Phucntg phap giai ToAn Ilinh hoc theo chuycn lic- Nguyen Pliii Khanh, Nguyen Tat Thii 3 T u E-6;3 ve hai tie'p tuye'n EA, EB A, B la tie'p diem den C.. Tu dp suy ra I, G, H thang hang;
Trang 1T R U N G T A M L U Y C N T H I D A I H O C V I N H V I £ N S A I G O N
Tdng chu bi§n: PHAM H 6 N G D A N H NGUYEN PHU KHANH - NGUYIN TAT THU NGUYEN TAN SIENG - TRAN VAN TOAN - NGUYEN ANH TRUCfNG
(Nhdm giao vien chuyen luyen thi B^i hpc)
PHUONG PHAP GIAI TOAN
HtNH HOC
Trang 2Chiu trdch nhiem xuat ban
Gidm doc - Tong bi&n tap : TS P H A M THj T R A M
Bien tap : N G Q C L A M
Che ban : C O N G TY K H A N G V I E T
Trinh bay bia : C O N G TY K H A N G V I E T
Tong phdt hanh va doi tdc lien ket xuat ban:
I Tpa dp trong mat phang
• Cho u ( x p y j ) ; v(x2;y2) va k e R K h i do:
1) u + v = (xi + X 2 ; y i + y 2 ) 2) u - v = ( x i - X 2 ; y i - y 2 )
3) k u = ( k x i ; k y i ) 4) Z=Jx\+y\) u=vc ^r^ "''""''^
6) U V = X]X2 + y ] y 2 = > u l v < ; : > u v = 0<=> \-^\2 + y ] y 2 = 0
• H a i v e c t a u ( x j , y j ) ; v ( x 2 ; y 2 ) ciing phirang v a i nhau <=>
• Goc giija hai vec to u ( x j , y j ) ; v ( x 2 ; y 2 ) :
U V X i X 2 + y i y 2
c o s ( u , v ) =
u V Cho A ( x ^ ; y ^ ) ; B(xB;yB) K h i do :
• Cho tarn giac A B C v o i A{x^;y^), B(xB;yB), C{x^;y^) K h i do trong tarn
G ( x( , ; y g ) ciia tarn giac A B C la :
V _ X A + X B + X C
X G - ^
II PhirotTg trinh duong thang ,, ,^,
1 'Phuang trinh duong thdng 1.1 Vec to chi phucmg (VTCP), vec to phdp tuyen (VTPT) cua duong thang:
Cho d u o n g thang d
• n = (a;b) ?t 0 goi la vec to phap tuyen cua d neu gia ciia no v u o n g v o i d
Trang 3• u = ( u j; u 2 ) ^ 0 goi la vec ta chi phuong cua d ne'u gia cua no trung hoac
song song voi duong thang d
Mot duong thang c6 v6 so VTPT va v6 so VTCP ( Cac vec to nay luon cung
phuong voi nhau)
• Moi quan he giua VTPT va VTCP: n.u = 0
• Ne'u n = (a; b) la mpt VTPT cua duong thang d thi u = (b; -a) la mot VTCP
cua duong thang d
• Duong thang AB c6 AB la VTCP
1.2 Phuwig trinh dumig thang
1.2.1 Phuatig trinh tong qudt cua duong thang:
Cho duong thMng d d i qua diem A(xQ;yQ) va c6 n = (a;b) la VTPT, khi do
phuong trinh tong quat ciia d c6 dang: a(x - X Q ) + b(y - yp) = 0
1.2.2 Phuovg trinh tham so cua duong thang:
Cho duong thSng d di qua diem A(xo;yo) va c6 u = (a;b) la VTCP, khi do
X = X Q + at phuong trinh tham so cua duong thang d la: , t G R
[y = y ( , + b t
2 Vi tri tuang doi giua hai duang thdng
Cho hai duong thcing dj : a^x + bjy + c^ = 0; d2 : a2X + b2y + C2 = 0 Khi do vi tri
|a,x + b,y + Cj = 0 tuong doi giua chung phu thuoc vao so nghiem cua h^ : < , (I)
