^ + = 1. Vict phiTdng trlnh tiep tuy^n d cua (E) biet d d l hai true toa dp Ox. Oy Ian lirpt tai A, B sao cho AO = 2B0.
Hi/dng d i n giai
* Do tinh cha't doi xiJng ciia clip (E), ta chi can xet triTcJng hdp x > 0, y > 0.
Goi A (2m; 0), B(0; m) la giao diem cua tiep tuye'n cua (E) vdi true tao (j^
( m > 0 ) . . '
ii •i X V ^
PhiTdng trinh diTcJng th^ng AB : + — = 1
2m m > '
o X + 2y - 2m = 0, AB tiep xiic vdi (E) o 64 + 4.9 = 4 m ' o 4m^ = 100 j o m ^ = 2 5 o m = 5 ( d o m > 0 ) . ' M S H v
Vay phi/tfng trinh tiep tuye'n la: x + 2y - 10 = 0. ' V i tinh chat doi xtfng nen ta c6 4 tiep tuyen la: ^ ^
X + 2y - 10 = 0; X + 2y + 10 = 0; X - 2y - 10 = 0; x - " 2 y + 10 = 0.
Bai 4. Trong mat phing vdi h§ tpa dp Oxy, hay viet phiTOng trinh chinh ta'c cila Elip (E) biet (E) c6 tam sai blng ^ va hinh ciia chOr nhat cd sd cua (E) co
chu vi b i n g 20. . M i l
Hifdng dSn giai
Goi phi/dng trinh chinh t^c cua Elip (E)lk: — + ^ = 1, a > b > 0.
a b
TCr gia thiet, ta c6 he phiTdng trinh:
qs 11-.
^5 a 3
2(2a + 2b) = 20. Suy ra a = 3; b = 2.
t
X y
Vay phiTdng trinh chinh t i c cua (E) IS: — + ^ = 1.
Bai 5. Parabol y^ = 2x chia diOn tich hinh tron x^ + y^ = 8 theo ti so nao?
Hifdng dSn giai Hinh tron x^ + y^ = 8 c6 R = 2>/2 ,
do CO dien tich la nR^ = 871.
S
Ta can tinh tl so — (hinh ve) trong do:
2 \
• I : S| = | ( > / 2 x - x ) d x + S 4 „ , „ r t n O A B . iit? i'^f \
- x V 2 x - X 2 _ 2
Q + S q u a l l r o n O A B ~ " J "
= > S 2 = 4 7 1 -'2 ^
— + 7t
l3 J
2
3 7t + 37t + 2
0ai 6- '^''""S '"^^ phing vdi he tpa dp Decac vuong g6c Oxy, cho Elip (E) c6 phi/dng trinh + = ^ • ' ^ i ^ ' " M chuyen dpng tren tia Ox va diem N chuyen dpng tren tia Oy sao cho diTcfng thing M N luon tiep xuc vdi (E). Xac dinh tpa dp cua M , N de doan M N c6 dp dai nho nhat. Tinh gia tri nho nha't do.
Hi^dng dSn giai
Cdch I : Gia suT M( m ; 0) va N( 0 ; n) \k hai diem chuyen dpng tren hai tia Ox
Oy. s
= 1.
Di/dng thing nay tiep xuc vdi (E) khi va chi khi: 16 Theo bat d i n g thtfc Cosi ta c6:
MN^ = m^ + n^ = (m^ + n^)
= 49 = > M N > 7
2 f r
, m ; + 9
r iA^_9_^ 2 2
= 25 + 1 6 + 9 ^ > 2 5 + 2 x/r6:9 m
Ding thiJc xay ra
16n^ ^ 9m^ i grcux ioJj ô '
m n ,2
m'+n^ = 4 9 o m = 2V7,n = N/2T .2
m > 0, n > 0 ^ô A '
Ke't luan: Vdi M( 2N/ 7 ; 0 ) , N ( 0 ; V' 2T) thi M N dat gid tri nho nhat vk gia tri nho nhat la ( M N ) = 7 .
Cdch 2: Gia sijT M (m; 0 ) va N ( 0 ; n) la hai diem chuyen dpng tren hai tia Ox v4 Oy. Dirdng thing M N c6 phiTdng trinh — + ^ -1 = 0
E^irdng thing nay tiep xuc vdi (E) khi \k chi khi: 16| — +9 -Ar =1
"Theo ba't ding thtfc Bunhia-copxki, ta c6:
MN^ = m^ + n^ = (m^ + n^)
( -
2
( ^ ] + 9
r 16 9 1 > ( 4 3^
m.— + n — I m ' n^J ^ m
= 4 9 o M N > 7 .
^^ng thij-c xay ra
m: — = m: — 4 3
m n a;nfjilq I lit!.: .,
+ n ^ = 4 9 o m = 2 V 7 . n = V2T nni'i
m >0,n >0 . < a / -J..,. j i i j t/
U^^n^^^jru^^ T O . / , hoc - N ^ W r T T ^
K6i luan: V d i M(2>/7;0) . N(0;V2r) thl M N dat gid trj nh6 nhaft gia trj nh6nha'tla(MN) = 7.
