x0x1x2x3
x0x1x2 x3x4x5x6x7x8x5x6
x0 x1 9–1
1,x7x8x9xaxbxc[O;·1;·2]xd[O;·01;·02]xexfx10x11x12,x13x11xaxbxc[O;·1;·2]x14,x15xaxbxcx16x17x18x19
x1a:
·01 =
p2
2p
2
2
!; ·02 =
p2
2p
2
2
!
:
(1)x1bx1cxaxbx1dx1exfx1f;
(2)x1bx1cx11xaxbxcx20x16x17x18x19·1 =
1
0
,·2 =
0
1
x13x15xaxbxcx14x16xaxbx21x19;
(3)x22x23x18x19?!v x13[O;·1;·2]x16x21x19x1a
1
1
,x24x25x13x15xaxbxc[O;·01;·02]x14x16x21x19.
x4,(1)x26x1a(·01;·02) = (·1;·2)T,x27x20T =
p2
2?
p2
2p
2
2
p2
2
!
,x28x29xaxbx1dx1exfx1fx1a
x
y
= T
x0
y0
=
p2
2?
p2
2p
2
2
p2
2
!
x0
y0
:
(2) x2a(1)x23,(·1;·2) = (·01;·02)T?1,x27x20T?1 =
p2
2
p2
2
p2
2
p2
2
!
,x28x29 ·1 =
p2
2 ·
01?
p2
2 ·
02,
·2 =
p2
2 ·
01 +
p2
2 ·
02,x2b,·1 =
p2
2
p2
2
!
,·2 =
p2
2p
2
2
!
.
(3)x2c
x
y
= T
x0
y0
x2dx2ex23
x0
y0
= T?1
x
y
,x2fx13?!v = ·1?·2,x28x29
x0
y0
=
p2
2
p2
2
p2
2
p2
2
!
1
1
=
0
p2
;
x30x31x32?!v x13x15xaxbxcx14x16x21x19.
2,x13x33x34x8x9xaxbxc[O;·1;·2]x20,x22x23x15x16x8x9xaxbxc[O0;·01;·02]x16x11x12O0x16xaxbx1a(3;2),x12
M(5;3)x13x15xaxbxcx16x0x35x36,x37x12Mx16x15xaxbx0 > 0,x38x39x3ax3bx3cx1fx1bx1cx2c[O;·1;·2]x3d[O0;·01;·02]x16
xaxbx1dx1exfx1f.
x4,x26x1aX0 =
3
2
,x37x2ax3ex3fx23!O0M =
5
3
3
2
=
2
1
,x40!O0Mx16x41x42x18x19x32
2p5
5p
5
5
!
,
x2bcos = 2
p5
5,sin =
p5
5, x28x29T =
2p5
5?
p5
5p
5
5
2p5
5
!
,x26x43x1dx1exfx1fx1a
0
@
x
y
1
1
A =
0
BB
@
2p5
5?
p5
5 3p
5
5
2p5
5 20 0 1
1
CC
A
0
@
x0
y0
1
1
A:
￠ 1 ￠
3,x44x45x46x47x48Cx13x8x9xaxbxc[O;·1;·2]x20x16x49x4ax32:
x2?xy +y2 +2x?4y +1 = 0:
(1)x4bx15x16x8x9xaxbxc[O0;·01;·02],x4cO0x13x4dxaxbxcx14x16xaxbx1a(0;2),x37xe8
<
:
·01 =
p2
2 ·1 +
p2
2 ·2
·02 =?
p2
2 ·1 +
p2
2 ·2;
x38x39x3ax3bx3cx1fx1bx1cxaxbx1dx1exfx1f;
(2)x24x47x48Cx13x15xaxbxcx14x16x49x4a.
x4,(1)x4ex3ex44,X0 =
0
2
,T =
p2
2?
p2
2p
2
2
p2
2
!
,x37x4fx50x1fjTj = 1,x28x29xaxbx1dx1exfx1fx1a:
0
@
x
y
1
1
A =
0
BB
@
p2
2?
p2
2 0p
2
2
p2
2 20 0 1
1
CC
A
0
@
x0
y0
1
1
A:
(2) Cx16x49x4ax1a x026 + y022 = 1.
4,x44xex33x34x8x9xaxbxc[O;·1;·2],x51x15x16x8x9xaxbxc[O0;·01;·02]x52x53,x0x35x54y0x35x13x4dxaxbxcx20
x16x49x4ax21x55x32x?2y +2 = 0x542x+y +4 = 0.
(1)x24x2cx4dxaxbxcx3dx15xaxbxcx16x1dx1exfx1f;
(2)x24x8x48x?y +2 = 0x13x15xaxbxcx20x16x49x4a;
(3)x24x8x483x0 +y0 +1 = 0x13x4dxaxbxcx20x16x49x4a.
x4,(1)x26O0x12x16xaxb(x0;y0)x32x49x4ax56
( x?2y +2 = 0
2x+y +4 = 0 x16x57,x2bX0 =
x
0
y0
=
2
0
.
x58x59x?2y+2 = 0x16x49x18xcx5ax1a2, 1,2x+y+4 = 0x16x49x18xcx5ax1a?1, 2,x28x29·01 = 2
p5
5 ·1+
p5
5 ·2,
·02 =?
p5
5 ·1 +
2p5
5 ·2,x2b(·
01;·02) = (·1;·2)
2p5
5?
p5
5p
5
5
2p5
5
!
