BC.BA _ 2.1 + 0 + 0 _ 72
" 7 5
BC.BA N/5.72
V i du 7: Trong khong gian toa do Oxyz, cho tam giac ABC co A ( l ; 2; -1), B(2; - 1 ; 3), C(-4; 7; 5).
a) Tinh do dai duong cao hA ciia tam giac ve tir dinh A.
b) Tinh do dai duong phan giac trong ciia tam giac ve tir dinh B.
Giai
a) Ta co AB = (1; - 3 ; 4) AC = (-5; 5; 6), BC = (-6; 8; 2) [ A B A C ] = ( - 3 8 ; - 2 6 ; - 1 0 )
Vay SABC = - [AB, AC] I = - 7382 + 262 + 102 /555 2S ABC 2N/555 /555
BC V104 726
b) Gpi D(x; y; z) la chan dudng phan giac ve tir B:
Ta co , DA BA 726 1 DC BC /104
V i D nam giua A, C nen DA = — DC
5 2
O CD = 2DA <=> \
2(1 - x) = x + 4 2 ( 2 - y ) = y - 7 o 2 ( - l - z) = z - 5
2 x = —
3 11
y = T
z = 1
Vi du 8: Cho hinh binh hanh ABCD co 3 dinh A(3; 0; 4), B ( l ; 2; 3), C(9; 6; 4).
a) Tinh goc B cua tam giac ABC
b) Tinh dien tich hinh binh hanh ABCD.
Giai a) Ta co BA = (2; - 2 ; 1), BC = (8; 4; 1)
Ta co cosB = cos( B A , BC) = -j=-.
75
b) Hinh binh hanh ABCD co dien tich: S = 2 SAB C = 2.- I [ B A , BC ] | 1 2
Taco [ B A , BC] = (-6; 6; 24) => S = 18 72
Vi du 9: Cho b6n diem co toa do A(2; 5; -4), B ( l ; 6; 3), C(-4; - 1; 12), D(-2; -3; -2)
a) Chung minh ABCD la mpt hinh thang.
b) Tinh dien tich hinh thang ABCD.
Giai
a) AB = ( - 1 ; 1; 7), AC = (-6; -6; 16), hai vecto nay khong ciing phuong vi toa dp khong ti le suy ra A, B, C khong thang hang va co: A_
DC = (-2; 2; 14) = 2 AB => AB // CD.
Vay ABCD la hinh thang
") SABCD = SABC + SADC D"
= - | [ A B , A C ] | + - | [ A D , A C ] | = 371046 V i d u 10: Cho hai diem A(2; 0; -1), B(0; - 2 ; 3) 2 2
a) Tim toa dp diem C e Oy de tam giac ABC co dien tich bang T i l va thoa man OC > 1.
b) Tim toa do diem D e (Oxz) &k ABCD la hinh thang co canh day A B . Giai
a) Gpi C(0; y; 0) => AB = (-2; - 2 ; 4), AC = (-2; y; 1).
11 o-v/(2 + 4y)2+36 + (2y + 4r = V l l
b)
Vi
a)
Ta co: SABC = V U
O - I [ A B , A C ] |
20y2 + 32y + 12 = 0 <=> y = -1 hoac y = -— (loai) VayC(0;-l;0)
Goi D(x; 0; z) e (Oxz) => DC = (-x; - 1 ; -z)
ABCD la hinh thang khi va chi khi A B , DC cung huong
_ x - 1 -z .
<=> — = — = — < = > x= l, z = - 2 .
- 2- 2 4
V a y D ( l ; 0; - 2 )
du 11: Cho b6n diem A ( l ; 0; 0), B(0; 1; 0), C(0; 0; 1) va D(-2; 1; -2) a) Chung minh rang A. B, C, D la bon dinh cua mot hinh tu dien.
b) Tinh goc giua cac duong thang chua cac canh doi cua tu dien do.
c) Tinh the tich tu dien ABCD va do dai duong cao cua tu dien ve tir dinh A.
Giai
Taco: AB = ( - 1 ; 1;0), AC = ( - l; 0 ; 1), AD = ( - 3 ; l ; - 2 ) nen [ A B . A C ] / 1 0 0 - 1 - 1 1 \
V 0 1 1 - 1 - 1 0 / 0 ; i ; i )
D o a i [ A B , A C ] . AD = - 3 + 1 - 2 = -4 * 0, suy ra ba vecto A B , AC, AD kho lg dong phang. Vay A, B, C, D la bon dinh ciia mot hinh tir dien.
Taco CD = ( - 2 ; l ; - 3 ) , BC = ( 0 ; - l ; 1), BD = ( - 2 ; 0 ; - 2 ) .