CdcT. 3; Phircing trinh tiep tuyd'n tai diem (x„; y.,) thuoc — + — -1 •
Suy ra toa dp cua M va N la M 2 Q2
16 va N f n 9 l
va N 0;—
.o' =>MN^ =
2 2 \
4 ^ y o fl6^
9']
l l 6 9 J V ^0 yl)
fffis'l' ,:
SuT dung baft d^ng thtfc Co-si hoSc Bunhia-c6pxki (nhiT cdch 1 \k 2) ta c6 M N ^ ^ 7 ^ ng thurc xSy ra o x„ = -j-; yo = —j-
Bki T.Trong mat phing vdi true toa dp Oxy, cho d i l m C(2; 0) Ik Elip (E):
= 1. T i m toa dp cdc diem A, B thupc (E) biet r^ng hai d i l m A, B doi 1
xiJng nhau qua true hoanh v^ tarn gidc ABC Ik tarn gidc deu.
Hifdng dSn giai
Gia suf A(x„; y„). Do A, B doi xtfng nhau qua Ox nen B(x„, -y,)).
Ta C O A B ' = 4 yl vk AC^ = (x„ - 2)' + yl.
2 2
V i A 6 ( E ) n c n ^ + y ? , = l = > y ? , = l - - i i - (1).
4 4 V i A B = A C n c n ( x o - 2 ) ' + y ? , = 4 y ? , (2).
Thay (1) vJlo (2) va nit gpn, ta drfdc: Tx^ - 16x„ + 4 - 0 o x „ = 2
2 7 V d i xo = 2 thay vao (1). ta e6 y,, = 0. Trir5ng hdp n^y loai vl A = C
J 4>/3
V d i X o = - thay vao (1). la CO y „ = ± — •
7 ^ / i - \
Vay A (2 4S]
V 7 ; . B
(2 4 ^ 1 hoac A (2 AS' , B (2 AS]
[r 7 J
x^ y _ = i B a i 8. Trong mat phJng vdi h? true Oxy, eho ehp (E) • j '
phUdng trinh hypebol (H) e6 hai diT^ng ti^m can la y = ±2x va c6 hai diem la tieu diem cua elip (E). ,
Hiring dSn giai
Elip (E): ^ + ^ = 1 c6 hai tieu diem la F,(-VrO;0), F2(Vl0;0). Hypebol (H) c6 cung tieu diem vdi elip, suy ra phiTdng trinh cua Hypebol (H) c6 dang
a b s ,
Hypebol (H) c6 hai tipm can la y = ±2x = ± - x o - = 2 o b = 2a (2).
Si H
T i r ( l ) va (2)suyraa^ = 2;b^ = 8. , ^
x^ ' Vay phiTdng trinh Hypebol (H). ^ = 1. > M :
2 8 :)ni;an£l rin, '
Bai 9. Trong mat ph^ng vdi hp tpa dp Decac vu6ng gdc Oxy cho Elip (E):
2 y2
- - - — = 1, va cdc diem M ( - 2 ; 3), N(5, n). Viet phi/dng trinh cdc di/dng 4 1
thing di, dj qua M va tiep xiic vdi (E). Tim n de trong so cdc tiep tuyen cua E di qua N c6 mpt tiep tuyen song song vdi d| hodc d 2 .
Hvtdng dSn giai x^ v^
(E): ^ + ^ = l , M ( - 2 ; 3 ) , N ( 5 ; n ) . . Mif
Nhan xet (E) cd hai tiep tuyen thing duTng x = ±a = ±2 va dp x = - 2 la tiep luyc'n cua (E) di qua M .
Gpi dj la difilng thing qua M cd hp s6 goc k ,j dj: y = k(x + 2) + 3 o kx - y + 2k + 3 = 0
dj: tiep xuc (E) ô a ^ A ^ + b^B^ = C^ <ằ 4k^ + 1 = (3 + 2k)^
o k = - — = - - . Vay dj: 2x + 3y - 5 = 0 12 3
De thay tiep tuyen A cua (E) qua N(5; n) khong song song d|
==> A/ / d 2 = > k A = k d 2 = - |
• 2 2 A qua N (5; n) cd hp so goc k a = — j , A : y = - j (x - 5) + n
=^ -2x - 3y + 10 + 3n = 0; A tiep xuc (E) o 4(-2)^ + l ( - 3 ) ' = (10 + 3n)^
<=> 3n^ + 20n + 25 = 0 o n = -5 V n = - |
n = loai v l khi do A = dz. Do dd N(5; - 5 ) . ^'^ *' 3 , ' .'i
Luyen giii dS trade thi DH 3 man Bdc. Trung, Nam Todn hoc - Nguyln Van Thdng
K H6I D A D I $ N , T H f i T f C H K H6I C H6P , K H6I L A N G T R V . T H E T f C H C A C K H6I T R6N X O A Y
1. T 6 M T A T L i T H U Y E T . v A . H i N H L A N G TRg
7. Dinh nghla:
Hinh lang tru la hinh da di?n c6 2 mat song song g p i 1^ day, cac canh khong thuoc 2 day song song vdi nhau.