,x2ax5b
flfl
flfl
fl
2p5
5?
p5
5p
5
5
2p5
5
flfl
flfl
fl = 1,·
01;·02x5cx5d
x5ex5fxc,x28x29xaxbx1dx1exfx1fx1a
0
@
x
y
1
1
A =
0
BB
@
2p5
5?
p5
5?2p
5
5
2p5
5 00 0 1
1
CC
A
0
@
x0
y0
1
1
A:
(2) x?y +2 = 0x13x15xaxbxcx20x16x49x4ax1a:
2p5
5 x
0?
p5
5 y
0?2
!
p
5
5 x
0 + 2
p5
5 y
0
!
+2 = 0;
x2bx0?3y0 = 0.
(3)x2a(1)x16xaxbx1dx1exfx1fx2dx29x60x3d
0
@
x0
y0
1
1
A =
0
BB
@
2p5
5
p5
5
4p5
5
p5
5
2p5
5?
2p5
50 0 1
1
CC
A
0
@
x
y
1
1
A;
￠ 2 ￠
x403x0 +y0 +1 = 0x13x4dxaxbxcx14x16x49x4ax1a:
3
2p5
5 x+
p5
5 y +
4p5
5
!
+
p5
5 x+
2p5
5 y?
2p5
5
!
+1 = 0;
x2b5x+5y +10+p5 = 0.
x0 x1 9–2
1,x61x62x45x46x47x48x16x49x4a
5x2 +4xy +2y2?24x?12y +12 = 0;
x63x64x1cx25x16x65x3cx29x66x15x16xaxbx35.
x4,x3ax3b A =
5 2
2 2
,B =
12
6
,x26x43 I1 = Tr(A) = 7 > 0,I2 = jAj = 6 > 0,I3 =
flfl
flfl
flfl
5 2?12
2 2?6
12?6 12
flfl
flfl
flfl =?108 < 0,x43x47x48x32x67x68,Ax16x69x6ax6bx32x49x4a?2?7? + 6 = 0x16x6c,x57x60?1 = 6,
2 = 1,x40x62x61x6dx16x49x4ax1a6x02 +y02? 1086 = 0,x2b x023 + y0218 = 1.
x1ax64x1cx27x6ex6fx65x3c,x70x71x24x1cxaxbx1dx1exfx1f,x72x73x5bx69x6ax6c 6 xd 1 x16x41x42x69x6ax18x19x21x55x32
2p5
5 ;
p5
5
xd
p5
5 ;
2p5
5
,x28x29T =
2p5
5?
p5
5p
5
5
2p5
5
!
,x37jTj = 1,x74x24x47x48x16x20x75(x2bx15
xaxbxcx16x11x12)O0(x0;y0),x57x48x76x49x4ax56(
5x0 +2y0?12 = 0
2x0 +2y0?6 = 0
x60X0 =
x
0
y0
=
2
1
,x26x43xaxbx1dx1exfx1fx32
0
@
x
y
1
1
A =
0
BB
@
2p5
5?
p5
5 2p
5
5
2p5
5 10 0 1
1
CC
A
0
@
x0
y0
1
1
A:
x24x1cx15xaxbxcx20x0x35xdy0x35x13x4dxaxbxcx20x16x49x4ax21x55x32x?2y = 0 (x2by0 = 0)xd2x + y?5 = 0 (x2b
x0 = 0).
x30x10x1x0x11x31x32x12
x11x34x12x13x12x13x38x14
x15x3cx18x43x1bx48x4cx4dx50x51x52x4ax56x50x54x5bx5ax5cx5fx61
x62x62x66
x67x69x6b
x6bx6dx6d
x6fx6fx70
x71x72x73
x74x74x75
x76x77x78
x79x79x6e
x2fx2ex6ex0x6ex2fx10xf
x10x2fx0x30xfx1x0
x10x30x2fx6fx10x7cx6e
x2fx6dx7ax79
x78x77x76
x76x74x74
x73x72x71
x71x6fx6f
x6ex6cx6c
x6bx69x68
x66x64x62
x5ex5cx58
x5bx5ax50x57x56x55x53x52x50x4ex4dx1dx48x45x19x17x16x3x15x39x14x13x1x13x35x12x11
x12x1x32x11x0x32x30x10
x11
x75x75x75x75x75x75x75x40x50x20x8
x74
x74
x74
x74
x74
x74
x74
x74
x74
x74
x70x0
x10x0
x30x30x30x30x30
x30x30x30x30x30
x30x30x30x30x30
x30x30x30x30x30
x30x30x30x30x30
x30x30x30x30x48
x1cx6
xex2cx70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70
x7x1fx3ax9x21x46
x7ax2fxfx1x11x6
x5x28x7x1dx37
x68xdx77x2dxf
x
y
x0
y0
O
O0
x01x1x2
￠ 3 ￠
2,x61x62x45x46x47x48x49x4a
x2?4xy?2y2 +10x+4y = 0;
x63x64x1cx25x16x65x3cx29x66x15x16xaxbx35.
x4,x3ax3bA =
1?2
2?2
,B =
5
2
,x26x43I1 = Tr(A) =?1 < 0,I2 = jAj =?6 < 0,I3 =
flfl
flfl
flfl
1?2 5
2?2 2
5 2 0
flfl
flfl
flfl = 6 > 0,x43x47x48x32x77x47x48,Ax16x69x6ax6bx32x49x4a?2 +6 = 0x16x6c,x57x60?1 = 2,?2 =?3.
x40x62x61x6dx16x49x4ax1a2x02?3y02?1 = 0,x2b x021
2
y021
3
= 1.
x1ax64x1cx27x6ex6fx65x3c,x70x71x24x1cxaxbx1dx1exfx1f,x72x73x5bx69x6ax6c 2 xd?3 x16x41x42x69x6ax18x19x21x55x32
2p5
5 ;?
p5
5
xd
p
5
5 ;
2p5
5
,x28x29T =
2p5
5
p5
5
p5
5
2p5
5
!
