phuong phap toa do trong khong gian
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (181.02 KB, 29 trang )
H×nh gi¶i tÝch_HHKg
!"#$%& '"(")
*+,"" /0 $+1-&+1
o
45
2
2 345((2
62 7$898):2
6
%;<
1 1 1 1
ABCD.A B C D
*+,
1 1 1 1
AA ,BB ,CC ,DD
-& =#&
*(2 93">,*
1
CC
?
1
CM MN NC= =
2@A/BC
D+E 95"
1
B
"3-&2
2 34 !
1
A
-&=/BC2
62 7$8 =#&)+F8/BC:2
G
%;<2HHHH =#&*+12*+,H"H"
H"H2/I>J* =K$L?MM"MM"MM"HMM2
2 7$-N<>O/0P Q0H2
62 C8IR%&/0+SFP Q0HT
α
%&U/0R
-&/0HH27$>VWS)
α
2
GXY
Z>F[K$L Q0#5
x y z 1 0
x y z 1 0
+ + + =
− + − =
\&/0
1
(P ) : x 2y 2z 3 0+ + + =
2
(P ) : x 2y 2z 7 0+ + + =
\N<>/B>, Q0#-&NK] /0
1 2
(P ),(P )
2
^XX
_2 _(K"($"*T%* '+12
2 Z898:K-&$2
62 \.K"$&98%&%.S`
a
#IK$L-&
1
8
Q>T <-V
2 2 2
x y z 1+ + =
"
x 0, y 0,z 0≥ ≥ ≥
>#IS$23/0RNK]
1
8
/BS$*3"bK"$"L%B%c
*""?(d"(+d"(d2P>15
2
2 2 2
1 1 1
1
a b c
+ + =
2
62
2 2 2
(1 a )(1 b )(1 c ) 64+ + + ≥
2Z-V>8 93 9 *#S 0P2
a
I* = '-[K$L2Z>,e>J* =K"$"L%S$
9<P6MM"M6+M"MM d"+d"d2
2 Z8Ffg N/0:"+"2
62 Z898FE #Ih> h%& Qh>
2
i
#I-[K$L2Z>,K"$"L%S$%B%c 9""(
"(+"("+"d2
2 34+j2
62 j%&>k27$8:"+"2
G2 34+<#I8+1l+<#I8/T
%*)P#I2
mCXm
Z>F[ I* = '>kK$L3M6Mn-& Q0
#<>5
x 1 y 2 z 2
3 2 2
+ − −
= =
−
j%& 9 EKP)3D Q0#27$8 =#&32
YCXY
Z>F[ Ij = '>koK$L Q0#-&/
0R<>5
x 1 2t
(d) : y 2 t (P) : 2x y 2z 1 0
z 3t
= +
= − − − + =
=
2 Z* = 9=#?Ffgp 9 .R+12
62 jC%& 9 EKP 6MnMGD Q0#27$K V* =C2
XCXX
Z>F[ I* = '>koK$L Q0#-&/
0R<>5
x 1 y 1 z 3
(d) :
1 2 2
(P) : 2x 2y z 3 0
+ − −
= =
−
− + − =
2 Z* = 9)#-&R2Z8U#-&R2
62 \N<>N-[#H)#>,/0R2%S$ 9
1>,#?(" %&?E#<>.2@A!?E
AB AM
BM
+
93
# =>,/0R234q*=-V>8)3 9!?E *>V%.
S-&>V%.SS$2
XC
Z>F[ I>J* = '>koK$L+E 9_GMMn6"
aMGMn"6MGMn^"M6M2
2 34_ $%& '-&+/+,%&
-[2
62 Z8* = 9 EKP 9D Q023%& 9+SF>,
/B%&"+F8
R 18=
93F[=/02@A
=#&*+1 =#& *03"3"32W
S$ / 9`
C
Z>F[ I>J* = '>koK$L+E 9MM"
MM"MM"MM %&?E2
2 Z8FfU Q0-&F(62
62 j%&N-[)>,2Z>V)?E 9#I
8 *>V%.S2
R\ZZXY
Z>F[K$L Q>0∆<>5
2x y 1 0
x y z 1 0
+ + =
− + − =
-& Q0∆H<>
3x y z 3 0
2x y 1 0
+ − + =
− + =
2 34 Q0 b2Z 9)]2
62 \N<>lD)/0β D Q0∆-&∆H2
G2 Z98BF[.*+rβ-&+/0j =2
6R\ZZXX
Q0∆-&∆H<>? $5
x 1 y 1 z 2
( ) :
2 3 1
x 2 y 2 z
( ') :
2 5 2
+ − −
∆ = =
− +
∆ = =
−
2 34 Q0∆-&∆HA2
62 \N<> Q-[)∆-&∆H2
G_
Q0
1
(d )
6
-&# s<>5
1 2
x 1 t x 0
(d ) : y 0 (d ) : y 4 2t '
z 5 t z 5 3t '
= + =
= = −
= − + = +
2 34 Q0A2
62 j Q-[)
1
(d )
6
-&#
%&3
1
M (d ),∈
2
N (d∈
2Z* =
)3"-&-N<>?E) Q032
^BZ<XY
_ $%&U;2tS$3"%B%c>,*
_"_"?