Goi a, p, y lan luot la goc tao boi cac cap duong thang: AB va CD, AC va BD, A D va BC thi ta co:
•01 3N/7
cos a , — . — ., 2 + 1 cos(AB,CD) V2.N/14
2 + 0-21 14 V2.V8 0
1-1-21 377 cosp = |cos(AC,BD)|
cosy = |cos(AD,BC)| =
Th£ tich tii dien ABCD la V = -1[AB AC] .AD | =
V i
yfe.yfU 14
"A B A C ] . A D I = - 6IL J I 3 Do dai duong cao ve tir dinh A la:
3V 3V 2^3
A — =
"i—^Zi
^BCD i | r :
du 12: Cho tir dien ABCD co: A ( - l ; 2; 0), B(0; 1; 1), C(0; 3; 0), D(2; 1; 0) 21 a) Tinh dien tich tam giac ABC va t h i tich tii dien ABCD.
b) Tim hinh chiiu ciia D len mat phang (ABC).
| [ B C , B D ] |
Giai
a) Taco AB = ( ! ; - ! ; 1), AC = ( 1 ; 1;0), AD = (3; - 1 ; 0) n e n [ A B , A C ] = ( - 1 ; 1;2) T S -r-x -, I V6
SA Bc = - | l [ A B , A C1 V a [ A B , A C ] . A D = - 4 ^ > VA BC D=± I [ A B . A C ] . AD | = - b) Goi H(x; y; z) la hinh chieu D tren mat phang (ABC) thi: 6 i
AH = (x + 1; y - 2; z ) , DH = ( x - 2 ; y - 1; z). Taco:
DH.AB = 0 DH.AC = 0
AB,AC] A H = 0
x - y + z x + y = 3 x - y
<=> i 2z
4 x = —
3 5
Y =3 4 z = —
3 Vay H 4 5 4
.3 3 3,
Vi du 13:Trong khong gian Oxyz, cho 4 diem A(l; 0; 3), B(-3; 1; 3), C(l; 5; 1) va M(x; y; 0). Tim gia tri nho nhat T = 2 | M A | + |MB + MC|
Giai Goi I la trung diem cua BC:
=> I ( - l ; 3 ; 2 ) = > M B + MC = 2MI =>T = 2(MA + M I )
zA = 3 > 0 v a z i = 2 > 0 = > A v a I nam ve ciing 1 phia doi vdi mp(Oxy) va M(x; y; 0) thuoc mp(Oxy) nen lay doi xirng I ( - l ; 3; 2) qua mp(Oxy) thanh J ( - l ; 3; -2) => M I = MJ => T = 2(MA + MJ) > 2AJ - 2 738
Dau = xay ra khi M la giao diem ciia doan MJ vdi mp(Oxy) la
M 1 9
5 ' 5 0 Vay minT = 2738.
Vi du 14: Cho hinh lap phuong ABCD.A'B'C'D' co canh bang a. Goi I , J lan luot la trung diem ciia A'D' va B'B.
a) Chiing minh rang IJ _L A C . Tinh do dai doan thang IJ.
b) Chung minh rang D'B 1 mp(A'C'D), mp(ACB').
c) Tinh goc giira hai dudng thang IJ va A'D.
Giai a) Chon he toa do Axyz sao cho
A ( 0 ; 0 ; 0 ) , D(a; 0; 0), B(0; a; 0), A'(0; 0;a). Ta co C (a; a; a), B'(0; a; a), D'(a; 0; a) nen:
l ( | ; 0 ; a ) ; J(0; a; | )
Taco: LJ = ( 0 - - ; a - 0 ; -
2 2 = (~ 2; a ;- 2}
AC' = ( a - 0 ; a - 0 ; a - 0 ) = (a; a; a) a
nen LJ. AC' = - - .a + a.a - - .a = -a2 2 + a2 = 0.
V a y l J l A C . Doan IJ = -\f \2 r \2 1 a— + a + 1 2 1 a 1
I 2) 1 2 j
b) DS chung minh D'B 1 mp(A'C'D), ta chung minh
D ' B I A ' C . D ' B l A ' D o D ' B . A ' C = 0, D ' B . A ' D = 0 Taco D ' B = (-a; a;-a), A ' C = (a; a; 0); A ' D = (a; 0;-a) Do do D ' B A ' C = 0, DTJ.AT) = 0
Tuong tu, ta cung chung minh duoc D'B 1 mp(ACB')
c) A ' D = (a; 0; -a). Goi cp la goc giua hai dudng thang IJ va A'D thi:
•— .a + a.O (-a)
I J . A ' D 2 2
coscp = cos(JJ, A ' D )
IJ.A'D 176
iV2 Vay cp = 90°
V i d u 15- Cho hinh lap phuong ABCD.A'B'C'D' canh bang a. Tren cac canh BB', CD, A'D' lan luot lay cac diem M , N , P sao cho B'M = CN = D'P = ka
: 0< k < 1)
a) Tinh dien tich tam giac MNP theo k va a.
b) Xac dinh vi tri M tren BB' de dien tich tam giac MNP co gia tri be nhat.
