Phucntg phdp giai Todn Hinh hpc theo chuyen de- Nguyen Phu Khanh, Nguyen Tat Tltu
Mat khac (C) tiep xiic vdi d nen suy ra d(l,(d)) = R o = Vâ + - c <=>â +b^ +2ab-2c = 0 (S)
Tir (l),(2),(3) tadúoc a = | ; b = ^;c = 2 hoac a = | ; b = - ^ ; c = 2.
Vay CO hai duong tron thoa yeu cau bai toan la:
+ y^ - 3x - y + 2 = 0 va x^ + y"^ - 3x + 7y + 2 = 0 .
Bai 1 . 4 . 4 . Trong mat phang voi he tpa do Oxy, cho duang tron (C): + y^ - 2x - 2y +1 = 0 va duang thang d : x - y + 3 = 0. Viet phuong trinh duong iron (C) c6 tarn M nam tren d, ban kinh bang 2 Ian ban kinh duong tron (C) va (C) tiep xiic ngoai voi duong tron (C).
Jiuang đn gidi
Duong tron (C) c6 tarn I(l;l) ban kinh R = 1 Ta CO M e d => M{x; x + 3).
Vi (C) va (C) tiep xiic ngoai nen ta c6 M I = 3R « ( x -1)^ + (x + 2)^ = 9 o x^ + x - 2 = 0 <r> X = l;x = - 2 .
Vay CO hai duong tron thoa yeu cau bai toan la:
(x - if + (y - 4)2 = 4 va (x + if +{y-lf=4.
Bai 1 . 4 . 5 . Trong mat phang vai he tpa dp Oxy, cho duong thang d : x - y + l- 7 ^ = 0 va diem A(-1;1) . Viet phuong trinh duong tron (C) di qua A, 0 va tiep xiic voi d.
Jiiedng đn gidi
Gpi (C): + y^ - 2ax - 2by + c = 0 la duang tron can tim
V i ( C ) d i q u a O , A ^ r " ° ^ (1) [a - b = 1 [a - b = 1 Do(C)tiepxucvai d : x - y + l-N/2 =Or:>d(l,(d)) = R a-h + l-yfl r - - = 7 a 2 + b 2- c (2) 4i
Tu (1) va (2) giai h? thu dupe a = 0,b = l,c - 0 ho5c a = l , b = 0,c = 0.
V^y CO hai duang tron thoa man la : x^ + y^ - 2y = 0 va x^ + y^ - 2x = 0 .
64
Cty TNHH MTV DWH Khang Viet
pai 1-4.6. Trong mat phang voi he tpa dp Oxy, cho duong tron . + y^ = 1 • Duong tron (C) tarn I(2; 2) c§t (C) tai hai diem A, B sao cho ^5 = . Viet phuong trinh duong thang AB .
Jlit&ng đn gidi
Ta CO OÂ + OB^ = AB^ = 2 =i> AOAB vuong tai Ọ Mat khac OI la duang
trung true cua doan thang AB nen A, B thupc cac tryc toa dp. Vay: • Ă1;0);B(0;]), phuong trinh duang thang AB:x + y - l = 0 ' • A ( - l ; 0); B(0; -1), phuong trinh duong thang AB: x + y +1 = 0 .
Bai 1 . 4 . 7 . Trong mat phang Oxy cho diem M(2;3). Viet phuong trinh
duang tron (C) di qua M va tiep xiic vai hai true tpa dp. y v
J^Iuang đn gidi
Gpi I(a; b) la tarn ciia duong tron (C). V i (C) tiep xiic vai hai tryc tpa dp va di qua M(2;3) nen ta suy ra dupe I nam trong goc phan t u thii nhat hay
a,b>0
Ta CO d(I,Ox) = d(I,Oy) <=> a = b = R .
Phuong trinh (C): (x -a)^ + (y -a)^ = ậ Vi M e (C) nen ta c6:
(2-a)2 + (3-a)2 = 3 ^ c^â -10a + 13 = 0 o a = 5±2^/3 . V§y phuong trinh duong tron (C) la:
(x - 5 - iSf + (y - 5 - iSf = (5 + iSf
hoac (x - 5 + iSf + (y - 5 + 273)^ = (5-273)^.
Bai 1 . 4 . 8 . Trong mat phang voi he tpa dp Oxy, cho duang tron
(C):{x-2f = - va hai duang thang A j: x - y = 0, A j : x - 7 y = 0 . Xac dinh 5
tpa dp tarn K va tinh ban kinh ciia duong tron (Ci); biet duong tron (Ci) tiep xiic voi cac duong thing Ai, A2 va tam K thupc duong tron (C).