,x37jTj = 1,x74x24x47x48x16x20x75(x2bx15
xaxbxcx16x11x12)O0(x0;y0),x57x48x76x49x4ax56(
x0?2y0 +5 = 0
2x0?2y0 +2 = 0
x60X0 =
x
0
y0
=
1
2
,x26x43xaxbx1dx1exfx1fx32
0
@
x
y
1
1
A =
0
BB
@
2p5
5
p5
5?1
p5
5
2p5
5 20 0 1
1
CC
A
0
@
x0
y0
1
1
A:
x24x1cx15xaxbxcx20x0x35xdy0x35x13x4dxaxbxcx20x16x49x4ax21x55x32x+2y?3 = 0xd2x?y +4 = 0.
x72x73x72
x73x73x73
x74x75x76
x2dx6cxfx6dx2fx6f
x35x32x48x45x19x18x3x14x2
x34x0x11
x10x12x0x1x38x16x18x1ax45
x48x49x4ax35x36x37x37x38x7
x10xfx0xfx2fx6dx2ex6bx76
x75x74x73
x73x73x72
x73x72x72
x72x72x72
x72x72x72
x72x72x69
x2c
x75x75x75x75x40x8
x74
x74
x74
x74
x74
x74
x74
x74
x74
x70x10x0
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x38x14
x50x50x50x50x50
x50x50x50x50x50
x50x50x50x50x50
x50x50xa
x7x1fx3ax9x21x46
x7ax2fxfx1x11x6 x1x13x9x3x18
x9x23x49xbx58
x
y
x0
y0
O
O0
x02x1x2
3,x61x62x45x46x47x48x49x4a
x2?3xy +y2 +10x?10y +21 = 0;
x63x78x1cx25x16x65x3cx29x66x15x16xaxbx35.
x4,x3ax3bA =
1? 3
2
32 1
!
,B =
5
5
,x26x43I1 = Tr(A) = 2 > 0,I2 = jAj =? 54 < 0,
I3 =
flfl
flfl
flfl
fl
1? 32 5
32 1?5
5?5 21
flfl
flfl
flfl
fl
=? 54 < 0,x43x47x48x32x77x47x48,Ax16x69x6ax6bx32x49x4a?2?2 54 = 0x16x6c,x57x60
￠ 4 ￠
1 = 52,?2 =? 12, x40x62x61x6dx16x49x4ax1a 52 x02? 12 y02 +1 = 0,x2b y022? x022
5
= 1.
x1ax64x1cx27x6ex6fx65x3c,x70x71x24x1cxaxbx1dx1exfx1f,x72x73x5bx69x6ax6c 52 xd? 12 x16x41x42x69x6ax18x19x21x55x32
p
2
2 ;?
p2
2
xd
p
2
2 ;
p2
2
,x28x29T =
p2
2
p2
2
p2
2
p2
2
!
,x37jTj = 1,x74x24x47x48x16x20x75(x2bx15xaxbxcx16
x11x12)O0(x0;y0),x57x48x76x49x4ax56 (
x0? 32 y0 +5 = 0
32 x0 +y0?5 = 0
x60X0 =
x
0
y0
=
2
2
,x26x43xaxbx1dx1exfx1fx32
0
@
x
y
1
1
A =
0
BB
@
p2
2
p2
2?2
p2
2
p2
2 20 0 1
1
CC
A
0
@
x0
y0
1
1
A:
x24x1cx15xaxbxcx20x0x35xdy0x35x13x4dxaxbxcx20x16x49x4ax21x55x32x+y = 0xdx?y +4 = 0.
x39x3ax4fx50x52x55x26x2axex2dxex0x2fxfx10
xcxbxc
xbxbxb
xbxbxb
xaxax9
x8x34x11x12x1x4
x31
x10x0x1x0x11x33x11x34x7
x9xaxa
xaxbxb
xbxbxb
xcxbxc
xcxbx36
x13
x10xfx0x2exfx2cxexdx26xbx23x53x50x4fx4ex3ax38x38x36x1e
x75x75x75x75x75
x74
x74
x74
x74
x74
x74
x74
x74
x74
x74
x74
x30x10x0
x73x73
x73x73
x73x73
x73x73
x73x73x71x71
x71x71
x71x71
x71x71
x71x71
x7x1fx3ax9x21x46
x7ax2fxfx1x11x6
xbx56xdx6a
x3x16x5x2a
x
y
x0
y0
O
O0
x03x1x2
4,x61x62x45x46x47x48x49x4a
4x2?4xy +y2 +6x?8y +3 = 0;
x63x64x1cx25x16x65x3cx29x66x15x16xaxbx35.
x4,x3ax3b A =
4?2
2 1
,B =
3
4
,x26x43 I1 = Tr(A) = 5 > 0,I2 = jAj = 0,I3 =
flfl
flfl
flfl
4?2 3
2 1?4
3?4 3
flfl
flfl
flfl =?25 < 0,x43x47x48x32x79x7ax48.Ax16x69x6ax6bx32x49x4a?2?5? = 0x16x6c,x57x60?1 = 0,?2 = 5,x72
x73x5bx69x6ax6c0xd5x16x41x42x69x6ax18x19x21x55x32
p
5
5 ;
2p5
5
xd
2
p5
5 ;
p5
5
,x28x29T =
p5
5?