SM SN
2
BM DN
= =
2
2 3/03b*_*R2Z8!?E
SP
CP
2
62 Z898_3R:98\)_
aBZ<XY
Z>F[K$L"/0R<>Ku$uLu(-& Q0
#<>
x 1 y 2 z 1
1 2 3
− − −
= =
\N<>N-[)#>,/0R2
i\\ZXY
!_" $%& Q>T+F8"'(Gv^
\& !_" $%&= %q*N2
2 Z8+F8/B=N2
62 N98FE+1^%B98FE"7$8#I8&B)
2
m\\ZXX
Z>F[ I* =K$L%;<2
1 1 1 1
A B C D
&MM"MM"MM"
1
D (0;0;a)
2j3%&> 9)"%&)
-[
1 1
CC D D
2Z+F8)/B D 9"
1
C
"3"2
Y\\Z
Z>F[ Q05
1 2
x 3 y 1 z 1 x 7 y 3 z 9
( ) : ( ) :
7 2 3 1 2 1
− − − − − −
∆ = = ∆ = =
− −
2 7$%;<>8b) Q0
3
( )∆
EKP
2
( )∆
D
1
( )∆
62 @A/0
α
5Ku$uLuG(2
\N<>N)
2
( )∆
:<
1
( )∆
%,/0
α
2
+ Z 93>,/0
α
9
1 2
MM MM+
uuuuur uuuuur
* c>VWS"
+N
1
M (3;1;1)
-&
2
M (7;3;9)
2
X\\Z
=U;2HHHH("(6"H(2
2 Z8FfU Q0H-&H2
62 j3%& 9 *:!?E
AM
3
MD
=
2Z8Ffg3 NH2
G2 Z898P#IHH2
6cXY
MM-& Q0
1 2
(d ),(d )
1 2
x y z 2 0
x 1 y 2 z
(d ) : (d )
x 1 0
3 1 1
+ − + =
− +
= =
+ =
t;<> Q0D"-[
1
(d )
-&b
2
(d )
2
6cXX
P#I+Nj = !6MGM"^MMn6"iMGMm"naMn
^MY2Z8 =#& Q)P#IKSg2
6c
Z>/0R-[*+12_%& 9+SF>, Q
0-[ R2
2 Z8:98B*N_2F_(62
62 3"%B%c%& 9# =>,*"3
∈
"
∈
-& /3("
(2Z=+9P%,IU-& 9/0_3-&_*
=
o
45
2
66&t*XX
_2 $%&U;"*_-[ $2
=#&*("(+"_(62j3%&> 9)_23/03b
:N#I`Z8#I8N#IS$2
6G&t*
=U;98+16m"#I8&B+1X-&*
%;&S?E2
2 Z8*)U;F(i2
62 @ 9q*=U;8S,>,2
6G&w
/0R<>
x 2y 3z 14 0− − + =
-& 9
3MnM
2 7$-N<>/0D3-&?? R2
62 7$j =N)3>,R2
G2 7$j = 9 EKP 3DR2
6^&w
P#I_2_(((
a 2
2_-[ "Z
-[" 93=_-&=?3((xx62
2 Z8 =#& *032
62 Z>V 93bS2
G2 C3bS7$P3%& Q-[)-&_2
6aZ\ZXm
Z>I* = '-[K$L+ 9
1 1 1
H( ;0;0),K(0; ;0),I(1;1; )
2 2 3
\N<>$N)/0C /0KuL(r#*8
b2
+ Z8?)0*+rC /0j =K$2
6iZ\ZXm
1>/0R2Z>, Q0-[ R*
%S$ 9_2j-&C%&N-[)%,_-&_2
2 34 9""""CO1>,=/B2
62 Z+F8)/B>,+N(6"(G"
·
o
BAC 60
=
2
6mZ\ZXY
\N < > / 0 N K] / B < >
2 2 2
x 2x y 4y z 6z 2 0− + − + − − =
-&?? /0R<>^KuG$n
6Lu(2
6YZ\ZXX
Z> I * = ' K$L / 0 R < >
16x 15y 12z 75 0− − + =
2
2 t;<>/B_%&Ej =-&NK] R2
62 Zj =N 9)R _2
G2 Z 9 EKP)Ej =DR2
6XZ\Z
%;<2HHHH"*) =#&+12Z>,
*H""HH%B%c%S$ 93""R?5H3((HR(xx2345
2
MN a.AB AD (a 1)AA'= − + + −
uuuur uuur uuur uuuur
62
AC'
uuuur
-[ /03R2
GZ\Z
'_2 !_* $ '+1" Q_(2
2 @N#I*+r /0R D* $-&-[
*+,_2
62 N!?E
h
3
a
=
/0R98:!?E&`
G\y
=U;2HHHH("(6"H(
a 2
-&3%&=
9= *"C%&> 9)H32
2 /3(
(0 m 2a)≤ ≤
2Z898FEP#IHC:-&> %&
)=2Z-V>8)3 998 *>V%.S2
62 C%&> 9)5
"WN#I)=b+r/0HC%&`
Z8#I8N#I :2
+"34 Q0H3NK] /B QF8H2
G6NXY
Z>F[ Ij =K$L Q05
1 2
x 2 2t x 1
( ) : y 1 t ( ) : y 1 t
z 1 z 3 t
= + =
∆ = − + ∆ = +
= = −
2 PW>1
1
( )∆
-&
2
( )∆
A2\N<>/0
( )α
P
1
( )∆
-&
??