[a2X + b2y+ C2 =0
• Neu (I) v6 nghiem thi d^ / /d2
• Ne'u (I) v6 so nghiem thi d^ = d j
• Ne'u (I) CO nghiem duy nha't thi dj va d2 cat nhau va nghiem ciia he la toa
do giao diem '
3 Goc giua hai dijcang thdng
Cho hai duong thang dj : a j X + b^y+ Cj =0; d2 :a2X + b2y + C2 = 0 Goi a
la goc nhon tao boi hai duong thang dj va d2
Ta CO : cosa = aja2 + bjb2
^/a^Tb^ ^/af+b
4 JChodng each tit mot diem den ducrng thdng
Cho duong th5ng A : ax + by + c = 0 va diem M ( X Q ; y ^ ) Khi do khoang each
tu M den A dugc tinh boi cong thuc:
Cty TNHH MTV DWH Khang Viet
d(M,(A)): axp + byp + c
Va^+b^
5 (phuong trinh duang phdn gidc cua goc tao boi hai duang thdng
Cho hai duong thang d^ : a^x + b^y + c^ = 0 va d2 : ajX + b2y + Cj = 0
Phuong trinh phan giac ciia goc tao boi hai duong thang la: - , v , •
a j X + b^y + Cj a2X + b2y + C2
+ ^ / a ^ ^ b [ • , i c - i ; ;
1 <Phuang trinh duang tron:
Cho duong tron (C) tam I(a; b ) , ban kinh R, khi do phuang trinh ciia (C) la: ( x - a) 2 + ( y - b ) 2 = R 2
Ngoai ra phuong trinh: x^ + y ^ - 2 a x - 2 b y + c = 0 voi a ^ + b ^ - o O cQng
la phuong trinh ciia duong tron c6 tam I(a;b), ban kinh R = Va^ + b^ - c
2 Phuang trinh tiep tuyen:
Cho duong tron ( C ) : ( x - a ) ^ + ( y - b ) ^ = R ^
• Tiep tuyen A ciia (C) tai diem M la duong thang d i qua M va vuong goc vai I M
• Duong thang A : Ax + By + C = 0 la tiep tuyen ciia (C) <=> d(I, A) = R '
• Duong tron ( C ) : (x - a ) ^ + (y - b)^ = R^ c6 hai tiep tuyen cung phuong voi
Oy la x = a ± R Ngoai hai tiep tuyen nay cac tiep tuyen con lai deu c6 dang:
y = kx + m
IV E lip
1 'i)inh nghra.-Trong mat phang cho hai diem co'djnh Fi ,F2 c6 Y^Yj =2c Tap
hop cac diem M cua mat phang sao cho MF^ +MF2 =2a (2a khong doi va
a > c > 0) la mot duong elip
• F,,F2 : la hai tieu diem va 2c la tieu cu ciia elip
• MF|,MF2 : la cac ban kinh qua tieu
2 Phuang trinh chinh tdc cua elip:
4 + 4 = ^ voi b^=a^-c^ K'
a 2 b^
Vay diem M(xo;y(,) e (E) • = 1 va <a Yo < b ,
Trang 4Phumtg phcip giiii Toan Hhih hoc theo chuyen tie- Nguyen Phu Khdnh, Nguyen Tat Thu
3 Tinh chat v>d hlnh dang cua elip: Cho (E): — + ^ = 1 , a > b
a b
• True doi xung Ox,Oy Tarn dói xiing O j ,
• Dinh: A[(-a;0), A2(a;0), 6^(0;-b) va 62(0; b ) ÂA2 = 2a goi la do dai
true Ion, B]B2 = 2b goi la do dai true bẹ
• Noi tiep trong hinh ehir nhat co so PQRS
C O kích thuoc 2a va 2b voi b^ = â - ệ
1 ^inh nghia: Trong mat phang voi h$ toa do Oxy eho hai diem Fi, F2 eo
FjF2 =2c Tap hop cac diem M ciia mat phSng sao eho MF^ - M F j =2a (2a
khong doi va c > a > 0 ) la mpt Hypebol
• Fp F2 : la 2 tieu diem va F|F2 = 2e la tieu eụ