2p5
5
2p5
5
p5
5
!
,
x37jTj = 1.
￠ 5 ￠
x7bx39Tx78x7cx7dxaxbx1dx1e,x2dx60
TT 0
0 1
A B
BT 3
T 0
0 1
=
0
@
0 0?p5
0 5?2p5
p5?2p5 3
1
A:
x2bb01 =?p5,b02 =?2p5,x4b
X0 =
0
@
b0222c
2b01?2
b
02
2
1
A =
p5
10
2p5
5
!;
x39X0x78x33x7exaxbx1dx1e
X0
1
=
E X
0
0 1
X00
1
;
x2dx60
E 0
XT0 1
0
@
0 0?p5
0 5?2p5
p5?2p5?2
1
A
E X
0
0 1
=
0
@
0 0?p5
0 5 0
p5 0 0
1
A;
x2bx49x4ax61x62x1a5y002?2p5x00 = 0,x27x62x61x49x4ax1ay002 = 2
p5
5 x
00,x7fx16xaxbx1dx1exfx1fx1a
X
1
=
T TX
0
0 1
X00
1
=
0
BB
@
p5
5?
2p5
5?
9
10
2p5
5
p5
5
1
50 0 1
1
CC
A
0
@
x00
y00
1
1
A:
x30x0x79x7ax48x16x1x12xaxbx32TX0 =
910 ; 15
·T
,x15xaxbxcx20x0x35xdy0x35x13x4dxaxbxcx20x16x49x4ax21x55x32
2x?y +2 = 0xd2x+4y +1 = 0.
x30x10x11x31x10x11x10x32
x10x32x4x5
x5x5x6
x6x7x7
x7x7x8
x8x8x8
x8x9x8
x9x9x9
x9xax9
xax9xa
xax9x35
x2
x10x30xfx0x10x7cx2fx6cx2e
x2dxex2dx67x29x28x25x24x54x50x4ex4bx4ax48x47x46x2ax26x24x23
x75x75x75x72x40x74
x74
x74
x74
x74
x74
x74
x74
x74
x70x0
x0
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x10x10x38
x14
x50x50x50x50x50
x50x50x50x50x50 x7x1fx3ax9x21x46
x7ax2fxfx1x11x6
x1x13x9x3x18
x49x23x9x58xb
x
y
x0
y0
OO0
x04x1x2
5,x61x62x14x50x45x46x47x48x16x49x4a,x63x2x1cx25x3x32x4x5x47x48:
(1) 4x2?4xy +y2 +4x?2y = 0;
(2) x2?2xy +y2 +2x?2y?3 = 0.
x4,(1) x3ax3bA =
4?2
2 1
,B =
2
1
,x26x43I1 = Tr(A) = 5 > 0,I2 = jAj = 0,I3 =
flfl
flfl
flfl
4?2 2
2 1?1
2?1 0
flfl
flfl
flfl = 0,x1ax6x7x47x48x16x8x9x70x71xaxbxcxdxe,Ax16x69x6ax6bx32x49x4a?2? 5? = 0x16x6c,x57x60
1 = 0,?2 = 5,x72x73x5bx69x6ax6c0xd5x16x41x42x69x6ax18x19x21x55x32
p
5
5 ;
2p5
5
xd
2
p5
5 ;
p5
5
,x28x29
￠ 6 ￠
T =
p5
5?
2p5
5
2p5
5
p5
5
!
,x37jTj = 1.
x7bx39Tx78x7cx7dxaxbx1dx1e,x2dx60
TT 0
0 1
A B
BT 0
T 0
0 1
=
0
@
0 0 0
0 5?p5
0?p5 0
1
A:
x2bb01 = 0,b02 =?p5,x4bX0 =
0
b
02
2
=
0p
5
5
,x39X0x78x33x7exaxbx1dx1e,x2dx60
E 0
XT0 1
0
@
0 0 0
0 5?p5
0?p5 0
1
A
E X
0
0 1
=
0
@
0 0 0
0 5 0
0 0?1
1
A;
x2bx49x4ax61x62x1a5y002?1 = 0,x2by00 = §
p5
5,x30x32xbx72x33x4fx8x48.x7fx16xaxbx1dx1exfx1fx1a
X
1
=
T TX
0
0 1
X00
1
=
0
BB
@
p5
5?
2p5
5?
2
5
2p5
5
p5
5
1
50 0 1
1
CC
A
0
@
x00
y00
1
1
A:
(2) x3ax3b A =
1?1
1 1
,B =
1
1
,x26x43 I1 = Tr(A) = 2 > 0,I2 = jAj = 0,I3 =
flfl
flfl
flfl
1?1 1
1 1?1
1?1?3
flfl
flfl
flfl = 0,x1ax6x7x47x48x16x8x9x70x71xaxbxcxdxe,Ax16x69x6ax6bx32x49x4a?2? 2? = 0x16x6c,x57
x60?1 = 0,?2 = 2,x72x73x5bx69x6ax6c0xd2x16x41x42x69x6ax18x19x21x55x32
p
2
2 ;
p2
2
xd
p2
2 ;
p2
2
,x28x29
T =
p2
2?
p2
2p
2
2
p2
2
!