2
( )∆
2
62 Z8FfU
1
( )∆
-&
2
( )∆
2
GGNXY
%z>J '2HHH* $+16-&'+12
2 kN#I)%z>J*+r/0 DH-&-[ *H2
62 8#I8)N#I>,2
G^N
Z>F[ Ij =K$L7$-N<>?E) Q0
1>/0$u6L(-&b Q05
1 2
x 1 t x 2 t
( ) : y t ( ) : y 4 2t
z 4t z 1
= − = −
∆ = ∆ = +
= =
GaN
_2%&=P#I%&-[ !-&(6M
*_-[ -&_(2
2 Z8Ffg N/0_2
62 j%&> 9)2Z8Ffg N_2
GiN
Z>F[ I>J* =K$L+ 9MM"M6M"MMG2
2 \N<>lD)/0""-&2
62 @* =)B=NP#I2
G2 Z* = 9{ EKP D2
GmN
P #I * " " [ = -[ -&
(((2C8I3""C%B%c%&> 9)*""2jh%&
9 EKP)DC-&%& 9)h 32
2 Ph-[ 32
62 Z8#I8)P3:2
GYN
_2 $%&U; (6"(2*
+,)+1-&+1
a 2
2
2 Z898)_22
62 j3""h"|%B%c%&> 9)*""_"_2P_
-[ 3h|2
G2 Z8Ffg N_2
GXCZyXm
'_2 Q_(-& $*+1
2 6
293"%&> 9)*"<P2Z898)_3-&
+F8B=N 2
^CZyXY
Z8FfU Q05
1 2
x 2y z 0
x 1 y 2 z 3
(d ) : (d ) :
2x y 3z 5 0
1 2 3
+ − =
− − −
= =
− + − =
^CZ>]Xm
Z>F[ I* =,K$L 9M6M-& Q05
x y 1
z 3
3 4
−
= = +
2
2 \N<>/0 D 9-&P Q02
62 Z8Ffg } N Q02
^6CZ>]XY
Z>F[ Ij = '>koK$LP#I_2
!_n6M6M^"n6M6M"naM6M"n6MM2
Z8Ff~z* E_-&2
^GCZ>]XX
Z>F[ Ij =-[K$L=P#I+E !
MM"iMGM"n6MXM"_MaMY2
2 P_-[ 2
62 34N)_%,-[ 2jC%& 9)
N 27$j =C2
G2 jR"y$ề%B%c%& 9U*_-&2Zj = 93>,_?
Ry-&C3b2
^^CZ>]
Z>F[ Ij =-[K$L 96MM"MGM"
MMG2 93"%B%c%&> 9)-&"R-&y%& 9>,-&
?
OP 2
OC 3
=
-& Q03"Ryb2\N<>/0
3Ry-&!?E
AQ
AB
2
^a\CZy_Xm
ZM6Ma-&<>>$N%&5
1 2
x 3 y 6 z 1 x 4 y 2 z 2
(d ) : (d ) :
2 2 1 1 4 1
− − − − − −
= = = =
− −
2 \N<>8b*)2
62 \N<>8b) Q>2
^i\CZy_XY
Z>F[ Ij = '-[^MM^"GMGM"MaMa"
MM2
2 ZN-[)%,/0-&898P#I2
62 \N<>?E Q0-[)-&2
^m\CZy_
Q05
1 2
x y 2 z 4 x 8 y 6 z 10
(d ) : (d ) :
1 1 2 2 1 1
− + + − −
= = = =
− −
2 \N<> Q0#?? K-&b
1
(d )
*3"b
2
(d )
*2
Zj =3"2
62 %& 9>,
1
(d )
"%& 9>,
2
(d )
"-[ f
1
(d )
-&
2
(d )
2\Nn
<>/B QF82
^Y\CZy_
Z>F[ Ij =>koK$L^MM"
o o
B(x ;y ;0)
o o
x , y 0>
?(Y-&
·
o
AOB 60
=
2 @ V>,L 998+1Y2
62 j%&>j)-& 93>,3(K2Z3 93
-[ 32
^Xt;XX
2 Z>I* = 'K$L/0R
x y z 3+ + =
-&/B
2 2 2
x y z 12+ + =
23/0Rb:$N n
Q>T2Z-&+F8) Q>T 2
62 Z>I* = 'K$LnM6MG-&/0
R5Ku6(-&y5$nLn(
\N<>/04D-[ fR-&y2
at;3
Z>F[ Ij = '-[K$L 9_MM"
MM2 93MM"MM$ l?u(-&d"d2
2 3498_23F[J=-&-&2
62 Z8Ffg N_32Zg ?$>_3NK] =/BE
V2
a3WVSXY
Z>F[ Ij =>koK$LKA Q0<>
x y 4 z 1
( )
4 3 2
− +
∆ = =
−
\&/0<>Kn$uGLuY(R
\N<>N-[)
( )∆
>,R2
a63WVSXX
Z>F[ Ij =>koK$L/B Q0
( )∆
-&
z0y%B%c<>5
2 2 2
(C) : x y z 2x 4y 6z 67 0
2x y z 8 0
( ) :
2x y 3 0
(Q) :5x 2y 2z 7 0
+ + − − − − =
− + − =
∆
− + =
+ + − =
2 \N<>Sf/0]
( )∆
-&NK] 2
62 \N<>N-[)
( )∆
%,y2
aG3WVS
Z>F[ Ij =>koK$LGM6MG" Q
1>, Q0
1
(d )
<>5
1
x 2 y 3 z 3
(d ) :
1 1 2
− − −
= =
−
\& Q>31>, <0
2
(d )
<>5
2
x 1 y 4 z 3
(d ) :
1 2 1
− − −
= =
−
Z8 =#&*)2
a^\&XY
Z>F[I* = '-[K$L-&-[
"-[*"1>/0K$& Q0?? >J
K-&(62@ V* = 9" 9"+N>1& =Kd-& =
$d2\N<>8b)/0 D 9MM"d"-[ n
Q0 D-&>j)P#I2
aa\&XX
%;<2HHHH*-&= 93>,*"3(K"
xKx2@A/0R D 93P QAHH)-[HH•H2
2 Z8#I8)N#I)%;<b+r/0R2
62 3/0R%;<&FE #I"7$K 998)
=>FE #I S [98)FE #IF2
ai\&3
P#I2jH"H"H"H<P%&>j)"
""2j%& 9)H"H2
2 P>15
AG 3
AA' 4
=
2
62 P>15H"H"H"H qD$2
am*UXm
Q0<>5
1 2
x 2 2t
x y 2z 0
(D ) : (D ) : y t
x y z 1 0
z 2 t
= − +
+ + =
= −
− + + =
= +
2 P
1
D
-&
2
(D )
A2
62 Z8FfU
1
D
-&
2
(D )
2
G2 \N<> Q0
( )∆
D 93MM-&b qQf
1
D
-&
2
(D )
2
am*UXX
,>>J>TK$=-[*=N& !
