Phuang phdp gidi Todn Hinh hqc theo chuyen de-Nguyen Phu Khdnh, Nguyen Tat Thu
X = 1 + at , voi a + b 5t 0. y = 2 + bt
Do A, Be A suy ra A = (1 + atpZ + b t j ) , B = (1 + at2;2 + b t j ) . I la trung diem ciia AB khi va chi khi
2x, = X A + Xg
2 X I = X A + X B
ăti+t2) = 0
O t j+ t 2 = 0 (1)
b ( t i + t 2 ) = 0
A, B e (E) nen t j , t2 la nghiem cua phuang trinh
i i l f i + (1±M£ = 1 « f9a2 + l6bMt2 + 2(9a + 32b)t -139 = 0
16 9 I I ^ '
Theo dinh ly Viet ta c6 tj + t2 = 0 <=> 9a + 32b = 0 Ta CO the chpn b = -9 va a = 32.
Vay duong thang d c6 phuong trinh : 9x + 32y - 73 = 0 .
Cdch 2: Vi I thuoc mien trong cua elip (E ) nen lay tuy y diem Ăx;y) e (E) thi duang thang I M luon cat (E) tai diem thu hai la B(x';ý).
I la trung diem diem AB khi va chi khi x' = 2 - X 2 X J = X A + X B 2x, = X A + X B Vi M , M ' € ( E ) o y ' = 4 - y x^ v ' + i - - l • M ' ( 2 - x ; 4 - y ) 16 16 Suyra i l ^ ^ l 6 - 8 y 9x + 32y-73 = 0 (*)
Toa do diem M , I thoa man phuang trinh (*) nen duang thang can tim la: 9x + 32y-73 = 0.
x2 v^
Vidu /.4.2S. Trong mat phang Oxy cho (E): - y + ^ a b
= 1 va hai diem M , N thuoc (E) sao cho O M vuong goc voi ON. Chung minh rang :
1) 1 1 1 1
2) Duong thang M N luon tiep xuc voi mot duong tron co djnh.
Xffigidị
1) • De thay mot trong hai diem trung voi bon dinh ciia (E) thi dang thiic hien nhien dving
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, Neu ca hai diem khong trung vai cac dinh ciia (E):
Gpi M(x^^;yM), N ( x i ^ ; y N ) va k (k?^0) la h§ so goc ciia duong thSng
Q M thi he so goc ciia O N la - — (vi O M vuong goc vai O N ).
D o M , N e ( E ) n e n ^ + y f = . l ( l ) , ^ + y f = l „ , , ( 2 ) a b a b
Duong thang O M c6 phuang trinh la y = kx suy ra y ^ = kxj^ (3) Duong thang O N c6 phuang trinh la y = - -^x suy ra y ^ = - — x, v j (4)
K K
Thay (3) vao (1) suy ra:
â b^ M 1 k • + 2^ ^2 b^ a b — K X » « — k^âb^ + 1 a V + b ^ M - ^ ; X ^ • Do do O M ^ = x^ . y ^ = âk^+b
Tuong tu thay (4) vao (2) suy ra:
âk^+b^ x2 a ^ ^ k V j âk^b^ •yN = 4 r x 2 , = âb^ k^ ^ " a ^ + k V .Do d6 0 N 2 = x 2 j + y 2 j = . , V( k 2 + i) â + k V Suy ra 1 1 b ^ + k V â + k V ( a ' + b 2 ) ( k 2 H - l ) 1 ^ O M 2 O N ^ âb^fk^+l Vay âb^ k^+l) a V ( k 2 + l a 2 • b2 1 1 _ 1 _1_ O M ^ O N ^ ~ â b^
2) Gpi H la hinh chiéu ciia O len duong thSng M N khi do O H la duong cao cua tarn giac vuong M O N . Theo he thuc lupng trong tam giac vuong ta c6
1 1 1
^• + i o O H = ab
O H 2 O M ^ • O N ^ a2 ' b^ ^ " Vâ+b
Suy ra M N luon tiep xiic voi duong tron co dinh tam O ban kinh â + b^
Phuang phiif) giai Toiin llhiU hoc thco chuyen de- Nguyen Íht't Klidnh, Nguyen Tat Thu
x2
Vi du 1.4.28. Cho hypebol (H): — - ^ = 1 c6 cac tieu diem , . Lay M la
a b
diem bat k i tren (H). Chiing minh rang tich khoang each t u M den hai
d u a n g tiem can la hang só. _
Xffi gidị
Phuang trinh hai d u o n g tiem can ciia (H) la:
Aj : y = : - x hay b x - a y = 0 va A2 : y = - - x hay bx + ay = 0
a a Gia su M ( x ^ ; y [ ^ ) khi do theo cong thiic khoang each tij- m o t diem toi d u o n g thang ta c6 d ( M ; A j ) = Suy ra d ( M ; A i ) d ( M; A 2 ) = JJ7 ; d ( M; A 2 ) - V + b ^ b x M - a y M J N M + a y^ l b ^ x ^ - a ^ y ^ â+b^ â+b^ X ^ V
Mat khac M thuoc (H) n e n : — ^ = 1 hay b ^ x ^ - ây^ = âb^ a b
D o d o d ( M ; A i ) d ( M; A 2 ) = — la hang so a + b
Vi dụ 1.4.29. Cho parabol (P): y^ = 2ax. D u o n g thSng A bat k y d i qua tieu
diem F c6 h$ so goc k ( k ; t O ) cat (P) tai M va N . C h u n g m i n h rang tich
khoang each t u M va N den true Ox la hang sọ
JCgigidị
T i e u d i e m F(a;0).