,x37jTj = 1.
x7bx39Tx78x7cx7dxaxbx1dx1e,x2dx60
TT 0
0 1
A B
BT?3
T 0
0 1
=
0
@
0 0 0
0 2?p2
0?p2?3
1
A:
x2bb01 = 0,b02 =?p2,x4bX0 =
0
b
02
2
=
0p
2
2
,x39X0x78x33x7exaxbx1dx1e,x2dx60
E 0
XT0 1
0
@
0 0 0
0 2?p2
0?p2?3
1
A
E X
0
0 1
=
0
@
0 0 0
0 2 0
0 0?4
1
A;
x49x4ax61x62x1a2y002?4 = 0,x2by00 = §p2,x30x32xbx72x33x4fx8x48.x7fx16xaxbx1dx1exfx1fx1a
X
1
=
T TX
0
0 1
X00
1
=
0
BB
@
p2
2?
p2
2?
1
2p
2
2
p2
2
1
20 0 1
1
CC
A
0
@
x00
y00
1
1
A:
6,x24x14x50x45x46x47x48x16xfx10x48:
(1) 6x2?xy?y2 +3x+y?1 = 0;
(2) 2xy?4x?2y +3 = 0.
￠ 7 ￠
x4,(1)x3ax3bA =
6? 1
2
12?1
!
,B =
3
21
2
!
,x26x43I2 = jAj =? 254 < 0,I3 =
flfl
flfl
flfl
fl
6? 12 32
12?1 12
3
2
1
2?1
flfl
flfl
flfl
fl
=
25
4 > 0,x43x47x48x32x77x47x48.x47x48x16x20x75(x0;y0)x52x53x48x76x49x4ax56(
6x0? 12 y0 + 32 = 0
12 x0?y0 + 12 = 0
x60X0 =
x
0
y0
=
1
53
5
!
,xfx10x48x49x4ax1a
6
x+ 15
2
x+ 15
y? 35
y? 35
2
= 0:
x36x1fx2dx21x57x1a
3
x+ 15
+
y? 35
2
x+ 15
y? 35
= 0;
x28x29xfx10x48x49x4ax1a3x+y = 0x542x?y +1 = 0.
(2)x3ax3bA =
0 1
1 0
,B =
2
1
,x26x43I2 = jAj =?1 < 0,I3 =
flfl
flfl
flfl
0 1?2
1 0?1
2?1 3
flfl
flfl
flfl = 1 > 0,x43x47
x48x32x77x47x48.x47x48x16x20x75(x0;y0)x52x53x48x76x49x4ax56
( y
0?2 = 0
x0?1 = 0
x60X0 =
x
0
y0
=
1
2
,xfx10x48x49x4ax1a2(x?1)(y?2) = 0,x2bx = 1x54y = 2.
7,x31?x16x6bx11x12x49x4a
x2?2xy +?y2?2x+2y +5 = 0
x28x13x14x16x47x48x3cx15.
x4,x3ax3bA =
1
1?
,B =
1
1
,x26x43I1 = Tr(A) = 2?,I2 = jAj =?2?1 = (1)(?+1),
I3 =
flfl
flfl
flfl
1?1
1? 1
1 1 5
flfl
flfl
flfl = (5?+3)(1),x21x16x17x18x19x1ax11x12.
(i)x1b? 6= §1x1c,I2 6= 0,x1dx2dx21x1ax7x17x18x19,(a)? > 1x1e? <?1,x43x1cI2 > 0,x1b? > 1x1c,I1xdI3
x1fx20,x47x48x32x21x67x68;x1b? <?1x1c,I1xdI3x22x20,x47x48x32x67x68,(b)?1 <? < 1,x43x1cI2 < 0,x1b? =? 35
x1c,I3 = 0,x47x48x1axbx72x23x24x8x48;x25x1b? 6=? 35 x1c,x47x48x7fx32x77x47x48.
(ii)x1b? =?1x1c,I2 = 0,I3 6= 0,x47x48x32x79x7ax48.
(iii)x1b? = 1x1c,I2 = I3 = 0,x26x39x27x28x1dx19K1 =
flfl
flfl1
1 5
flfl
flfl+
flfl
flfl? 1
1 5
flfl
flfl = 102 = 8 > 0,x2dx23x47
x48x32xbx72x21x33x4fx8x48.
8,x31?x16x6bx11x12x49x4a
x2 +2?xy +y2?2x?2?y +? = 0
x28x13x14x47x48x16x3cx15.
x4,x3ax3bA =
1
,B =
1
,x26x43I1 = Tr(A) = 1 +?,I2 = jAj =?(1),I3 =
￠ 8 ￠
flfl
flfl
flfl
1
1
1
flfl
flfl
flfl =?(1)2(2?+1),x21x16x17x18x19x1ax11x12.
(i)x1b? 6= 1;0x1c,I2 6= 0,x1dx2dx21x1ax7x17x18x19,(a) 0 <? < 1,x43x1cI2 > 0,I1 > 0,I3 < 0,x47x48x32x67
x68; (b)? < 0x1e? > 1,x43x1cI2 < 0,x29x1b? =? 12 x1c,I3 = 0,x47x48x1axbx72x23x24x8x48;x25x1b? 6=? 12 x1c,x47
x48x7fx32x77x47x48.
(ii)x1b? = 0x1c,I2 = 0,I3 6= 0,x47x48x32x79x7ax48.
(iii)x1b? = 1x1c,I2 = I3 = 0,x26x39x27x28x1dx19K1 =
flfl
flfl 1?1
1 1
flfl
flfl+
flfl
flfl 1?1
1 1
flfl
flfl = 0,x2dx23x47x48x32xbx72x2a
x2bx8x48.
9,x22x23x49x4a(A1x+B1y+C1)2+2(A2x+B2y+C2) = 0,x27x20A1B2?A2B1 6= 0,A1A2+B1B2 = 0.