%,N"1>, Q>T $PS)>J" !T%*1>, Q
>T $P)>J23/0-[* $)>J=
o
45
2
Z8#I8KD-&98)>J2
aY*U
Z>F[ Q0A5
x 1 3t
2x 3y 1 0
(a) : (b) y 2 2t
y z 1 0
z 1
= − +
+ − =
= +
+ + =
=
Z8FfU-&2
aX*U
Z>F[K$L+E 96MM"M6M"MM6"6M6M"
d2
2 jh%&> 9) *"7$* = 9|) *0h
/02
62 Z8982
G2 Z* = 9H EKP D Q02
i*Z<XY
#I-[K$L2Z>,K"$"L%B%c%S$ 9""2
2 Z8#I8:("(+"(2
62 f?e""$ l%[uuuuu(FF51?E27$
K V>V%.S)98P#I2
i*Z<3
%;<2HHHH*+12f?e3-&%B%c%&>
9)-&H2
2 P3?? H2
62 Z8FfU Q0-&3:2
i6Xm
9M6MG-&^M^Ma>F[ I* =-[K$L2
2 \N<> Q02Z 9R) /0K$2P
W>1 j 9y=K$"+9P
QA QB−
>V%.SFy
>OR2
62 Z 93>,K$?l =#&3u3WS2
i6XX
Z>I* =>koK$L Q0#-&/0R<>
x 1 y 2 z
(d) :
3 1 1
− +
= =
(P) : 2x y 2z 2 0+ − + =
2 t;<>/B1>, Q0#"NK] R-&
+F8+12
62 j3%& 9)R #"Z%&N 9)/B R2Z83Z2
iG[t3
<05
x 1 3t
2x 3y 4 0
(d) : (d') : y 2 t
y z 4 0
z 1 2t
= +
+ − =
= +
+ − =
= − +
2 34 <0#-&#HA2
62 Z8FfU Q0 2
G2 9"F-&E V>,= Q0#?
AB 117=
2C
# =>,#H">VWS)#I82
i^\yyZXm
=U;2HHHH H("(+"(2Z898P
#IHH:"+"2
ia\yyZXY
%;<2HHHH *+12
2 7$8FfU Q0H-&H2
62 34 QAH-[ /0HH2
ii\yyZXX
P#I '*+12
2 f?e%&= 9$ l>,*27$K V-V>8) 9#I8
%&WS2
62 f?e3%&= 9=*2y 93#k/0??
-&23/0&$b*-&"%B%c*"R"y2ZP3Ry
%&`7$K V-V>8)3 9#I8P3Ry%&%.S2
im\yyZ
%;<2HHHH *+12f?e3""R"y%B%c%&
> 9)*HH"HH"H"H2
2 34+E 93""R"yO1>,=/02Z8-)P
3Ry:2
62 Z8#I8P3Ry:2
iY\yyZ
=U;2HHHH ("(+"H(2
2 Z8#I8)H:"+"2
62 f?e3"%B%c%&> 9)-&27$898P#IH3
:"+"2
iX\y€
_2 $%&-[*"*_-[
$2yF}-[ _"C-[ _2P_-[
C-&8#I8C+N>1("
BC a 3=
-&
SB a 2=
2
m\y€
e/0R"y-[ :$N
( )∆
2Z>,
( )∆
%S$
(%& =#&>.2Z>,e#Q0K-[
( )∆
-&r>y%S$
9?
2
2
a
BN
b
=
2
2 Z8Ffg '3:"+2
62 Z83:"+2\.>V&)3 =#&k92Z8 =#&k
9 2
m\y€
Z>Ij =K$L Q0
m
(d )
<>
mx y mz 1 0
x my z m 0
− − + =
+ + + =
2 \N<> Q0
( )∆
%&N-[)
m
(d )
%,K$2
62 34 Q0
( )∆
%[NK] = Q>TE V%&Ej =2
m6yXm
%& Q-[) Q0K-&$A"=K"=
$2/(#"%&= 9$ l=K"%&= 9$ l=$2/3("
(
(m 0,n 0)≥ ≥
2f?e%[
2 2
m n k 0+ = >
"FF[ l2
2 @ V" 9 =#& *3 *>V%.S"WS2
62 Z>>Qc Q0K"$-[ -&
mn 0≠
"7$K V"
:F-&# 998P#I3 *>V%.S-&8>V 2
mGyXm
*23= 93$ l>, Q0-[
*3F[>O
2 ZD•8>j-&>k)32
62 j%&>k)"7$K V-V>8)3 998P#I
*>V%.S2
m^yXm
-[*"2e Q0K"$-[
-&rO8 /0 2 93F[>O >,K" 9
F[>O >,$2/3("(2
2 Z898)232
62 Z83:""-& 'FI E "" 93-[2
mayXY
Z>F[ Ij = '-[K$L 9MM"
M+M"MM"+"d2k=U;;"""%&+E !-&j
%& ! E#I !)= 2
2 Z8Ffg N2
62 Z8* =N-[)KE/02Z 'FI E
"+" 9N 1>/0K$2
miyXY
Z>F[ Ij =,-[K$LKA '>
K$*+1" Q0?? >J$" 9=B
PS)K$2@A 9
a
S(0;0; )
3
2
2 @j =) 9"-&> 9h)"? -N<>)
RP_h-&KK K2
62 Z8Ffg NR"g ?$>FfU Q0K-&
_h2
mmyXY
Q>T+F842@A_2_-[
/0 $_-&E V"_(>."#$%&P‚ƒ=N Q
>T 7& QA-&-[ 2
2 Z8+F8)/B*N_22
62 $%& 998 *>V%.S`
mYyXX
Z>F[ I* ='-[K$L 9MM"
MM"MM"MM#d"#d2jH"H:Pk%&N-[)
KE Q0"2
2 \N<>/0P Q0H"H234/0
-[ Q02
62 Z8#: 9HH?E +1
o
45
2
mXyXX
%;<2HHHH2k/0P QA)
-[-& D> 93)*HH23/0 -[&
B2Z8!?E98) 2
Yy
9MMnG"6MMn-&/0R<>5
3x 8y 7 1 0− + − =
2 Zj = 9)/0R-& Q0 D 9"2
62 Zj =)1>,R?%& '2
Yy
Z>F[ Ij =>koK$L 9
A(1; 3;0)−
"
B(5; 1; 2)− −
-&/0R<>5
Ku$uLn(
2 34 Q0D-&bR*= 9= *2Z* = 92
62 Z>,R 93?
MA MB−
>V%.S2
Y6y
=%z>J P2HHH $%& !"