Giai:
Chon he true toa do Axyz nhu hinh ve A(0; 0; 0), B(a; 0; 0)
C(a; a; 0), D(0; a; 0) A'(0; 0; a), B'(a; 0; a) C(a; a; a), D'(0; a; a)
a) ECM = k B T l => M(a ; 0; a - k a ) CN = kCD => N(a - ka; a ; 0) D ' P = k D ' A ' = > P ( 0 ; a - k a ; a)
=> M N = (-ka; a; -a + ka), MP = (-a; a - ka; ka) nen:
[ M N MP ] = (k2a2 - ka2 + a2; k2a2 - ka2 + a2; k2a2 - ka2 + a2) SMNP=i |[MN, MP] | =^(k2-k+l)vdike (0; 1)
z z
b) Taco: k2- k + 1 = ( k - - )2+ - > | 2 4 4
Dau = khi k = - e (0; 1) nen SMNP be nhat khi M la trung diem BB'.
2
V i d u 16: Cho hinh lap phuong ABCD.A,B,C,Di canh a, tren BC, lay diem M sao cho DTM, DA^, AB[ dong phang. Tinh dien tich S cua A M A B j .
Giai
Chon he Oxyz sao cho B = O, B,(a; 0; 0), C,(a; a; 0), C(0; a; 0), A(0; 0; a), Ai(a;0;a),Di(a; a; a), D(0; a; a).
V i M G BC| nen goi M(x; x; 0) Ta co L \ M = (x - a; x - a; -a)
DAX = (a; -a; 0) A BX= (a;0;-a)
Vi DjM, DAX, ABj dong phang nen 3a [ D ^ D A ^ A B j = 0 => x = —-2
^ --(3a 3a Do do M —,
D D,
/ i / i / i
/ i ' l A, c > X . _ _
/ằ \_——
/ \ Ci
B X
1 2 2 0 nen MA
Vay: S [ M A , M B ]
M B: = a2Vl9
3a 3a
2 2
Vi du 17: Lang tru tii giac diu ABCD.A1B1C1D1 co chieu cao bang nua canh day Diem M thay doi tren canh AB. Tim gia tri Ion nhat ciia goc
ATM C
Giai Chon he true nhu hinh ve (Aixyz) Dat A M = x, 0 < x < 2 .
Ta co: M(x; 0; 1 ) ; Ai(0; 0; 0); Ci(2; 2; 0) nen M A i = (-x; 0; -1);
MCj = ( 2 - x ; 2 ; - l ) Dat a = A^MCTi thi:
A D
A
/ \
\ \
\
\
/ \
A ' \
cosa = c o s ( M A i , M CX)
x2 - 2 x + l ( x-l )2
>/x2+l.>/(2-x)2+5 V x2+ l . v / ( 2 - x )2+ 5 >0
Do do a < 90° Vay goc a = A 7 M CI lan nhat khi x = 1 tuc M trung diem AB.
V i d u 1 8 : C h o h i n h c h 6P S.ABC co duong cao SA = h, day la tam giac ABC
vuong tai C, AC = b. BC = a. Goi M la trung diem ciia AC va N la diim sao cho SN = - S B
a) Tinh do dai doan thang MN. 3
b) Tim su lien he giua a, b, h d l M N vuong goc vdi SB.
Giai
Ta chon he true toa do Oxyz co g6c O triing vdi A, tia Ox triing vdi tia AC, tia Oz triing vdi tia AS sao cho d i i m B nam trong goc xOy. Khi do:
A(0; 0; 0), C(b; 0; 0), B(b; a; 0), S(0; 0; h). M ( - ; 0; 0) SB = (b; a; -h)
Goi N(x; y; z) thi SN = (x; y; z - h) Tir dieu kien SN = - SB nen
3
_ b a , - h
X = 3 'Y = 3 V a Z ~ h = T 3
a) Taco M N
b a 2h 3 ' 3 ' l i b _ b a 2h 3 2 ' 3 ' T
l - f _ b . a 2h^
J ~ l 6 ' 3 b_2
36
4h5
- V b2 + 4 a2 +16h2 6
b) M N vuong goc vdi SB khi va chi khi MN.SB = 0 - b2 -2h2
0 cs> 4h2 = 2a2 6 3 3
Vằ d u 1 9: Cho tir dien SABC co SC = CA = A B = aV2, SC 1 (ABC), ta
giac ABC vuong tai A. Cac diim M e SA, N e BC sao cho:
A M = CN = t ( 0 < t < 2a).
a) Tinh do dai doan M N . Tim gia tri t d l M N ngan nhat.
b) Khi doan M N ngan nhat, chung minh M N la dudng vuong goc chung ciia BC va SA.