(1)x2cx2dx43x49x4ax13x14xbx0x79x7ax48;
(2)x24x1cx72x2ex35x16x49x4a.
x4,(1) x44x43x47x48x49x4ax32x2fx5bx8x9xaxbxc[O;·1;·2]x16,x2ax5bA1A2 + B1B2 = 0,x26x43x8x48L1,
A1x + B1y + C1 = 0 xdx8x48 L2, A2x + B2y + C2 = 0 x30x23x31x24,x28x32x44 A1B2? A2B1 > 0,x33
1 = pA21 +B21,?2 = pA22 +B22,x34L1;L2x16x41x42x49x18x18x19·01 =
A
1?
1; B1?
1
·
xd·02 =
A
2?
2; B2?
2
·
x5c
x5dxbx35x36x37x31x24x56.x33T =
A1
1
A2
2
B1
1
B2
2
!
,x34xe(·01;·02) = (·1;·2)T,Tx32xbx35x31x24x3ax3b,x37jTj = 1,x26
x43·01;·02x5cx5dxbx35x5ex5fxc.
x38x39x33 8
<
:
x0 = A1?
1
x+ B1?
1
y + C1?
1
y0 = A2?
2
x+ B2?
2
y + C2?
2;
x31xe
0
@
x0
y0
1
1
A =
0
B@
A1
1
B1
1
C1
1
A2
2
B2
2
C2
20 0 1
1
CA
0
@
x
y
1
1
A;
x30x32xbx35x8x9xaxbx1dx1exfx1f,x13x15x16xaxbxc[O0;·01;·02]x20,x47x48x16x49x4ax61x62x1a?21x02 +2?2Y0 = 0,x58x59
1 6= 0,x26x43x02 + 2?2?2
1
y0 = 0,x30x32xbx0x79x7ax48.
(2)x43x79x7ax48x16x72x2ex35x32y0x35,x49x4ax1ax0 = 0,x2bA1x+B1y +C1 = 0.
10,x44x45x46x47x48x49x4ax1a
a11x2 +2a12xy +a22y2 +2b1x+2b2y +c = 0:
x2cx2d:
(1)x45x46x47x48x1axbx0x3ax35x77x47x48x1ex7x0x23x30x3bx8x16x8x48x16x3cx21x3dx71x0x3ex32I1 = 0;
(2)x45x46x47x48x1ax68x16x3cx21x3dx71x0x3ex32I21 = 4I2,I1I3 < 0;
(3)x45x46x47x48x51x13x14xbx35x67x68,x38x24x3fx67x68x34x40.
x2x3,(1) x51x43x45x46x47x48x1ax3ax35x77x47x48,x34x2ax5bI1x32x31x24x28x1dx19,x2cx3ax35x77x47x48x16xbx41x49x4ax42x23
I1 = 0;x51x32x7x0x30x23x3bx8x16x8x48,x34x27xbx41x49x4ax1a?1x02 +?2y02 = 0,x37?1?2 < 0,x2by0 = §
r
1
2 x
0
x13x14x7x0x30x23x3bx8x16x8x48,x26x43
r
1
2 ￠
r
1
2
=?1,x2ex601?
2
= 1,x2bI1 = 0.
￠ 9 ￠
x43x44,x51I1 = 0,x34x26I1 =?1 +?2 = 0,x2dx23?2 =1(6= 0),x63x37I2 =21 < 0,x2ax62x61x49x4a
1(x02?y02)+ I3I
2
= 0x2dx23,x1bI3 6= 0x1cx47x48x32x3ax35x77x47x48;x1bI3 = 0x1cx47x48x32x7x0x30x23x3bx8x16x8x48.
(2)x51x43x45x46x47x48x32x68,x34x3dxeI2 > 0,I1 ￠I3 < 0(x67x68x9),x45x46x47x1bx48x49x8x9xaxbxcx23x62x61x49x4a
x1a?1x02 +?2y02 + I3I
2
= 0,x26x1ax32x68,x28x29?1 =?2,x5bx32I1 = 2?1,I2 =?21,x28x29I21 = 4I2.
x43x44,x51I1 ￠I3 < 0,I21 = 4I2,x2dx23I2 > 0,x47x48x32x67x68,x1dx26I1 =?1 +?2,I2 =?1?2,x2aI21 = 4I2
x2dx60?1 =?2,x40x43x47x48x32x68.
(3)x13xbx41x67x68x49x4a x2a2 + y2b2 = 1 (a > 0,b > 0)x20,x67x68x16x34x40S = …ab,x51x47x48x13x14xbx35x67x68,
x34x47x1bx48x49xaxbxcx44x6dx2dx60x25x16x62x61x49x4ax1a?1x02 +?2y02 + I3I
2
= 0,x2b x02?I
3I
2
￠ 1?
1
+ y02?I
3I
2
￠ 1?