·
ABC = α
"
Hc $
β
2j%&> 9)H2N
·
BIC
%&-[2
2 34-[2
62 345
2 2
tg tg 1α + β =
2
YGy
Z>F[ Ij =,-[K$L/0??
1 2
(P ),(P )
<><P%&5
1
2
(P ) : 2x y 2z 1 0
(P ) : 2x y 2z 5 0
− + − =
− + + =
-& 9nMM1>FfU/0 2j_%&/B+SFD
-&NK] f/0
1 2
(P ),(P )
2
2 34+F8)B_%&=1?E-&8+F8 2
62 j%&)B_2P>1== Q>TE V2@j
=-&+F8) Q>T 2
Y^y"
_2 !_" $%&((G"(62N>1
/+,_"_"_ 'c / $=
o
60
2C} Q_)
2
2 PW>1%&-T>T=N-&_-[ 2
62 Z898)2
Yay3XY
Z>F[ Ij = '-[K$L Q0#-&/
0R<>2
x z 3 0
(d) : (P) : x y z 3 0
2y 3z 0
+ − =
+ + − =
− =
Z<>N-[)#>,R2
Yiy3XY
e Q0K"$A-&-[ "%& Q
-["(2Z%S$ 93>,K">,$ 3(K"($2
2 34/)P#I3%&-[2
62 Z898-&#I8&B)P#I3:"K"$2
Ymy3
_2 $%&-[*"_-[
"
SA a 2=
2Z>,*%S$ 93$ l2/3+1
α
2*_
-[ 32
2 P>1%[== T>TE V-&898P#I_
:-&
α
2
62 *-[ _"C-[ _2P_-[
C-&8 =#&C2
YY_R
Z>F[ 9"":Pk=K"$"L-[
g [=?(d"
OB a 2=
"(d2j%& ! E
#I )U;-&3%&> 9) *2R%&/0
D"3-&b:= Q0-[ Q032
2 jh%& })R "8 =#& *h2
62 Z8!?E98)FE #I c*&FbFE2
+rR
G2 Z8Ffg N/0R2
YX_R
Z>F[ Ij =K$L%;<2HHHH?
>O Ej ="MM"MM"HMM2j3%&> 9) *"
%&)-[HH2
2 \N<>/B_ D 9"H"3"2
62 Z8+F8 Q>T)_ /B D 9H""H"2
G2 Z8#I8N#I)%;<b+r32
X_R
U;-&h|F[O1>=/0-&f
7 'FI5("
AD AF a 2= =
" Q0-[ |2jC%& n
Q-[)-&|="C=|2
2 j%& 9) Q0| /0P-&?? |2
Z8!?E
DI
DF
2
62 Z8 =#& *C2
G2 Z8+F8/B=NP#IC2
X_R
=U;2HHHH("(6"
AA' a 2=
"3%&=
9= *"C%&> 9)H32
2 /3(
0 m 2a≤ <
2Z898FEP#IHC:-&"> %&
)=2Z-V>8) 93 998P#I *>V%.S2
62 C3%&> 9)5
WN#I)=b+r/0HC%&`Z8#I8
N#I :2
+ 34 Q0H3NK] /B QF8H2
X6_RXY
Z>F[ Ij =>koK$L Q0<>n
<P5
x 2 t
x 2z 2 0
(d) : y 1 t (d ') :
y 3 0
z 2t
= +
+ − =
= −
− =
=
2 P>1#-&#HA27$-N<> Q-[
)#-&#H2
62 \N<>#*lD)/0 '#-&#H2
XG_R
Z>F[ Ij =>koK$LMnM-& Q0:
Pk<>5
1 2
x t
3x y z 3 0
(d ) : y 1 2t (d ) :
2x y 1 0
z 3t
=
+ − + =
= − −
− + =
= −
P>1
1 2
(d ),(d )
-&O==/02
X^_R
P '_2" Q_-&/0
( )α
D-[
*+,_2N/0
( )α
b_
1
H
&
1
SH 1
SH 3
=
-&b*+,_"
_"_%B%c*H"H"H2
2 Z8!?E#I8N#IHHH-&#I8 $2
62 +N* $+12Z898)_2HHH2
Xa_RR
Z>I* =K$L Q0
1 2
x y 2z 0
x 2 y z 2
(d ) : (d ) :
x y z 1 0
1 2 1
+ + =
+ −
= =
− + + =
−
2 @A-V>8< EU Q0
1 2
(d ),(d )
2
62 \N<>N-[)
1
(d )
>,K$-&-N<
>N-[)
2
(d )
>,5
(P) : x 2y z 3 0− + + =
2
Xi_Ry$<XX
Z>F[ Q0<>5
1 2
x 1 3t
x y 0
(d ) : (d ) : y t
x y z 4 0
z 2 t
= +
+ =
= −
− + − =
= +
2 7$PW Q0
1 2
(d ),(d )
A2
62 Z8FfU Q0
1 2
(d ),(d )
2
Xm_Ry$<XX
_2 $%&e%J ' (6"(((
-& Q
SO a 3=
"> %&> 9)2
2 Z898)_22
62 j
α
%&/0D-&-[ _27$K VN#I)
Fb+r
α
XY_R3
Z>F[ I>Jj =K$L Q0
1 2
x 2y z 0
x 1 y 2 z 3