Giai
a) Ta chpn he true Oxyz sao cho g6c toa dp O = A. True Ox chua AC, true Oy chira A B va true Oz 1 (ABC). Khi do canh SC song song vdi true Oz va ta co:
r'," J •- * $
A ( 0 ; 0 ; 0 ) B(0; aV2 ; 0), C(aV2 ; 0; 0), S(aV2 ; 0; aV2 )
M M N M N
t\/2 „ tj2] ( r- tJ2 tJ2 ) 2 2 ) { 2 2
V2(a-t,;^;-^)
2(a2 - 2 a t + t2, t2) 2 t2
\/3t 4at + 2a2 2a 2a<
Vay M N ngan nhat bang khi t b) Khi M N ngan nhat thi: M as/2 Q aV2
2a 3
\
va N 2ax/2 aV2
^ f aV2 aV2 aV2^
_ , |MN.SA = 0 ia co {
[MN.BC = 0
=> MN la duong vuong goc chung ciia SA va BC.
Vi du 20: Cho h'nh chop tii giac deu S.ABCD canh day a, mat ben tao voi
day goc a. Tim tana dk SA vuong goc SC.
Giai
Chon he true Oxyz co O la tam day ABCD. tia Ox chiia A, tia Oy chiia B. tia Oz chiia S. Ta co-
*V2 ^ 'aV2
; 0;0 , B 2
as/2
va S 0 ; 0 ; - t a n a 2
0: ;0
>
0;0 , D O ; -a ^ ; o ]
) I 2 J
nen SA aV2 a .
———, U — tan a 2 2
, SB 0; lV2
2 ' 2 — tan a
SC = aV2 a
, U — tan a v.2 2
,SD U, , tan a 2 2 Ta co S A ' . SC
tan2 a - 1 = 0 2 2 2
ô SA.SC = 0 o - — + — tan2 a = 0 ô —
2 4 2 v2
<=> tan2 a = 2 <=> tan a = \J2
Vay neu tana = %/2 thi hai canh ben doi dien ciia hinh chop vuong goc voi nhau.
D A N G 3: MAT C A u
Phirong trinh mat cau: (S) tam I(a, b, c) ban kinh R:
(x - a)2 + (y - b)2 + (z - c)2 = R2 hay:
x2 + y2 + z2 + 2Ax + 2By + 2Cz + D = 0, A2 + B2 + C2 - D > 0 co tam I ( - A , - B , - C ) va ban kinh R = %/A2 + B2 + C2 - D
Chii y: - Tam duong tron I(x; y; z) ngoai tiep tam giac ABC trong khong gian:
f l A = IB = IC [ I G (ABC)
Tam I(x; y; z) ciia mat cau ngoai tiep tii dien ABCD la diem each deu 4 I A = IB f A I2 = B I2
IA = IC o IA = ID
dinh: I A = IB = IC = I D <=> A I2 = C I2 A I2 = D I2
Ung dung toa dp trong khong gian de giai cac bai toan hinh khong gian co dien, quan he song song, vuong goc, dp dar, goc , khoang each, v j tri tuong doi,...
V i du 1: Tim toa dp tam va tinh ban kinh ciia moi mat cau sau day:
a) x2 + y2 + z2 - 8x + 2y + 1 = 0.
b) 3x2 + 3y2 + 3z2 + 6x - 3y + 15z - 2 = 0.
b) 9x2 + 9y2 + 9z2 - 6x + 18y + 1 = 0.
d) x2 + y2 + z2 - x + y - 2z + 100 = 0.
Giai a) Taco: a = - 4 , b = l, c = 0, d = 1.
nen a2 + b2 + c2 - d = 16 + 1 - 1 = 16 > 0
Mat cau (S) co tam I(-a; - b ; -c) nen 1(4; - 1 ; 0) va ban kinh R = V a2+ b2+ c2- d = Vl6 = 4.
b) 3x2 + 3y2 + 3z2 + 6x - 3y + 15z - 2 = 0
<=> x2 + y2 + z2 + 2x - y + 5z
Do do, mat cau co tam 11 -1;
c) 9x2 + 9y2 + 9z2
3 1 5 2' 2 6x + 18y + 1 = 0
= 0
va co ban kinh R _7V6
176'
<=> x2 + y2 + z2 x + 2 y + - = 0 3 ^ 9
Do do mat cau co tam I(—; -1; 0) va co ban kinh R= 1.
d) Taco: a = --,b = -,c = -l,d = 100 ' 2 2
nen a2 + b2 + c2 - d = - + - + 1 - 100 < 0 Vay do khong la phuong trinh mat cau. 4 4
V i du 2: Lap phuong trinh mat cau trong cac truong hop sau:
a) Co tam 1(5; - 3 ; 7) va co ban kinh R = 2.
b) Co tam 1(4; - 4 ; 2) va di qua goc toa do
c) Di qua diem A(2; - 1 ; -3) va co tam 1(3; - 2 ; 1).
