan A c6 phifOng trlnh * = " ' ^
- Phirpng trinh c u a parabol \k: y^ =2px v^ goi Ik phiTdng trinh chinh tac cua
parabol
- p > 0 goi la tham s6 tieu (chu y ring p b ^ng khoang cdch tif tieu diem F den
diTiJng chuan A).
- Neu M(x; y) nkm tren parabol y^ = 2px. thi MF goi Ik hkn kinh qua tieu CD^
diem M. Ta co cong thiJc de tinh ban kinh qua tieu: MF = x +
3. Vdi dQng phuang trinh khdc cua parabol
• y^ = -2px
Tieu diem F ; dirCJng chuan " = ^
• x^ = 2py
Tieu diem F 0;il
, 2)
; diftlng chuan y = —j
:-2py
jgu diem F ; dirdng chuan y = —
^ piSu ki$n tiip xuc ciia dudng thdng vdi parabol
' Ax + By + C = 0, (A^ + B^ > 0) Ik tiep tuyen cua parabol y^ = 2px dieu Ici^ncan vadu ia:pB^ = 2AC
Chti y: Vdi parabol c6 phiTdng trlnh dang y^ = -2px, (p > 0) thi dieu kipn ticp
xdc la-pB^ = 2AC. 1
PHl/CfNG TRiNH C A C D l / d N G C d N I C