2
= 1.
x28x29x34x40S = …
rI2
3
I22 ￠
1
1?2 =
…jI3jpI2
I22,
11,x22x23x67x68x4ax35x54x4bx35x21x55x13x8x48x+y?1 = 0x54x?y +1 = 0x36,x37x4ax4bx35x4ax21x55x1a4xd2.
x24x43x67x68x16x49x4a.
x4,x28x32x44x4ax35x13x0x35(x2by0 = 0)x36,x4bx35x13y0x35(x2bx0 = 0)x36,x78x1dx1e8
<
:
x0 =
p2
2 x?
p2
2 y +
p2
2
y0 =
p2
2 x+
p2
2 y?
p2
2 ;
x1bx5dx3ax3bx3cx1fx1a
0
@
x0
y0
1
1
A =
0
BB
@
p2
2?
p2
2
p2
2p
2
2
p2
2?
p2
20 0 1
1
CC
A
0
@
x
y
1
1
A;
x27x4cx36x9x16x4dx3ax3b
p2
2?
p2
2p
2
2
p2
2
!
x32x4fx50x1fx3ax5b1x16x31x24x3ax3b,x26x43x30x32x5ex5fx8x9xaxbxcx4ex16xaxbx1d
x1e,x13x15x16xaxbxcx4fx47x48x16x49x4ax73x1a x024 + y021 = 1,x26x39xaxbx1dx1exfx1f,x2dx60x13x4dxaxbxcx14x16x49x4a:
1
4
p
2
2 x?
p2
2 y +
p2
2
2
+
p
2
2 x+
p2
2 y?
p2
2
2
= 1,x2bx1a5x2 +6xy +5y2?6x?10y?3 = 0.
x0 x1 9–3
1,x50x55x14x50x72x73x51x34x32x52x1ax53x5ax54Rx3dx55x56x16x57x58,x63x2x1cx59x5ax32x41x58?x52x58?
(1) x 7! x2; (2) x 7! x3; (3) x 7!jxj;
(4) x 7! 2x; (5) x 7! sin(x2); (6) x 7! tanx.
x4,(1)–(5)x5bx32Rx3dx55x56x16x57x58,x27x20(2),(4)x32x41x58,(2)x32x52x58,(6)x28x32x57x58.
2,x44Sx13x14x33x34x36x28xex12x56x5dx16x5cx2b,Lx32xbx0x8x48,x5dx33x34x36x5ex35x12P(x;y)x72x73x3dx25x2fx5bLx16
x72x2ex12P0(x0;y0),x30x32Sx3dx55x56x16xbx35x1dx1e,x2ex1ax2fx5bx8x48Lx16x43x58,x2eLx32x43x58x35.
(1)x24x1cx33x34x2fx5bx8x48y = xx16x43x58xfx1f;
(2)x44x43x58x35x1aAx+By +C = 0,x24x1cx30x1cx16x43x58xfx1f;
(3)x44x53 1;x53 2x32x2fx5bx33x34x36x7x0x33x4fx8x48L1;L2x16x43x58.x2cx2dx53 2x53 1x32xbx35x33x7e.
x4,(1)x22x23P(x;y)x2fx5bx8x48y = xx16x72x2ex12x1aP0(x0;y0),x348
<
:
x+x0
2 =
y +y0
2
y?y0
x?x0 =?1;
￠ 10 ￠
x57x60x43x58xfx1fx1a (
x0 = y
y0 = x:
(2)x12P(x;y)x2fx5bx8x48Ax+By +C = 0x16x72x2ex12x1aP0(x0y0),x348
<
:
A
x+x0
2
·
+B
y +y0
2
+C = 0
A(y?y0) = B(x?x0);
x57x60x43x58xfx1fx1a (
x0 = 1A2 +B2?(B2?A2)x?2ABy?2AC￠
y0 = 1A2 +B22ABx+(A2?B2)y?2BC￠:
(3) x29L1 x78x1axx35x5fx60xaxbxc,x34L1;L2 x16x49x4ax21x55x1ay = 0xdy + C = 0,x27x20C 6= 0,x61
x53 1(P) = P0(x0;y0),x53 2(P0) = P00(x00;y00),x2a(2)x23,(
x0 = x
y0 =?y;
( x00 = x0
y00 =?y0?2C:
x62x63x6dxex60 (
x00 = x
y00 = y?2C:
x2dx64x53 2x53 1x32xbx35x33x7e.
3,x44x4d x32x1dx1e(x;y) 7! (x0;y0),(
x0 = 2x+3y +2
y0 =?x+2y?3
x65:
(1)x12(?1;1)x66x1dx5dx67x4x5x12?
(2)x8x48y = 2x66x1dx5dx67x4x5x65x3c?
(3)x12(9;?3)x32x2ax59x35x12x1dx46x1ax16?
x4,(1)x12(?1;1)x66x1dx1a(3;0).
(2)x8x48y = 2x36x16x12x32(t;2) (t 2R),x25x4d((t;2)) = (2t+8;?t+1),x26x43x1dx1ex6dx16x12x5dxbx0x8x48,
x27x68x5ax49x4ax1a
( x = 8+2t
y = 1?t;,x1ex+2y?10 = 0.
(3)x57x49x4ax56
( 2x+3y +2 = 9
x+2y?3 =?3 x60x = 2,y = 1,x40x4d((2;1)) = (9;?3).
4,x13x8x9xaxbxc[O;·1;·2]x20,x24x1cx33x34x69x12M0(x0;y0)x7cx7d 0x9x16x1dx1exfx1f.
x4,x29M0(x0;y0)x1ax11x12x5fx60x15x8x9xaxbxc[M0;·1;·2],x34x69x12M0x7cx7d 0x9x16x1dx1ex13x15xaxbx14x16
x1dx1exfx1fx1a (
x0 = cos 0?x?sin 0?y
y0 = sin 0?x+cos 0?y:
x44x12P(x;y)x45x7cx7dx1dx1aP0(x0;y0),x6cx4ex33x7exaxbx1dx1ex16xfx1f,P;P0x12x16x15xaxbx73x1a(x? x0;y? y0),
(x0?x0;y0?y0),x62x63x36x34x16xfx1fx2bx60(
x0 = cos 0(x?x0)?sin 0(y?y0)+x0
y0 = sin 0(x?x0)+cos 0(y?y0)+y0:
5,x51x5dx47x482xy = a2x69x11x12x7cx7d …4,x24x15x16x47x48x49x4a.