Giai
a) Phuong trinh mat cau tam I(a; b; c), ban kinh R:
(x - a)2 + (y - b)2 + (z - c)2 = R2 nen phuong trinh mat cau can tim la:
(x - 5)2 + (y + 3)2 + (z - 7)2 = 4.
b) Mat cau tam I di qua goc toa do nen co ban kinh R = 10 = \/l6 + 16 + 4 = 6.
Vay phuong trinh mat cau: (x - 4)2 + (y + 4)2 + (z - 2)2 = 36.
c) Ban kinh R = I A = 3 V2 nen co phuong trinh:
(x - 3)2 + (y + 2)2 + (z - l )2 = 18.
Vi du 3: Lap phuong trinh mat cau:
a) Nhan doan A B lam duong kinh voi A(6; 2; -5), B(-4; 0; 7).
b) Di qua ba diem A(0; 8; 0), B(4; 6; 2), C(0; 12; 4) va co tam nam tren mp(Oyz).
Giai
a) Mat cau nhan A B lam duong kinh nen tam I la trung diem cua A B , do do AR 1
1(1; 1; 1) va co ban kinh R = = - V l 0 0 + 4 + 144 = V62 2 2
Phuong trinh mat cau: (x - l )2 + (y - l )2 + (z - l )2 = 62.
hay: x2 + y2 + z2 - 2x - 2y - 2z - 59 = 0
b) Tam I ciia mat cau nam tren mp(Oyz) nen 1(0; b; c).
Ta co I A = IB = IC nen:
J l A2 = I B2 f ( 8 - b )2 + c2 = 42+ ( 6 - b )2+ ( 2 - c )2 [ I A2 = I C2 ° [(8 - b)2 + c2 = (12 - b)2 + (4 - c)2 Giai ra duoc b = 7 va c = 5.
Vay 1(0; 7; 5), R = I A = VO + 1 + 25 = V26
Mat cau co phuong trinh: x2 + (y - 7)2 +(z - 5)2 = 26.
Y j du 4: Cho A(0; - 2 ; 1), B ( - l ; 0; 1), C(0; 9; -1). Lap phuong trinh mat cau co duong tron Ion la duong tron ngoai tiep tam giac ABC.
Giai:
Goi I(x; y; z) la tam duong tron ngoai tiep tam giac ABC.
Ta cc^AB_= ( - 1 ; 2; 0), AC = (0; 2; -2), A I =(x; y+ 2; z - 1)
=> [ AB , AC ] = (-4^-2; -2)
nen I e(ABC) ô [ A B , AC] Al=0 o 2 x + y + z + l = 0 _ 1
A I2 = B I2 Ta co A I2 = C I2 o
I e (ABC)
-2x + 4y = -3 y - z = - 1 <=> -j 2x + y + z = - 1
6 5 6 1 z = — 1 5 1 6
nen tam 11 —; — ; — I va ban kinh R = A I 6 6 6,
Vay PT mat cau la X + T; + y + 7 + z
5^3 6 75
~36' V i du 5: Lap phuong trinh mat cau:
a) Co ban kinh bang 2, tiep xuc voi mat phang (Oyz) va co tam nam tren tia Ox.
b) Co tam 1(1; 2; 3) va tiep xuc voi mp(Oyz).
Giai
a) V i tam I ciia mat cau nam tren tia Ox va mat cau tiep xiic voi mp(Oyz) nen diem tiep xiic phai la O, do do ban kinh mat cau la:
R = IO = 2 v a I ( 2 ; 0 ; 0 ) .
Mat cau co phuong trinh: (x - 2)2 + y2 + z2 = 4.
b) V i mat cau co tam 1(1; 2; 3) va tiep xiic voi mp(Oyz) nen ban kinh R ciia mat cau bang khoang each tir I toi mp(Oyz), vay R = I xT j = 1.
Mat cau co phuong trinh: (x - l )2 + (y - 2)2 + (z - 3)2 = 1
V i du 6: Cho A ( l ; 2; -4), B ( l ; - 3 ; 1), C(2; 2; 3), D ( l ; 0; 4). Lap phuong trinh mat cau ngoai tiep tii dien ABCD.