Jiuang đn gidi
Gpi K(a;b), K e ( C ) « ( a - 2) 2 + b 2 = i
0
|a-b a- 7 b Ta CO (Cj) tiep xiic voi Aj, A2 o ^ =
Tu (1) va (2) ta c6: 5 ( a - 2) 2 + 5 b 2= 4 (1) (2) a - b = a - 7 b 5( a- 2 f + 5 b 2 = 4 5(a-b) = a- 7 b (I) 65
Phumtg phapgiai Tonn llhih hoc theo chuyett de- Nguyen Phil Khdnh, Nguyen Tat Thu Hoac (I) (II) 5 ( a - 2) 2 +5b^ = 4 5(a - b) = - a + 7b 25â - 2 0 a + 16 = 0 b = -2a 'a = 2b 2 5 b ^ - 4 0 b + 16 = 0 (11) v6 nghi^m. <=>(a;b) = 8 4 5 ' 5 Ban k i n h ( C j ) : R = Y 8 4 a - b 2V2 Vay K V5 5 va R = 2^2
Bai 1.4.9. Trong mat phang voi he toa d o Decac v u o n g goc Oxy, cho hai
d u o n g t r o n : ( C i ) : x ^ + y^ - 1 0 x = 0 va (C2) :x^ + y^ + 4 x - 2 y - 20 = 0
Viet p h u o n g trinh d u o n g tron (C) d i qua cac giao d i e m ciia (Ci), (C2) va c6
tarn nam tren d u o n g thang A : x + 6 y - 6 = 0 .
Jiu&ng dan gidi
Xet he p h u o n g t r i n h x^ + y ^ - 1 0 x = 0
x^ + y ^ + 4 x - 2 y - 2 0 = 0 . Giai he nay ta duoc hai cap nghiem ( x ; y ) = ( l ; - 3 ) , ( 2 ; 4 ) . Suy ra ( C j ) va (C2) c a t n h a u tai Ă1;-3),B(2;4).
Goi I la tam cua (C), ta c6 1 ( 6 - 6 m ; m )
V i l A = IB nen ta c6: (5 - 6m)^ + ( m + 3)^ = (4 - 6m)^ + ( m - 4)^ o m = - 1 Suy ra 1(12;-1), R = I A = 5V5 .
Vay p h u o n g t r i n h (C): (x -12)2 + (y +1)2 ^ 125 .
Bai 1.4.10. Trong mat phang v o i he tpa dp Oxy , cho d u o n g tron (C) c6 p h u o n g t r i n h : + y^ - 2x - 6y + 6 = 0 va diem M ( - 3 ; l ) . Gpi T ^ T j la cac tiep d i e m ciia cac tiep tuyen ke t u M deh ( C ) . Viet p h u o n g t r i n h d u o n g thSng di
q u a T p T j .
Jiu6ng dan giai
D u o n g tron (C) c6 t a m 1(1; 3) va ban k i n h R = 2. D o I M = iS > R nen diem
M 6 ngoai d u o n g tron (C)- Gpi T(XQ; yg) la tiep diem ciia tiep tuyen ke t u M .
Ta c6: T € ( C ) M T 1 I T '
T 6 ( C )
M T . I T = 0
Cty TNHH MTV DWH Khang Viet
Ma M T = ( X o+ 3 ; y o - l ) ; I T = ( x o - l ; y o - 3 ) D o do ta c6; ^ô+Yô - 2 x ( , - 6 x 0 + 6 = 0 ( x o + 3 ) ( x o - l ) + ( y o - l ) ( y o - 3 ) = 0 o 2 x o + y o - 3 = 0 (1) ' ^ o ^ + y o ^- 2 x 0 - 6 x 0 + 6 = 0 ' ' o ^ + y ( ) ^ + 2 x o - 4 y o = 0
Tpa d o cac tiep d i e m Tj,T2 thoa man dang thuc (1). / ' Vay p h u o n g trinh d u o n g thSng d i qua Tj ,T2 la: 2x + y - 3 = 0 .
Bai 1.4.11. Trong mat phang Oxy cho hai duong thang dj : m x + ( m - l ) y + m = 0 va d j : (2m - 2)x - 2 m y + 1 = 0. C h i i n g m i n h rang di va d2 luon cat nhau tai mpt diem nam tren m o t d u o n g tron c6' dinh.