(d ) : (d ) :
2x y 3z 5 0
1 2 3
+ − =
− − −
= =
− + − =
Z8FfU Q0
1
(d )
-&
2
(d )
2
XX_R3
Z>F[ I>Jj = '-[K$L Q0#5
x 1 y 3 z 2
1 2 2
+ + +
= =
-& 9GM6M2@ 9 EKP)D#2
XX_R3
P ' _2 $ %& -[ * -&
_(_(_(_(2
2 Z8#I8&B-&98)_2:2
62 8?)V#I_"_2
_R3
#I-[K$L2Z>,+*K"$"L%B%c%S$ 9""
?("(+"("+"d2
2 j%&N-[)>,2P%&>k)
2Z8:"+"2
62 P>1
2 2 2 2
ABC OAB OBC OAC
(S ) (S ) (S ) (S )= + +
ABC
S
"
OAB
S
"
OBC
S
"
OAC
S
%B%c%&#I8)"""
_R\Xm
I>JK$L-&%;<2HHHH !>O E* ="
!MM"MM"HMM2 93"$ l>, *0H"n
<P?3((
0 a 2< <
2 \N<> Q032
62 Z 9 Q03 qQ-[ Q0H-&2
G2 @ V 9 *03 =#&+AS-&8 =#&+AS 2
^2 345C$ l Q03%[?? =/0E V2
7$-N<>)/0 2
6_R\XY
Z>F[ I* ='-[K$L 9MM"
M+M"MM> "+"%&?E#<2
2 34+j2
62 @+F8-&j =)/B*NP#I2
G2 Zj =) 9H EKP D2
G_R\XX
Z>F[ Ij =,-[K$LM6Mn6-&/0
R56Ku6$uLua(
2 t;<>/B_? 9)_-&R%& Q>T
-+1
8π
2
62 34/B_>BNK] Q0#<>56Kn
6($uG(L2
G2 t;<>/0P#-&NK] _2
^_R\XX
P#I23=
α
?? -&b*"""
<P* 93""R"y2
2 34P3Ry%&+&2
62 @-V>8)
( )α
9#I8)P3Ry *>V%.S2
a_R\
%;<2HHHH*+162jh"|<P%&>
9)*-&H2
2 34 Q0h|?? H-&8 =#&h|2
62 jC%&> 9)HH2Z8Ffg ! NhC|-&@
U Q0h|-&2
i_R\
Z>/0Re Q>T QF8"%&= 9=2
Z>,e Q0K-[ R%S$ 9_?_("jC"%B%c%&
Q-[*gKE_"_2
2 34_"C%&-[2
62 Z8 =#&)C:-&2
G2 @-V>8)>,?l#I8_-&%.S2Z
>V%.S 2
m_R\
%;<2HHHH*+12 93"$9 =
>, *-&H<P?3(H(2j
α
-&
β
%B%c%&*+r
3 Q0-&H2
2 Z8 =#&3:-&2Z 93 *>VWS2
62 Z8
α
-&
β
F3WS2
G2 Z>>QclD3IP5
2 2
1
cos cos
2
α + β =
2
YZCZXX
Z>F[ Ij =>koK$L Q0#-&/0R
<>5
x 1 y 1 z 2
(d) : (P) : x y z 1 0
2 1 3
+ − −
= = − − − =
Z<>8b) Q0
( )∆
DMMn6?? R-&
-[ #2
XZCZ
96MGMa-&R<>
2x 3y z 17 0+ + − =
2 \N<> Q0#D-&-[ R2
62 34 Q0#bL"#93)# L2
G2 ZH EKP DR2
Z$,Xm
Z> F[ I j = , -[ K$L %; <
2HHHH HMM"HM6M"H6MM2j3""R"y:Pk%&> 9
) *HH"HH"H"2
2 Zj =N)%,2
62 34 Q03y-&RO1>=/0-&8#I8P
3Ry2
Z$,
Z>F[ Ij =,-[K$L+E 9M6M6"n
M6Mn"MiMn"nMiM62
2 P>1%&=P#I-&/* E+12
62 Z8FfU Q0-&2
G2 \N<>*NP#I2
6Z3Xm
Q0A<>5
x 1 x 3u
(m) : y 4 2t (n) : y 3 2u
z 3 t z 2
= = −
= − + = +
= + = −
2 Z8FfU Q0-&2
62 \N<> Q-[) Q0-&2
GZ3XY
Z>F[K$L/0R D+ 9MM"nMn6M"6M
Mn2
2 \N<>lD)/0R2
62 \N<>?E) Q0# D>j)-&
-[ R2
G2 @ Q*gKE-&898P#I2
^Z3XX
Z>F[ I>Jj =K$L Q0#-&/0Rn
<>2
2x y 2z 3 0
(d) : (P) : x 2y z 3 0
2x 2y 3z 17 0
− − − =
− + − =
− − − =
2 Z 9 EKP)GMnM6D Q0#2
62 \N<>N-[)#>,R2
aZ3
\N<> Q0 D 936MnM-[-&b Q0#
<>5
5x y z 2 0
x y 2z 1 0
+ + + =
− + + =
iZ3
Z>F[ Ij =>koK$L Q0#<>5
x.cos y.sin z.sin 6sin 5cos
x.sin y.cos z.cos 2cos 5sin
α + α + α = α + α
α − α + α = α − α
\.