Giai
Goi I(a ; b; c) la tam mat cau ngoai tiep tii dien
_6_
~ I T i_
~~ TT TT
Do doi
riA = IB A I2 = B I2 - y + z = - 1 IA = IC • A I2 = C I2 ằ < x + 7z = -2 a- IA - I D A I2 = D l2 j - 4z = 1
6_ 1_
• I ' l l ' 11 v a R = I A = V 2 6 . Vay (S): (x + 2)2 + (y - l )2 + z2 = 26.
178
Vi du 7: Cho tii dien OABC co A(2; 0; 0), B(0; 4; 0), C(0; 0; 4).
a) Lap phuong trinh mat cau ngoai tiep tii dien OABC.
b) Tinh t h i tich hinh chop O.ABC.
Giai
Phuong trinh ciia mat cau (S) can tim co dang:
a)
x2 + y + z + 2ax + 2by + 2cz + d = 0 Ta co (S) qua g6c O(0; 0; 0) nen d = 0.
Vi (S) qua cac diem A(2; 0; 0), B(0; 4; 0), C(0; 0; 4) nen ta co:
4 + 4a = 0 fa = - l
• 16 + 8b = 0 <=> | b = -2 16 + 8c = 0 [c = -2 Vay phuong trinh ciia mat cau la:
x2 + y2 + z2 - 2x - 4y - 4z = 0.
Cach khac: Ta co OA, OB, OC doi mot.
vuong goc nen goi M la trung diem AB thi M I = - O C
2
Tir do suy ra tam I( 1; 2; 2) b) V - - SoAB
o
O C = - .OA.OB.OC = —
Vi du 8: Lap phuong trinh cac mat cau doi xiing ciia mat cau 6 3 (S): x2 + y2 + z2 + 4x + 8y - 2z - 4 = 0.
a) Qua mp(xOy) b) Qua mp(yOz).
Giai
Mat cau (S) da cho co tam I(-2; -4; 1), R = 5.
a) Goi (S') la mat cau doi ximg cua (S) qua mp(xOy) thi co t i m I'(-2; - 4 ; —1) va R' = R = 5 nen co phuong trinh:
(S'): (x + 2)2 + (y + 4)2 + (z + l )2 = 25.
b) Goi (S") la mat cau doi xiing ciia (S) qua mp(yOz) thi (S") co tam I"(2; -4; 1) va R" = R = 5 nen co phuong trinh:
(S"): (x - 2)2 + (y + 4)2 + (z - l )2 - 25.
Vi du 9: Cho diem A(3; 0; 0), B(0; 4; 0). Lap phuong trinh mat cau co tam la hinh chieu H cua goc O len duong thang A B va ban kinh R = \/l975
Giai Ta co A thuoc true Ox, B thuoc true Oy.
Trong tam giac AOB: A O2 = AH.AB A H OA'
AB
A H AB Do do A H
25 AB = OA2 A B2
27 36 25 25
9^
25
;0 H 48 25 ^ ; 0
25
Vay phucmg trinh mat cau can tim: [ x - — I 25, V i du 10: Cho phuong trinh:
x2 + y2 + z2 + 2xcosa - 2ysina - 4z - (4 + sin2a) = 0.
Tim a de phuong trinh tren la phuong trinh mot mat cau va tim a de ban kinh mat cau la nho nhat.
Giai
Ta co: a = cosa, b - -sina, c = - 2 , d = -(4 + sin2a)
=> a2 + b2 + c2 - d = cos2a + sin2a + 4 + 4 + sin2a = 9 + sin2a > Va Vay phuong trinh da cho la phuong trinh mat cau voi moi a
Ta co R = \/9 + sin2 a > 3 => Rmin = 3 khi a = kn, (k e Z).
V i du 11: Cho sau diem A(a; 0; 0), B(0; b; 0), C(0; 0; c), A'(a'; 0; 0), B'(0; b'; 0), C'(0; 0; c') vdi aa' = bb' = cc' * 0; a * a', b * b\ c * c'.
a) Chung minh co mot m'at cau di qua sau diem noi tren.
b) Chung minh dudng thang di qua goc toa do O va trong tam tam giac ABC, vuong goc vdi mat phang (A'B'C).
Giai
Ta xac dinh tam va ban kinh R cua mat cau qua 4 diem A, A', B, C.