V i di qua tieu diem F c6 he so goe k nen c6 phuong trinh: A : y = k Hoanh do giao diem ciia A va (P) la nghiem ciia phuang trinh:
N2 a a X — V 2 , a X — 2 = 2 a x« 4 k 2 x 2 - 4( 2 a + k2a)x + k 2 a 2= 0 (*) Á = 4 ( 2 3 + k^a)^ - 4k'*â = 16â ( l + k^) > 0
Theo d j n h ly Viet c6 Xj^^.Xj^ = —
M ^ t k h a c t a c o d ( M ; O x ) - | y ^ i | ; d ( N ; O x ) = |yN
Suy ra d ( M ; O x ) . d ( N ; O x ) - : | y ^ i . y ^ , | = j4a^|xj^.Xj^
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Vidu ị4.30. Trong mat phang Oxy cho elip (E) ep hai tieu diem Fi|(-V3;0);
p2(^/3;0) va d i qua d i e m A ^/3;^ . Lap phuong trinh chinh t^c ciia (E) va v6i moi diem M tren elip, hay tinh bieu thue:
P = FjM^ + F^M^ - 30M^ - FiM.F2M .
Xffi gidị
x2 y2
G i a s u (E): — + ^ = 1 v a i a , b > 0 . a b
eo gia thiet bai toan ta c6 h$ x2 v^ S u y r a (E): — = a ^ - b ^ + S 3 1 « a 2 = 4 , b 2= l . + — - = 1 . a2 4b2 v2 v 2 Xr> •> 0 X r X e t M ( x o ; y o ) 6 ( E ) = ^ - ^ + y 2= l= ^ y 2= i - ^ . • % "nisi r/ M S - - cf^
Suy ra P = (a + exo )^ + (a - ex,, )^ - 2(x^ + y^) - (â - êx^) = 1 .
Vi du 1.4.31. Tren mat phSng tpa dp Oxy, cho duong thang A : x - y + 5 = (T
x2 y2 2 2
va hai elip ( E j ) : — + ^ - = 1; (E2): — + ^ = l ( a > b > 0 ) c6 ciing tieu diem. 25 16 a b
Biet rang (E2) d i qua diem M thupc A . Lap phuang trinh ( E 2 ) , biet (E2) CO dp dai true Ion nho nhat.
Xffi gidị
V i hai E Iip c6 cung tieu diem nen ta c6: â - b^ = 25 - 1 6 = 9 Gpi M ( m ; m + 5) e A , do M thupc (Ej) nen: m^ (m + 5)^ - 1
« m2(a2 - 9 ) + ( m + 5)^3^ = d?-{â - 9 ) o á* - 2{m^ + 5 m +17)3^ + 9m2 = 0 V i > 9 n e n t a c o â = m^ + 5m +17 + J( m ^ + 8m + 17)(m2 + 2m +17) = f(m) • N e' u m 2 + 5 m> 0 = > f ( m ) > 1 7 • V o i m e( - 5; 0 ) , taco: ' f(m) = ("^^ + 8 m + 17)(m^ + 2 m +17) - (m^ + 5m)^ ^ yj{m^ + 8 m + 17)(m^ + 2 m +17) - ( m ^ + 5m) 61