C. CAU HOI VA BAITAP
h)Phuong trinh chfnh tac ciia (F) cd dang ^ +^ =1 a fe
Vi (F) ed mdt tieu diim Fj i-j3; o) ntn c = j3. Ta cd
' J-3^ 1; € ( F ) = > - 2 - + - y = l 1 3 1; € ( F ) = > - 2 - + - y = l 1 3 a" 4fé â=b'^ + c'^=b^+3. (1) (2)
Thé(2) vio (1) tadugc - 2 ^ + - ^ = l « 4fe2+3(fê +3) = 4fe2(fê +3) fê+3 4fe"
« 4 f e % 5 f e ^ - 9 = 0 » fê=l. Tiir (2) suy ra 0^=4.
2 2
vay phuong trinh chfnh tic ciia elip (F) la : — + ^ = 1.
Vi du 2. Lap phuong trinh chfnh tac cCia elip (£) trong mdi trudng hop sau : a) Mdt dinh tren true Idn la diem (3 ; 0) va mot tieu diem la diem (-2 ; 0); b) (E) di qua hai diem M(0 ; 1) va A/
GiAi
r,.v3^
' 2
a) Ta cd a = 3 ; e = 2.
Suy ra fê = â - c^ = 9 - 4 = 5. vay phuong trinh chfnh tic cua elip la
2 2
^ + ^=1. 9 5
b) Phuong trinh chfnh tic ciia (F) cd dang :
2 2
^ + ^ = 1
2 , 2
a b
Do (F) di qua hai diim M(0 ; 1) va TV vao phuong trinh cua (F) ta dugc :
1
1; J~3
ntn thay toa đ ciia M vk N
I 1 3 , M Ab <:> Ife2=l \â=Ạ 2 2
VAN dg 2
Aac dinh cac thanh phan cua mot elip khi biet phuong trinh chinh tac cua
elip do
1. Phuang phdp
2 2
Cdc thanh phdn ciia elip (E): --- + ^ =1 (h.3.7) a fe
- True Idn cua (F) nim tren Ox, ÂÂ = 2a ; - True nhd cua (F) nim tren Oy, B^B^ = 2fe ; (adsbygoogle = window.adsbygoogle || []).push({});
- Hai tieu diim : F, (- e ; 0), F2 (e ; 0)
- Tieu cu: - Bdn dinh - Ti so — vdi c P^^2 = : A j ( - a B^(0; < 1 ; = ^a- 2c; ;0), -b). '-b' Â(a; 8^(0 » 0), •,b); lb; -a  ' -c \ F, ~b c ^ 0 Si F2 J a  X Hinh 3.7
- Phuong trinh cac dudng thing chiia cac canh ciia hinh chii nhat co sd la
X = + a ; y = ± fẹ
2. Cdc vidu
Vi du 1. Xac djnh do dai cac true, toa do cac tieu diim, tea đ cac dinh va
2 2
ve elip (E) cd phuong trinh — + ^ = 1.
GlAl 2 2 Phuong tnnh (F) cd dang : ^ + ^ = 1- Do do a fe c = \lá-h^ =4. |â=25 a = 5 Ife2=9 1^ = 3
vay (F) cd:
- True Idn : AjA2 = 2a = 10 ; - True nhd : BjB2 = 2fe = 6 ;
- Hai tieu diim : Fj (- 4 ; 0), F^ (4 ; 0);
-Bdndinh: Aj(-5;0), A2(5;0), B j ( 0 ; - 3 ) , B2(0;3). Hinh ve ciia (F) nhu hinh 3.8.
Vi du 2. Cho elip (£) cd phUOng trinh
2 2
^ +^ = 1.
Hinh 3.8
^
100 36
Hay Viet phUOng trinh dudng trdn ('g') cd dUdng kfnh la F.,F2 trong đ F., va FJ la hai tieu diem ciia (£).
GiAi
2 2
X y
Phuong trinh (F) cd dang — + " ^ = 1 •
a fe
Tacd â= 100, fê = 36. Suy ra c^ = â - fê = 64
e =8.
Ducmg trdn dudng kfnh FjF2 cd tam la gd'c toa do va cd ban kfnh B = c = 8. vay phuong tiinh cua ('^) la : x^ + y^ = 64.
VAN dE J
Diem Md\ dong tren mot elip Ị Phuang phdp (adsbygoogle = window.adsbygoogle || []).push({});
Cdch 1 : Chiing minh tdng khoang each
tiJT M din hai diim cd dinh F,, F2 la
mdt bang sd 2a (F^F^ < 2a).
Khi đ M di đng tren elip (F) cd hai tieu diim FJ, F2 va true Idn la 2ạ
Cdch 2 : Chiing minh trong mat phing toa đ Oxy diem M(x ; y) cd toa đ thoa
man phuong trinh- 2 2
^ + ^-=1 a fe
vdi a, fe la hai hing sd thoa man 0 < fe < ạ
Hinh 3.9
2. Cdc vi du
Vi du 1. Cho hai dudng trdn '^'^(F, ; R.^) va '^2('^2 • ^2)- ^ ^ t ^ "^"^ *^°"9 (•^2) v3 F| ^ F^. Ggi M la tam cQa dudng trdn ( ^) thay đi nhi/ng ludn tiep xuc ngoai vdi C^.,) va tiep xuc trong vdi (^2)• ^^^ chimg to diem Mdi
đng tren mot elip.
GIAI
Ta cd MFj = B + /?j; MF2 =R2-R- Suyra MF^+MF^= R^+R^.
vay M di đng tren elip cd hai tieu diim la Fj va F2 va cd true Idn la
2a = R^+R^.
Vi du 2. Trong mat phSng toa do Oxy cho diem M(x ; y) di đng cd toa do
ludn thoa man