Goi I(x; y; z) la t i m mat cau do, ta co: LA2 = LA'2 = I B2 = I C2 (x - a)2 + y2 + z2 = (x - a')2 + y2 + z2
(x - a)2 + y2 + z2 = x2 + (y - b)2 + z2 (x - a)2 + y
2 + z
2 = x2 + y2 + (z - c)2
y - | | |
<=> <
-2ax + a = -2a' x + a' -2ax + a2 = -2by + b2 -2ax + a2 = -2cz + c2 Do do x = a + a
y = aa Tam I a + a' b + aa
R = I B ' Va I B '2 =
2b a + a
a + a
2b va z c + aa 2c c + aa 2 ^
2c~
Ta co:
aa J2Y 2b
( 2 c + aa
2c aa
2b - b ' c2 aa 2c a + a
IB
^b2 aa 2b
2 A c +aa
2c
(vi aa' = bb1)
Tuong tu I C2 = IC'2 = I B2
Vay B',C ciing thuoc mat cau noi tren: dpcm.
b) Goi G la trong tam AABC => OG = f - ; - ; | )
\S 3 3J Ta co: A ' B ' = (-a'; b'; 0 ) , A T T = (-a'; 0; c'j nen OG.ATT = - — + ^ _ + o = 0 ;
3 3 OG.ATT = - ^ - + ^ + 0 = 0
3 3
Do do OG 1 A'B', A' C => OG _L mp(A'B'C).
Vi du 12: Cho bon diem A ( l ; 2; 2), B ( - l ; 2; -1), C(l; 6; -1), D( - l ; 6; 2) a) Chung minh ABCD la hinh tir dien va co cac cap canh doi bang nhau.
b) Tinh khoang each giira hai duong thang A B va CD.
c) Viet PT mat cau ngoai tiep tir dien ABCD.
Giai
a) Taco AB = (-2; 0; -3), CD = (-2; 0; 3), BD = (0; 4; 3),AC = (0, 4. -3)
=> [ A B , CD].BD * 0 nen ABCD la tu dien.
Ta co: AB = 74 + 0 + 9 = CD, AC = 70 + 16 + 9 = BD, A D = 74 + 16 + 0 = BC .
Vay tir dien ABCD co cap canh doi bang nhau.
b) Trung diem ciia A B la E(0; 2; -) , ciia CD la F(0; 6; -)
2 2 Ta co EF = (0; 4; 0) nen EF. AB = -2.0 + 0 . 4 - 3 . 0 = 0, E F. CD = 0.
Do do EF la doan vuong goc chung ciia A B va CD.
Vay d(AB, CD) = EF = 70 + 16 + 0 = 4 c) Trung diem ciia EF la I (0; 4; — ).
729 2
Ta co I A = IB = IC = ID = nen I la tam mat cau ngoai tiep ABCD.
2
1 29 Vay phuong trinh x2 + (y - 4)2 + (z - — )2 = —
d* T:
Vi du 13: Trong khong gian Oxyz cho b6n diem AQ; - 1 ; 2), B(l; 3; 2), C(4; 3; 2), D(4; - 1 ; 2).
a) Chiing minh A, B, C, D la bon diem dong phang.
b) Goi A' la hinh chieu vuong goc cua diem A tren mat phang Oxy. Hay v i i t phuong trinh mat cau (S) di qua bon diem A', B, C, D.
Giai
a) Ta co AB = (0; 4;0), AC = (3; 4;_0), AD = (3; 0; 0) n e n [ A B , AC] = (0; 0;-12) => [AB A C ] . A D = 0.
Vay A , B, C, D dong phang.
b) Hinh chieu cua A len mp(Oxy) la A ' ( l ; - 1 ; 0) Goi phuong trinh mat cau (S): x2 + y2 + z2 + 2ax + 2by + 2cz + d = 0
(S) qua A', B, C, D nen:
T + l + 0 + 2a-2b + 0c + d = 0 l + 9 + 4 + 2a + 6b + 4c + d = 0 16 + 9 + 4 + 8a + 6b + 4c + d = 0 16 + l + 4 + 8 a - 2 b + 4c + d = 0
a
<=> \
Vay mat cau (S): x2 + y2 + z2 5x - 2y
5 2 b = - 1 c = - 1 d = 1 2z + 1 = 0.
V i du 14: Cho mat cau x + y + z - 2x + 6y + 2z - 14 = 0.
a) Xet cac diem M ( l ; - 2 ; 1) va N(-3; - 1; 3) diem nao nam trong va diem nao nam ngoai mat cau?
b) Lap phuong trinh cua mat cau doi xung voi mat cau da cho qua mat phang (xOz). Hai mat cau do co cat nhau khong?
Giai
a) Ta co the viet phuong trinh mat cau da cho duoi dang:
(x- l )2 + (y + 3)2 + (z + l )2 = 25.
Mat cau nay co tam 1(1; - 3 ; -1), R = 5.