• 1VIF[,MF2 : la eac ban kinh qua tieụ
2 'Phimng trinh chinh idc cua hypebok x^ y^
â
= 1 voi h^=c^-ậ
3 Tinh chat vd hlnh dang cua hypebol (fi):
• True doi xung Ox (true thuc), Oy (true ao) Tam doi xung O
• Dinh: Aj(-a;0), A2 (a;0) D Q dai true thuc: 2a va do dai true ao: 2b
• Tieu diem Fi(-e; 0), Fj ( c; O)
• Hai tiem can: y = ± —x
a
• Hinh eho nhat co so PQRS c6 kieh thuoe 2a, 2b voi b^ = c^ - ậ
• Tam sai: e = — =
a
• Hai duong chuan: x = ±— = ± —
Cty TNHH MTV DWH Khang Viet
• D O dai cac ban kinh qua tieu cua M ( x o ; y ( , ) e ( H ) : +) MF^ = ex„ + a va MF2 = e X ( , - a khi X Q > 0 +) MFj = -exp - a va MF2 = -exp + a khi X Q < 0
Parabol la tap hop cae diem M cua mat phang each deu mot duong thang
A c o ' d i n h v a m o t diem F co dinh khong thuoe A
A : duong chuan; F : tieu diem va d(F,A) = p > 0 la tham sótieụ
2 'Phuxmg trinh chinh tdc cua ^arabd: = 2px 3.jrinh dang cua Parabol (<P):
• True Ox la true dói xung, dinh Ọ Tieu diem F ( ^ ; 0 )
• Duong chuan A : x =
• M ( x ; y ) e ( P ) : MF = x + ^ voi x > 0
B, CAC BAI THlfONG GAP
§ 1. cAc B A I T O A N C O B A N
1 Xg.p phuang trinh duang thang
De lap phuong trinh duong thang A ta thuong dung cac each sau
• T i m diemM(xo;yo) ma A di qua va mot VTPT n = (a;b) Khi do phuong trinh duong thang can lap la: ăx - X Q ) + b ( y - yp) = 0
• Gia su duong thang can lap A : ax + by + e = 0 Dua vao dieu kien bai toan ta tim dugc a = mb,c = n b Khi do phuong trinh A : m x + y + n = 0 Phuong phap nay ta thuong ap dung doi voi bai toan lien quan den khoang each va goe
• Phuong phap quy tich: M(xQ;yQ)e A:ax + by + e=^Oc:> axy + by^ + e = 0
Vidu 1.1.1.Trong mat phSng voi he toa do Oxy cho duong tron
( C ) : ( x- ] ) 2 + ( y - 2 ) 2 = 2 5
1) Viet phuong trinh tiep tuyen ciia (C) tai diem M(4;6), ' 2) Viet phuong trinh tiep tuyen cua (C) xuát phat t u diem N ( - 6 ; l )
Trang 5Phucntg phap giai ToAn Ilinh hoc theo chuycn lic- Nguyen Pliii Khanh, Nguyen Tat Thii
3) T u E(-6;3) ve hai tie'p tuye'n EA, EB (A, B la tie'p diem) den (C) Viet
phuong trinh d u o n g thang A B
D u o n g tron (C) c6 tam 1(1; 2), ban kinh R = 5
1) Tie'p tuyen d i qua M va vuong goc v o i I M nen nhan I M = (3;4) lam VTPT
Nen p h u o n g trinh tie'p tuye'n la: 3(x - 4) + 4(y - 6) = 0 <=> 3x + 4y - 36 = 0
2) Gpi A la tie'p tuye'n can t i m
Do A d i qua N nen p h u o n g trinh c6 dang
A : a ( x + 6) + b ( y - l ) = 0<=>ax + by + 6 a - b = 0, a^ + b^ (*)
Ta c6:
7a+ b d(I,A) = R o
l A N A = 0 [(a - l)(a + 6) + (b - 2)(b - 3) = 0
= ^ 7 a - b + 20 = 0
T u do ta suy ra duoc A e A : 7 x - y + 20 = 0
Tuong t u ta cung c6 dug-c B e A = > A B = A = > A B : 7 x - y + 20 = 0