α
%&?E2
2 P>1#?? /05
x.sin 2 y.cos 2 z 1 0α − α + − =
62 j#H%&N-[)#>,/0K$234F
α
$ l"
Q0#H%[NK] = Q>TE V2
mZ%cXm
\N<> T0 DGMn6Mn^"?? /0<
>GKn6$nGLnm(" qQb Q0
x 2 y 4 z 1
3 2 2
− + −
= =
−
YZ%cXY
Z>F[/0R<>
2x 5y z 17 0+ + + =
\& Q0#<>
3x y 4z 27 0
6x 3y z 7 0
− + − =
+ − + =
2 @ 9) Q0# /0R2
62 \N<> Q0 D"-[ #-&1>R2
XZ%cXX
Q0
k
(d )
<>5
x 3 y 1 z 1
k 1 2k 3 1 k
− + +
= =
+ + −
"F%&?E2
2 P
k
(d )
%[1>=/0E V2\N<>/0
2
62 @ VF 9
k
(d )
?? /05
iKn$nGLnG(
\&Kn$u6LnG(2
6Z%c
Z>F[ Ij =>koK$L/B_-&/0
<>5
2 2 2
(S) : x y z 4
(P) : x z 2
+ + =
+ =
2 P>1Rb_2@-&+F8) Q>T%&$N)
R-&_2
62 \N<> Q
1
(C )
%&N-[)>,/0
K$2
6Z%c
Z>F[ Ij =>koK$L2
2 t;<>lD)/0 D 93MM"GMM-&*
/0K$=
3
π
2
62 9MM"M+M"MM "+"%&+?E#<$ l-&%[
f75
2 2 2
a b c 3+ + =
2@ V"+"?FfgMM N/
0%&%.S2
66\z
P_2 $%&-[*-&"
(("(62*+,_-[ / $-&
SD a 3=
%&?E#<>n
.2Zg> 9h)#khC-[ _C=_2
2 Z898_2:-&P_-[ hC2
62 34 9_"""h"C"O==/B2@-&+F8)/
B:2
G2 Z8Ffg> 93) *0_ N/0_:2
6G@
Z>F[ Ij =,-[K$LP '
_2"_GM6M^"M6MG"GMMG2
2 t;<> Q-[) Q0-&_2
62 j%&/B*N_22t;<>/0D
-&?? 2
G2 j%&>#9)"%&>k)2Z8 =#&2
6^€XX
_2_%& Q-& $%&-[*2
·
o
BSC 45=
2/
·
ASB = α
"
α
9V#I_+1
o
60
2
6a€
P '_2 =#&* $(-&
·
SAB = α
2Z89
8_2:-&
α
2
6i€
P#I"> ("((+"(("
α
%&0V#I
*
2
π
α <
2
\. 'FI& E +" Q0E 9Uh) 9U|
)%& Q-[)-&`\. 'FI-g c"7$P
B QF8 Dh"|-&898P#I 72
6m€Z
=U;2HHHH("(+"H(23"%B%c%&
> 9*HH-&2
2 \N<>/0 D3-&?? Q0H-&H2
62 Z898H2
G2 %& 9+SF‚=H2Z8!?E98HH-&%z>J
2HHH2
6Y€c3XY
Z>F[ Q0>2
1 2
x 7 y 3 z 9 x 3 y 1 z 1
(d ) : (d ) :
1 2 2 7 2 3
− − − − − −
= = = =
−
2 PW>1 %& Q0A2
62 t;<> Q-[) Q0 2
6X€c3
Z>F[ Q0
m
(d )
<>5
x my z m 0
mx y mz 1 0
− + − =
+ − − =
2 \N<>N
m
( )∆
)
m
(d )
%,K$2
62 34F$ l
m
( )∆
%[NK] = Q>TE V>K$2
6X€c3
P#I2
2 34 Q0Ep !)P#I >j)/ E#I q
D$*= 92j 9 %&2
62 34 ! $%&/)P#I98+1
2
G'6
'_2 !_" =#&* $+12j3"%B%c%&
> 9)*_-&_2Z8:#I8)3+N
3-[ _2
Z>F[ Ij =,-[K$L Q0<
>5
1 2
x 1 t
x 2y z 4 0
( ) : ( ) : y 2 t
x 2y 2z 4 0
z 1 2t
= +
− + − =
∆ ∆ = +
+ − + =
= +
\N<>RP
1
( )∆
-&?? 2
2
( )∆
+ 36MM^2Zj ==
2
( )∆
?3 =#&WS2
G'6
%;<2HHHH*+12
Z8:FfU Q0H-&H2
j3""R%B%c%&> 9)*H""HH2Z8U n
Q03R-&H2
G6'6
Z>F[ I* =,-[K$L/0R-& Q
0
m
(d )
2
m
(P) : 2x y 2 0
(2m 1)x (1 m)y m 1 0
(d ) :
mx (2m 1)z 4m 2 0
− + =
+ + − + − =
+ + + + =
@ V 9
m
(d )
?? R2
GG'G
2 %;<2HHHH2Z8?E )0V#I„"H"…2
62 Z>F[ I j = ,-[ K$L = U ;
2HHHH>O Ej ="MM"MM"HMM+"+d2j3
%&> 9)H2
Z898FEP#IH3:-&+2
+ @†?E
a
b
9/0H-&3-[ 2
G^'G
2 %z>J P2HHHH $%&*"
+1
o
60
2j3%&> 9)*H-&%&> 9)H234+E
9H"3""O==/027$8 =#&H: 9P
H3%&-[2
62 Z>F[ Ij = ,-[ K$L 96MM"
MMY-& 9?