75 < R nen diem M o trong mat cau va 6 > R nen diem N nam ngoai mat cau.
Ta co I M I N
7 l2 + 22 b)
3)2 + (z + l )2 = 25.
716 + 4 + 16
Goi I' la diem doi xung Qiia I qua mat phang (xOz) thi diem I' co toa do la (1; 3; - 1 ) . Do do mat cau doi xung voi mat cau cho truoc qua mat phang (xOy) co phuong trinh la:
(x - l )2 + (y
Hai mat cau nay co khoang each d giira hai tam la 6 va ta co d < II' = 10 nen hai mat cau do cat nhau theo giao tuyen la mot duong tron.
V i du 15: Xet v i tri tuong doi ciia 2 mat cau:
(Si): x2 + y2 + z2 - 4x + 6y + 2z + 5 = 0 (S2): 2x2 + 2y2 + 2z2 - 2x + 8y - 6z - 1= 0
Giai
Mat cau (Si) co tam 1(2; - 3 ; - 1 ) , ban kinh R = 3.
Va (S2 2 2 2
<=> x + y + z x + 4y 3z = 0 . 2
nen co tam J(—; - 2 ; —), r = 1 3
v 2 2 z
[9 25 Ta co khoang each 2 tam IJ = J— + 1 + —
38 R + r = 3 + R
'4
= 3 26
V i R + r < IJ < I R - r | nen 2 mat cau cat nhau.
V i du 16: Xet v i tri tuong doi giua hai mat cau (S) va (S') co phuong trinh lan luot la:
(S): x2 + y2 + z2 + 2ax + 2by + 2cz + d = 0 (1) (S'): x
2 + y2 + z2 + 2a'x + 2b'y + 2c'z + d' = 0 (2) Giai
Xet he phuong trinh (1), (2). Trir hai phuong trinh do ve voi ve ta duoc he phuong trinh tuong duong:
x2 + y2 + z2 + 2ax + 2by + 2cz + d = 0 (1) 2(a - a )x + 2(b - b ')y + 2(c - c ')z + d - d1 = 0 (2')
V i tam I , I1 cua (S) va (S') theo thu ur co toa do (-a; - b ; - c ) , (-a'; -b'; -c') va ban kinh cua (S) va (S') theo thir tu la:
R = \/a2 + b2 + c2 - d, R' = Va ,2+ b ,2+ c '2- d' nen co cac truong hop:
a) a = a',b = b',c = c',d = d ' < * I = I ' , R = R <=> (S) = (S').
b) a = a' b = b', c - c', d ± d' o I = I ' , R * R' <=> (S) va (S') dong tam, (S) n (S1) = 0 , tuc la hai mat cau khong co diem chung.
c) I a - a' I + | b - b ' | + | c - c ' | * 0 <=> I * I' <=> (2') la phuong trinh ciia mot mat phang (a) vuong goc voi duong thang II1.
Vi (S) n (S') = (S) n (a) nen niu (S) n (S*) = (S) n (a) * 0 thi giao do hoac la mot diem thuoc duong thang II', hoac la mot duong Uon nam trong mat phang vuong goc voi II' (tam la giao cua mat phang do voi II').
V i du 17: Cho duong tron (C) co phuong trinh:
x2 + y2 + z2 + 2axx + 2hxy + 2cxz + d: = 0 (Sx) x2 + y2 + z2 + 2a2x + 2b2y + 2c2z + d2 = 0 (S2) Chung minh rang moi mat cau (S) di qua duong tron (C) deu co phuong trinh dang:
O O O 0 0 0 X(x + y + z + 2aix + 2biy + 2ciz + di) + u(x + y + z" + 2a2x + 2b2y
+ 2c2z + d2) = 0, trong do X ^ - p . Giai
Gia sir (S) la mat cau bat ki qua duong tron (C).
Lay mot diem M0( x0; y0; Zo) e (S) nhung M0 £ (C). Chon:
X = x2 + y2 + z2 + 2a2xa + 2b2y0 + 2c2z0 + d2 va p = - ( x2 + y2 + z2 + 2 al X o + 2 bi y o + 2cl Z p + dx) Tir do duoc dpcm.
V i du 18: Cho mat cau (SI co phuong trinh x2 + y2 + z2 - 4x + 4y - 2z = 0 va hai d i i m A(4; 2; 0), B(2; 1; 2). Chimg minh rang duong thang A B va mat cau (S) khong co diem chung.
Giai
Mat cau c6 tam 1(2; - 2 ; 1) va ban kinh R = 3.
Taco S iA B= i | [ I A , I B ] | = - I H . A B = — , A B = 3.
2 2 2 Dod6IH=^S- = ^>3 = R
AB 3