2 Cdch lap phimng trinh dizcrng tron
De lap p h u o n g trinh d u o n g tron (C) ta thuong su dung cac each sau
Cdch 7;Tim tam I(a;b) va ban kinh ciia d u o n g tron K h i do p h u o n g trinh
d u o n g tron co dang: (x -a)^ + ( y - b)^ =
Cdch 2;Gia su p h u o n g trinh d u o n g tron co dang: x^ + y^ - 2ax - 2by + c = 0
8
Cty TNHH MTV DWH Khang Viet
Dua vao gia thie't cua bai toan ta tim dugc a,b,c Cach nay ta t h u o n g ap dung khi yeu cau viet phuong trinh d u o n g tron di qua ba diem
Vi du 1.1.2. Lap p h u o n g trinh d u o n g tron (C), bie't 1) (C) d i qua A(3;4) va cac hinh chie'u ciia A len cac true toa do
2) Goi I(a;b) la tam ciia d u o n g tron (C), v i l € ( C i ) nen: ( a - 2 ) + b = - (1)
Do (C) tie'p xuc voi hai d u o n g t h i n g A ^ A j nen d(I, A j ) = d(I, A2)
a - b a - 7 b
<=>b = -2a,a = 2b
• b = -2a thay vao (1) ta CO dugc:
(a - if- + 4a^ = - <=> 5a^ - 4a + — = 0 phuong trinh nay v6 nghiem
• a = 2b thay v a o ( l ) taco: ( 2 b - 2 r + b ' ' = - < : : > b = - , a = -
o 0 0
Suy ra R = D ( I , A , ) = Vay p h u o n g trinh ( C ) :
3 Cac diem, ctqc biet trong tam gidc
Cho tam giac ABC K h i do:
( 8l 2 r 4^ ' 8 -:l.:J
x — + y
-I 5j 5 , 25
Trang 6Phumig phdpgidi Todn Hiith hoc theo chiiyen de - Nguyen Phi't Klidnh, Nguyen Tat Thu
• Trong tam G
• True tam H :
3 ' 3 AH.BC = 0
BH.AC = 0 Tam duong tron ngoai tiep I: lA^ = IB^ lA^ = IC^
• Tam duong tron noi tiep K :
Chu y:C6 the tim K theo each sau:
* Ta CO AK = KD tu day ta c6 K
BD ^ Tam duong tron bang tiep (goc A) J: AB.AJ AC.AJ AB AC
BJ.BC AB.BJ
BC AB l?jdui.i.3.Cho tam giac ABC c6 A(1;3),B(-2;0),C 5 3
1) Tim toa do true tam H, tam duong tron ngoai tiep I va trong tam G cua
tam giac ABC Tu dp suy ra I, G, H thang hang;
2) Tim toa do tam duong tron noi tiep va tam duong tron bang tiep goc A
cua tam giac ABC
AH = (x-l;y-3),BH = (x + 2;y),BC = 21 3 ,AC = ( 3 _21 8' 8
CUj TNHH MTV DWH Khang Viet
Ma < AH.BC = 0 nen ta eo BH.AC = 0
3 1
7(x-l) + (y-3) = 0 j7x + y-10 = 0 (x + 2) + 7y = 0 [x + 7y + 2 = 0
3
X = —
2
y = -: 1 2' 2
Suy ra H Goi I(x;y), taeo:
Ma AK = (x-l;y-3),BK = (x + 2;y),AB = (-3;-3) nen (*) tuong duong voi -3(x-l)-3(y-3) - 8 ^ ^ - ^ ) - f ^ y - ' ^
3(x.2).3y 8 ^ - " ' ^ " ^
2x - y = -1 x = 0 x-2y = -2 [y = l
8 ^ Vay K(0;1)
Goi J(a;b) la tam duong tron bang tiep goc A eiia tam giac ABC Ta co:
Trang 7Phuvng phlip gidi Todn Hinh hoc theo chiiyen dc- Nguyen Phu Khdnh, Nguyen Tat Tltu
BC ~ AB
2a - b = - 1 2a + b = -4
4 Cdc duang ddc hiet trong tam gidc
4.1. D u a n g trung tuyen cua tam giac: K h i gap duong trung tuyen cua tam