AC (0;6;0)=
uuur
2Z8Ffg> 9) N
Q02
Ga'G
2 Z>F[ Ij =,-[K$L Q05
k
x 3ky z 2 0
(d ) :
kx y z 1 0
+ − + =
− + + =
ZF 9
k
(d )
-[ /0R5Kn$n6Lua(2
62 /0R-&y-[ "$N%& Q0
( )∆
2
Z>,
( )∆
%S$ 9" (2Z>/0R%S$ 9">/0
y%S$ 9?-&O-[
( )∆
-&((2Z8+
F8/B*NP#I-&8Ffg N/0
:2
Gik+V6
_2 $%& '*-&
_ ⊥
2Z8
Ffg 9 N_:"+N>1
i
_
6
=
2
Gmk+V6
Z>F[ Ij ='-[K$L/0R5
K $ L G − + + =
-& 9
M 6M G" aMmM6− − − −
2
2 Zj = 9H EKP DR2
+2 f?e3%&= 9*$>,/0R">VWS)+9P5
3 3+
2
GYk+V6
Z>F[ Ij ='-[K$L Q05
( )
6K $ L
5
K $ L 6
+ + + =
∆
+ + + =
-&/0R5
^K 6$ L − + − =
\N<>N-[)
( )
∆
>,R2
GXk+V6
Z>F[ Ij ='-[K$L Q05
K L
# 5
$ L
− − =
− + =
-&
6
K G$ G
# 5
K GL i
+ − =
+ − =
2 Z 9
6
# "#
b2
+2 \.(6"-N<>/0RP Q0
6
#
-&??
#
2Z8FfU
6
# "#
F(62
^k+V6
Z>F[ Ij ='-[K$L Q05
6K 6$ L
# 5
K 6$ 6L ^
− − + =
+ − − =
-&/B
6 6 6
_ 5 K $ L ^K i$ + + + − + =
2Z 9 Q0#b/B
_* 93"?FfU 9 +1X2
^k+VG
Z>F[ Ij ='-[K$LP#I
6MGM6"
iM M 6− −
"
M ^MG− −
"
MiM a−
2Z8U Q0-&2
Zj = 93= Q0?3-WS2
^6k+VG
Z>F[ Ij ='-[K$L Q0
K $ L
# 5
6
+
= =
-&
6
GK L
# 5
6K $
− + =
+ − =
2 P>1
6
# "#
A-&-[ 2
+2 \N<>lD) Q0#bf Q0
6
# "#
-&?
? Q0
K ^ $ m L G
5
^ 6
− − −
∆ = =
2
^Gk+VG
Z>F[ Ij ='-[K$LP#I
MM G
"MM"
M GM
d2j3%&> 9)2Z8Ff
U Q0-&32
^^k+VG
Z>F[ Ij ='-[K$L 9
MM
"
CGMM
2\N<>/0 D 9"C-&* /0K$=
+0
G
^a'G
Z>F[ Ij ='-[K$L Q05
F
K GF$ L 6
# 5
FK $ L
+ − + =
− + + =
ZF 9 Q0
F
#
-[ /0R5
K $ 6L a − − + =
^ik+VG
Z>F[ Ij ='-[K$L/0R5
6
6K 6$ L G + + − − =
-&/B_5
6 6 6
K $ L X− + + + − =
2Z 9/0RNK] /B
_2\. c7$K V* =N 9)R-&_2
^mk+VG
Z>F[ Ij ='-[K$L 96MM-&
M MG−
-& Q0#5
GK 6$
$ GL Y
− − =
+ − =
2
2 \N<>/0R D> 9) *-&-[ "
jC%& 9)#-&R"P>1#-[ C2
+2 \N <> N -[ )#>, / 0 < >5
K $ L + − + =
2
^Y'^
Z>F[ Ij =,-[K$L_2
$%&"b*E* =2N6MM"MM"
MM 2200S
2j
3%&> 9 *_2
2 Z8-&FfU Q0_"32
+2 f?e/03b Q0_* 92Z898FE
_232
^X'^
Z>F[ Ij =,-[K$L
MM 424A −−
-& Q
0#<>5
+−=
−=
+−=
t41z
t1y
t23x
\N<> Q0
∆
D"b-&-[ #2
a'^
Z>F[ Ij =,-[K$L%z>J P
111
CBAABC2
2NMM"nMM"MM"
MM b0aB
1
−
"
0b0a >> "
2
2 Z8FfU Q0
CB
1
-&
1
AC
:-&+2
+2 "+$ l%[f7u+(^2Z"+ 9F&U n
Q0
CB
1
-&
1
AC
%.S2
a'^
Z>F[ I j = ,-[ K$L + 9
MM 102A
"
MM 001B
"MM-&/0R5
02zyx =−++
2\N<>/B D+
9-&=/0R2
a6'a
Z>F[ Ij =,-[K$L Q0#-&/
0R5
09z2yx2P
1
3z
2
3y
1
1x
d
=+−+
−
=
+
=
−
−
5
5
2 Z* = 9=#?Ffg NR+162
+2 Z* = 9)#-&R2\N<>?E) Q0
∆
1>/0R+N
∆
D-&-[ #2
aG'a
Z>F[ Ij =,-[K$L%z>J P
111
CBAABC2
MM 030A −
"^MM"MGM"
MM 404B
1
2
2 Z* = !
11
CA "
2\N<>/B%&-&NK] /
0
11
BBCC
2
+2 j3%&> 9)
11
BA
2\N<>/0R D"3-&?
?
1
BC
23/0Rb Q0
11
CA
* 92Z8 =#& *32
a^'a
Z>F[ Ij =,-[K$L Q0
=−+
=−−+
+
=
−
+
=
−
012y3x
02zyx
2
1z
1
2y
3
1x
d
1
5#M5
6
2 P
1
d
-&
2
d
?? 2\N<>/0RPf
Q0
1
d
-&
2
d
2
+2 3/0* =KLb Q0
1
d
-&
2
d
%B%c* 9-&2
Z8#I8%&E* =2
aak+Va
Z>F[ Ij =, -[ K$L + 9 6MM"
M^M"_MM^2
2 Z* = 9=K$?P-?%&U;2\N<
>/B D+E 9"""_2
+2 Z* = 9
EKP 9D Q0_2
aik+Va
Z>F[ Ij =, -[ K$L + 9 MM"
M6M"MM62