2. Ttnhchdt
Trong hinh 13ng t r u :
Cdc canh ben song song va b i n g nhau ""^^
- Cac mat ben, mat ch<5o la hinh binh hanh ^ - Hai day c6 canh song song va b^ng nhau
3. Lang tru dting diu, lang tru diu, lang tru xi§n
Lang tru du'ng la lang tru c6 canh ben vuong gdc vdi day;
Lang tru deu la lang tru diJng C O day la da gidc deu; ^ Lang tru deu c6 cac mat ben la hinh chO" nhat b^ng nhau; . ^ . . J
Lang tru xien c6 canh ben khong vuong goc vdi day. *
4. HlnhhOp^'i^^ >i c j w iBit r
Hinh hop la lang tru c6 ddy la hinh binh hanh. '
• Hinh hop c6 cac mat doi dien la hinh binh hanh song song va b^ng nhau - Cac du'dng chco hinh hop c^t nhau tai trung diem
Hinh hop du'ng c6 canh ben vuong goc vdi day.
Hinh hop xien c6 canh ben khong vuong goc v d i day.
Hinh hop chu" nhat la hinh hop duTng c6 day la hinh chfl" nhat.
Hinh hop chff nhat c6 cac mat la hinh chu" nhat.
Ba do d a i cac canh xuat phat tif 1 dinh g p i la kich thUdc cua hinh hop i^hO' n h a t a , b, c.
Cac du'dng cheo hinh hop chuf nhat b^ng nhau va c6 dp dai:
rp v O . i '
d - Va^ + b^ + c^
4 Hinh l a p phU'dng la hinh h o p c6 6 mat la hinh v u o n g Cac c a n h c u a hinh l a p phUdng c6 dp dai: d = aVs 5. Diin ttch xung quanh va di$n tfch todn phdn
Sxq = Ph la chu v i thie't di$n th^ng 1 la dp dai canh ben
r
Lang tru diirng: ,..
p i a c h u v i d d y ' h la chieu cao
Hinh hpp chiynhSt:
COng(y TNIIIIMr\Klu^n, Vie,
S,p = 2(ab + be + ca) V ; a, b, c la kich thirdc cua hinh hpp chff nh§t:
6. Thi ttch • „„„^,, . . . T h ^ tich cua hinh hop chff nhat: I v^ a b c l a. b. c la kich thffdc - . The tich cua hinh laip phiTdng: V = a3 A la canh
. T h ^ tich lang tru: |V = B.h| B la d i ^ n tich d^y, h la chieu cao;
Hoac |V = S.l| S la dien tich thiet dipn t h i n g , I la canh h t n j . The tich cua lang tru tam gidc cut: Lang tru tam gidc cut la hinh da di$n c6
hai day la tam giac c6 canh ben song song khong b^ng nhau ,
•| \
S la d i ^ n tich thiet di?n t h i n g
a, b, c la dp dai cdc canh b6n • ' •
B. HiNH C H O P xti!!;; ^ ; I Dinh nghla
• Hinh ch6p la hinh da di$n c6 mpt mat la da gidc, c^c mat khdc la tam giac c6 chung dinh. Chieu cao h la khoang cdch tir dinh tdi ddy.
• Hinh chdp deu 13 hinh ch6p c6 day la da gidc deu va cdc canh ben b^ng nhau.
Binh cOa hinh ch6p deu c6 hinh chieu la tam cua ddy.
• Hinh chdp tam giac c6n gpi la hinh tff di$n
• Hinh tff d i ^ n la hinh ch6p tam gidc c6 &iy la mat nao cung difpc, dinh la
<Jiem nao cung difpc.
• Hinh tff dien deu la hinh tff dipn c6 cic canh b^ng nhau. '''^
^^^n tich xung quanh cua hinh chdp d^u: S,^ = Inad
"^sS'canh day '
^- d6 dai trung doan
^9 dai canh ddy
Luyen gUli dS trudc k9 thi DH 3 mijn Bdc. Trung. Nam Todn hpc - Nguyen van T/tang
Di0n lich loan phan:
3. TMttch
The tich hinh ch6p V^-i-B.h 3
The tich tiir di^n: V = —abdsina 6
' ; ' i t ? ti
: J | ? ! f s t J r i o qu.^
a, b: do dai hai canh doi
d: do d^i doan vuong goc chung a: g6c giffa hai canh doi.
Ti so the tich cua hai hinh chop tarn gidc c6 chung dinh 3 canh ben
V s A B T ' SA'.SB'.SC
^ S A B C SA.SB.SC