giac, ta chu yeu khai thac tinh chat d i qua dinh va trung diem cua canh do'i dien
4.2. D u o n g cao cua tam giac: Ta khai thac tinh chat d i qua d i n h va vuong
goc voi canh do'i dien
4.3. D u o n g trung true cua tam giac: Ta khai thac tinh chat d i qua trung
diem va vuong goc voi canh do
4.4. D u o n g phan giac trong: Ta khai thac tinh chat ne'u M thuoc AB, M ' doi
xung voi M qua phan giac trong goc A thi M ' thuoc A C
Vidu 7.i.4.Trong mat ph^ng v o i he tpa do O x y , hay xac d j n h toa do d i n h C
cua tam giac ABC bie't rang hinh chie'u vuong goc cua C tren d u o n g thang
AB la diem H ( - l ; - l ) , d u o n g phan giac trong cua goc A c6 p h u o n g trinh
x - y + 2 = 0 va d u o n g cao ke t u B c6 phuong trinh 4x + 3y - 1 = 0
JCffigidi
K i hi?u d , : X - y + 2 = 0, d2 : 4x + 3y - 1 = 0
Goi H ' la diem doi x u n g voi H qua d j K h i do H ' E A C
Goi A la d u o n g thang d i qua H va vuong goc v o i d j
x + y + 2=::0 Phuong trinh cua A : x + y + 2 = 0 Suy ra A n d j = I :
x - y + 2 = 0 I(-2;0)
Nen A C n d j = A :
Ta CO I la t r u n g diem ciia H H ' nen H ' ( - 3 ; l )
D u o n g thang A C d i qua H ' va vuong goc voi d j nen c6 p h u o n g trinh :
•A(5;7)
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Cty TNHH MTV DWH Khang Vie
Vi du 1.1.5. Trong mat phang vai he toa do Oxy , cho tam giac ABC biet
A ( 5 ; 2 ) Phuong trinh d u o n g trung true canh BC, d u o n g t r u n g tuyen C C Ian l u ^ t la x + y - 6 = 0 va 2 x - y + 3 = 0 T i m toa do cac d i n h B,C cua tam giac ABC
Xgfi gidi
Goi d : x + y - 6 = 0, C C : 2 x - y + 3 = 0 Ta c6: C(c;2c + 3) Phuong trinh BC : x - y + c + 3 = 0
Goi M la t r u n g diem ciia BC, suy ra M :
3 ' 3 , C
14 37
3 ' 3
5 Mot sobdi todn dung hinh ca ban
5.1. H i n h chie'u v u o n g goc H cua diem A len d u o n g thang A
• Lap d u o n g thang d d i qua A va vuong goc v o i A
• D u n g r doi x u n g v o i I qua d u o n g thang A
• D u o n g tron ( C ) c6 tam I ' , ban kinh R
Chii y: Giao diem ciia (C) va ( C ) chinh la giao diem cua va A
5.4. D u n g d u o n g thang d ' doi xung voi d qua d u o n g thang A
• Lay hai diem M , N thuoc d D u n g M ' , N ' Ian luot d o i x u n g v o i M , N qua A
'if!', r<(.:
• d ' = M ' N '
Trang 8Phumig phdp gidi Todii Uinh hoc theo chuyen dc - Nguyen Pliii Khdnh, Nguyen Tat Thti
Vidu 1.1.6.Trong mat phang Oxy cho d u o n g thang d : x - 2 y - 3 = 0 va hai
diem A(3;2), B ( - l ; 4 )
1) T i m diem M thuoc d u a n g thang d sao cho M A + M B nho nhat,
2) Viet p h u o n g t r i n h d u a n g thang d ' sao cho d u o n g thang A : 3x + 4y + 1 = 0