Jihcang dan gidi ; i ! M
Ta thay d^ l d 2 nen d j va d2 luon cat nhau tai I .
M a t k h a c : d j d i qua diem códjnh A ( - 1 ; 0 ) , d2 d i qua diem c6'dinh ^ ( ^ ' ^ )
Do d o k h i m thay doi t h i I luon nam tren d u o n g tron d u o n g k i n h A B .
Bai 1.4.12. Trong m a t phang v o i he toa do O x y , cho d u o n g tron (C): x^ + y2 - 2x + 4y = 0 va d u o n g thang d : x - y = 0 . T i m toa dp cac diem M tren d u o n g thSng d, biet tir M ke dupe hai tiep tuyen M A , M B den (C) ( A, B la
< 3
ac tiep diem) va khoang each t u diem N ( 1 ; - 1 ) den A B bang —j= .
v 5
Jiuang dan giai
V i M £ d = ^ M ( m ; m ) Gpi A ( x o ; y o ) . K h i do, ta c6: cac l A . M A = 0 ' ^ o + y o - 2 x o + 4 y o = 0 A e ( C )
Suy ra ( m - l)Xo + ( m + 2)yo + m = 0 .
Do do, ta CO p h u o n g t r i n h A B la: ( m - l ) x + ( m + 2)y + m = 0.
^ a t khac: d ( N , AB) = —j= nen ta c6 p h u o n g trinh: m - 3
Giai p h u o n g trinh nay ta t i m dupe m = 0, m =
Ta loai m = 0, v i k h i d o M 6 ( C ) .
7 ( m - l) 2+ ( m + 2)2
58 13
Phucntg phdp gidi Tpdti llttih hoc theo chuyen de"-Nguyen Phti Khanh, Nguyen Tat Thu
Vay C O m o t d i e m M thoa yeu cau bai loan: M 58 58
13 13 y^ y^
Bai 1.4.13. Cho elip ( E ) : — + ^ = 1 . Viet p h u o n g t r i n h d u o n g thSng d (tj
qua M(1;1) va cat ( E ) tai hai d i e m A , B sao cho M la t r u n g d i e m ciia A B .
JIu&ng đn gidi
D u o n g thang A d i qua M , v o i vec to chi p h u o n g u = (a;b) c6 dang:
X = 1 + at / 2 , 2 ^\ â > 0 . = 1 + bt V / A , B e Ari> va <^ X g = 1 + at2 y A = l + b t i [ y 3 = l + bt2 M la t r u n g d i e m ciia A B k h i va chi k h i ^A + '^B - a ( t i + t 2 ) = 0 ( 0 0 \ y ^ { c ^ t , + t 2 = 0 d o a 2 + b 2 > 0 y A + y B = 2 y M [ b ( t i + t2) = 0 ' ^ \ A , B e ( E ) => t j , t 2 la nghiem ciia p h u o n g t r i n h
4(at +1)^ + 9 ( b t +1)^ = 36 (^4â + 9h^y^ + (8a + 18b) t - 23 - 0 t j +12 0 <=> 4a + 9b = 0
ChQn a = 9,b = -4r=> A : x = l + 9t hay 4x + 9 y - 1 3 = 0.
y = l- 4 t
Bai 1.4.14. Trong mat phang voi he toa do O x y , cho diem C(2;0)va elip
x^
(E): — + ^ = 1 . T i m tga do cac diem A , B thuoc (E). Biet rang A , B dói xung
i\hau qua true hoanh va t a m giac ABC la tarn giac deụ
Jiuung đn gidi
Gpi A(Xo; y Q ) , do A, B dói x u n g nhau qua tryc hoanh nen B(xQ;-yo).
A B 2 = 4 y o 2 ; A C 2= ( 2 - x „ ) ' + y( , 2
2
(1)
V i A e ( E ) ^ 4 . > : f = l . . y o ^ = i ^
4 1 4
A B C la t a m giac deu nen A B - A C o 4yô = (XQ - 2)^ + yô(2)
T u (1) va (2) ta co : 7\Q^ - 16xo + 4 = 0 <=>
X o = 2
CtyTNHH MTV i> l 17/ KItang Viet
Voi X Q = 2 => yo = 0 A s C loai
2 , 2 48 4V3
Vai xo = - thay vao (1) ta dugc y^ = — <=> XQ = + —
Vay A ( 2 4 S ] J 2 -4V3 7 ' 7 ' 7 ' 7 hoac A