la d u o n g phan giac ciia goc tao boi hai d u o n g thang d va d '
JCffigidi
1) Ta tha'y A va B n a m ve m o t phia so v o i d u o n g thang d Goi A ' la diem doi
x u n g v o i A qua d K h i do v a i m o i diem M thuoc d, ta l u o n c6: M A = M A '
V i A la phan giac cua goc h g p bai giiia hai d u a n g thang d va d ' nen d va
d ' do'i x u n g nhau qua A , do do l e d '
'3 _ 1 6 ' 5 ' " 5
Lay E(3;0) G d , ta tim dugc F la d i e m do'i x i i n g v a i E qua A , ta c6
F e d ' Suy ra FI = (2 U
5 ' 5 , do do p h u o n g trinh d ' : l l x - 2y - 1 3 = 0
Cty TNHH MTV DWH Khang Viet
CP BAI TAP Bai l - l - l - Trong mat phang Oxy cho tam giac ABC CO A ( 2 ; l ) , B(4;3), C ( - 3 ; - l )
1) T i m toa do true tam, tam d u o n g tron ngoai tiep tam giac A B C 2) Viet p h u o n g t r i n h d u o n g tron ngoai tiep tam giac ABC
Jiuang ddn gidi
1) Goi H ( x ; y ) la true tam tam giac ABC, ta c6: A H B C = 0
BH.AC = 0
'(x - 2)(-7) + (y - 1 ) ( - 4 ) = 0 J7x + 4y - 1 8 = 0 (x - 4)(-5) + (y - 3)(-2) = 0 ^ [Sx + 2y - 26 = 0 ^
X = 34
y = -46 Vay H 34 46
Goi I ( x ; y ) la tam d u a n g tron ngoai tiep tam giac ABC, ta c6:
Bai 1.1.2. Trong mat phang toa do Oxy cho tam giac ABC c6 A(3;2) va
p h u o n g t r i n h hai d u a n g trung tuyen B M : 3x + 4y - 3 = 0 , C N : 3x - lOy - 1 7 = 0 Tinh toa do cac diem B, C
Jiuang dan gidi : ? ; • ;
Goi G la trong tam ciia tam giac, suy ra toa do ciia G la nghiem cua he '3x + 4y - 3 = 0
3 x - 1 0 y - 1 7 = 0
7
^ = 3 [ y = - l
> ; r J J ' I ' i
Trang 9-Phumig phdpgiiii Toan Hitih hoc theo chuyen de- Nguyen Phi'i Khanh, Nguyen Tat Thu
Goi E la trung diem ciia BC, suy ra EA = - G A => E(2;
Gia sir B(a;b), suy ra C ( 4 - a ; - 5 - b ) T u do ta c6 h^:
Bai 1.1.3. Trong mat phang toa do Oxy cho tam giac A B C c6 A ( - 3 ; 0 ) va
p h u o n g trinh hai d u o n g phan giac trong B D : x - y - 1 = 0,CE : x + 2y +17 = 0
Tinh toa do cac diem B, C
Jiu&ng ddn gidi
Gpi A^ d o i x i i n g v o i A qua BD, suy ra A j e BC va A ^ ( l ; - 4 )
A j do'i x u n g v o i A qua CE, suy ra A 2 e BC va A 2 ( - — ; - — )
5 5 Suy ra p h u o n g trinh BC : 3x - 4y - 1 9 = 0
x - y - l = 0 Toa dp B la nghi^m cua he:
Toa do C la nghiem cua he:
Bai 1.1.4.Trong mat phSng toa do Oxy cho tam giac A B C c6 C(5;-3) va
p h u o n g trinh d u o n g cao A A ' : x - y + 2 = 0 , d u o n g trung tuyen
B M : 2x + 5y - 1 3 = 0 Tinh toa do cac diem A , B
Jiixang ddn gidi
Ta CO p h u o n g trinh BC: x + y - 2 = 0
fx = - l Suy ra toa do ciia B la nghiem cua he: x + y - 2 = 0
2x + 5 y - 1 3 = 0 l y = 3 • B ( - l ; 3 ) Gpi A(a;a + 2), suy ra toa do ciia trung diem A C la M + 5 a - 1 ^
M a M e B M nen 2 ^ y ^ + 5 ^ - 1 3 = 0 « a = 3 =^ A ( 3 ; 5 )
Vay A(3;5),B(-1;3)
Bai 1.1.5. Trong mat phang toa dp Oxy cho tam giac A B C CO B(l; —3) va
p h u o n g trinh d u o n g cao A D : 2 x - y + 1 = 0, d u o n g phan giac C E : x + y - 2=::0
.Tinh toa dp cac diem A , C
Cty TNHH MTV DWH Khang Viet Jiic&ng ddn gidi
Ta CO p h u o n g trinh B C : x + 2y + 5 = 0
[x + y - 2 = 0 [x = 9 Tpa dp diem C la nghiem '^"^ L ^ 2y + 5 = 0 ^ |y = - 7 Gpi B' la diem d o i x u n g v o i B qua CE, suy ra B'(5;l) va B' e A C
Bai 1.1.6. Trong mat phang v o i h^ tpa dp Oxy, cho tam giac ABC co M (2; 0)
la trung diem cua canh AB D u o n g trung tuyen va d u o n g cao qua d i n h A Ian
lupt CO p h u o n g trinh la 7x - 2y - 3 = 0 va 6x - y - 4 = 0 Viet p h u o n g trinh
duong thang A C
Jiu&ng ddn gidi
| ' 7 x - 2 y - 3 = 0 Toa do A thoa m a n he: <^
• • [ 6 x - y - 4 = 0
V i B do'i xiing v o i A qua M nen suy ra B = (3; - 2 )
D u o n g thSng BC d i qua B va vuong goc v o i d u o n g thSng: 6x - y - 4 = 0 nen suy ra
Phuong trinh B C : x + 6y + 9 = 0
' 7 x - 2 y - 3 = 0 ' x + 6 y + 9 = 0 Tpa dp trung diem N cua BC thoa man he: •N Suy ra A C = 2 M N = (-4; - 3)
Trang 10Phumig phtip giai Toan Hinh hoc theo chuyen dS"- Nguyen Phu Khdnh, Nguyen Tat Thii
1) Gia six A : ax + by + c = 0 la tiep tuyen ciia (C)
Bai toan co ban ciia phuong phap toa do trong mat phang la bai toan xac
dinh toa do ciia mot diem ChSng han, de lap phuong trinh duong thang can
tim mot diem di qua va VTPT, voi phuong trinh duong tron thi ta can xac djnh
tarn va ban kinh Chung ta co the gap bai toan tim toa do ciia diem dugc hoi
true tiep hoac gian tiep
• Ve phuong dien hinh hgc tong hgp thi de xac dinh toa do mot diem, ta
thuong chiing minh diem do thugc hai hinh (H) va (H') Khi do diem can tim
chinh la giao diem ciia (H) va (H')
• Ve phuong di^n dai so, de xac dinh toa do ciia mot diem (gom hai toa do) la
bai toan di tim hai an Do do, chiing ta can xac djnh dugc hai phuong trinh
chiia hai an va giai he phuong trinh nay ta tim dugc toa do diem can tim Khi
thiet lap phuong trinh chiing ta can luu y:
+) Tich v6 huong ciia hai vec to cho ta mgt phuong trinh,
+) Hai doan thang bang nhau cho ta mgt phuong trinh,
+) Hai vec to bang nhau cho ta hai phuong trinh,
Cty TNHH MTV DWH Khang Viet
+) Neu diem M e A : ax + by + c = 0,a ^ 0 thi M - b m - c - ; m , liic nay toa
do ciia M chi con mgt an va ta chi can tim mgt phuong trinh
Vi da 1.2A. Trong mat phang Oxy cho duong tron (C): (x - 1 ) ^ + (y - 1 ) ^ = 4
va duong thang A : x - 3 y - 6 = 0 Tim tga dg diem M nam tren A , sao cho tvr M ve dugc hai tiep tuyen MA, MB (A,B la tiep diem) thoa AABM la tam giac vuong
Xgigiai
Duong tron (C) co tam 1(1; 1), ban kinh R = 2
Vi AAMB vuong va I M la duong phan giac ciia goc AMB nen A M I = 45°
Trong tam giac vuong l A M , ta co:
IM = 2V2, suy ra M thugc duong tron tam I ban kinh R' = 2 Mat khac M e A nen M la giao diem ciia A va (I,R') Suy ra tga do ciia M la nghiem ciia he
x = 3 y + 6
x - 3 y - 6 = 0 ( x - i ) 2 + ( y - i ) 2 =8 " [(3y + 5)2 + ( y - l ) ' =8 'x = 3y + 6
Vay CO hai diem M j (3; - l ) va M 2 - ; — thoa yeu cau bai toan
Vi du 1.2.2. Trong mat phSng voi he tga do Oxy cho cac duong thang
d i : x + y + 3 = 0, d j : x - y - 4 = 0, dg : x - 2 y = 0 Tim tga do diem M nam tren duong thSng sao cho khoang each tu M den duong thang d^ bang hai Ian khoang each tu M den duong thang d2
Xffi gidi
Taco M e d 3 , s u y r a M(2y;y) Suy ra d(M,di) = — ^ ; d ( M , d 2 ) = ^ ^ Theo gia thiet ta co: d(M,di) = 2d(M,d2) <^ 3y + 3 ^ 2 l y - 4
Trang 11Phuvng plidp gidi Toiin Hiith hoc theo chuyen dc- Nguyen Phii Klidnh, Nguyen Tn't Thu
Vi du 1.2.3 Tron g he toa do O x y , cho die m A(0; 2) va d u o n g th3ng
d : x - 2 y + 2 = 0 T i m tren d u o n g thang d hai diem B, C sao cho tam giac
Vay CO hai bp d i e m thoa yeu cau bai toan la:
Vay B ( 3 ; - 1 ) ; C ( 5 ; 3 ) hoac B ( - 1 ; 3 ) , C ( 3 ; 5 )
( b - l f - ( c - 4 f 3
xy = 2 x = 2 x - - 2 x ^ - y ^
<=> • V <
x ^ - y ^ = 3 y = i y = - 1
Cty TNHH MTV DWH Khang Viet
Vi du 1.2.5 Cho parabol (P): y^ = x va hai diem A(9; 3), B ( l ; -1) thupc (P)
Gpi M la diem thupc cung A B cua (P) (phan ciia (P) bi chan b o i day A B ) Xac djnh tpa dp d i e m M nam tren cung A B sao cho t a m giac M A B c6 dien tich ion nha't
JCgi gidi
P h u o n g t r i n h A B : x - 2y - 3 = 0
Vi M G (P) => M ( t ^ ; t) t u gia thiet suy ra - 1 < t < 3
Tam giac M A B c6 dien tich ion nha't o d ( M , AB) Ion nha't
Vi du 2.6 T r o n g mat phang Oxy cho d u o n g tron (C): (x - 1 ) ^ + y^ = 2 va
hai diem A ( l ; - 1 ) , B(2;2) T i m tpa diem M thupc d u o n g tron (C) sao cho dien tich tam giac M A B bang ^
Xffi gidi
Ta CO A B = Vio va S^^^AB = - d ( M , A B ) A B = d ( M , A B ) =
Lai CO A B = (1;3) nen n = ( 3 ; - l ) la VTPT ciia d u o n g thang A B Suy ra p h u o n g t r i n h A B : 3(x - 1 ) - ( y +1) = 0 hay 3 x - y - 4 = 0 Gpi M ( a ; b) e (C) => (a - i f + b^ = 2
b = 3a - 5 ( a - 1 ) 2 + ( 3 a - 5 ) 2 = 2
b = 3a - 5
3 a - b - 4 = l
( a - l ) 2 + b 2 = : 2 i ) k > , J
3 a - b - 4 = - l ( a - l ) 2 + b2 = 2
b = 3a - 3
hoac ( a - 1 ) ' + ( 3 a - 3 ) 2 = 2 v i , h ; ^ / , „
b = 3 a - 3
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