￠ 11 ￠
x4,x51P(x;y)x12x7cx7dx3dx67P0(x0;y0)x12,x34Px12x2dx2aP0x12x45x7cx7d?…4 x60x3d,x26x43
8<
:
x = cos
…4
·
x0?sin
…4
·
y0 =
p2
2 (x
0 +y0)
y = sin
…4
·
x0 +cos
…4
·
y0 =
p2
2 (?x
0 +y0);
x62x63x11x49x4a2xy = a2x2bx60y02?x02 = a2.
6,x33x34x16x3ax6ax1dx1ex4d x51xex7x35x28x6bx12A;B,x34x8x48ABx36x5ex35x12x5bx32x28x6bx12.
x4,x44C x12x13x8x48AB x36,x34x6cx4eC x32x52x42x5bx48x6cAB x36,xed(A;C) + d(C;B) = d(A;B)x1e
jd(C;A)?d(C;B)j = d(A;B),x33x4d(C) = C0,x34x26A;Bx32x4dx16x28x6bx12,xed(A;C0)+d(C0;B) = d(A;B)
x1ejd(C0;A)?d(C0;B)j = d(A;B),x38x39A;B;C0x28x10x48,x34x25x3x5cx5dxbx35x6dx9x3c,x25x6dx9x3cx7x6ex44x54x6e
x5bx6fx6dx6e,x7x6ex44x70x71x5bx6fx6dx6e,xdx36x72x3ax1fx73x74.
7,x24x14x72x75x58x1dx1ex16x28x6bx12,(
x0 = 3x?y?5
y0 = 2x+y +1:
x4,x57x49x4ax56
( x = 3x?y?5
y = 2x+y +1; x60x =?
1
2,y =?6,x2bx28x6bx12x29xe
12 ;?6
·
xbx35.
8,x51x13x75x58x1dx1ex41x14xbx0x8x48x16x76xdx27x55x56x2ax2b,x34x2ex30x0x8x48x1ax41x16x28x1dx8x48.x24x14x72x75x58x1d
x1ex16x28x1dx8x48,(
x0 = 7x?y +1
y0 = 4x+2y +4:
x4,x75x58x1dx1ex5dx8x48x1dx5dx8x48.x44x28x1dx8x48x1aL, Ax+By +C = 0,x51P(x;y) 2 L,x34P0(x0;y0) =
x41(P) 2 L,x2bAx0 +By0 +C = 0,x62x63x61x62x6dx60
(7A+4B)x+(2B?A)y +(A+4B +C) = 0:
x26x1aLx36x16x77x3fx12x5bx52x53x43x49x4a,x78x2dx30x35x49x4ax79x32Lx16x49x4a.x26x43xe
7A+4B
A =
2B?A
B =
A+4B +C
C = k:
x57x60k = 6x1ek = 3,x28x29 (
A =?4B
C = 0 ·
( A =?B
C = 32 B;
x40x28x1dx8x48xex7x0,?4x+y = 0x54?2x+2y +3 = 0.
9,x44x13x33x34x36x7ax1cx67x7x35x6dx9x3cABCx54DEF,x65xex16x35x75x58x1dx1ex5d4ABCx1dx5d4DEF?
x4,x2ax7bx3e3.3(9)x2dx23xe3! = 6x35x28x1fx16x75x58x1dx1e.
10,x2cx2d,x33x34x36x77x7ax7x35x8x9xbx7c(I)x54(II),x7fx7dx13x7exbx16x3ax6ax1dx1ex5d(I)x1dx5d(II).
x2x3,x7bx3e3.2(6)x22x45x7fx0x67x52x53x0x3ex16x3ax6ax1dx1ex16x7exbx76,x44xex8x9xbx7c[O;·1;·2]xd[O0;·01;·02],
x34x36x37x31x24x17·1;·2xd·01;·02x2dx29x6x7x7exbx16x31x24x1dx1ex41 x4cx60x41(·i) = ·0i,i = 1;2,x44x18x19– =!OO0,x1d
x2dx7x1xbx35x33x7ex1dx1ex54 –,x5bx32x2x2bx1dx1ex54 –x41x32xbx35x3ax6ax1dx1e,x25x5dx8x9xbx7c[O;·1;·2]x57x3d[O0;·01;·02].
x7dx13x76x3x2c.
x0 x1 9–4
1,x2cx2d,x33x34x36x69xbx35x4x7x12x7d90–x5 180–x5 270–x16x6dx35x7cx7dx521x5 x522x5 x523x54x6x1fx1dx1ex45x56x5dxb
x35x1dx1ex7.
￠ 12 ￠
x2x3,x61x520 = E,x34x26x52ix52j =
( x52
i+j x51i+j 6 3,
x52i+j?4 x51i+j > 4,x7x16x76x8(1)x66x52x53;x76x8(2)x34x32x58x59
x16;x1dx26x52?1i = x524?i,x76x8(3)x79x66x52x53,x28x29x30x32xbx35x7.
2,x1ba;bx4bx1ax77x3fx16x28x9x1axax16x5ax1c,x14x50x28xex16x75x58x1dx1ex56x5dx16x5cx2bx32x52x1axbx35x7?(
x0 = ax+by
y0 = bx+ay:
x4,x30x35x75x58x1dx1ex72x73x16x3ax3bx32A =
a b
b a
,x44A1 =
a
1 b1
b1 a1
,A2 =
a
2 b2
b2 a2
,x34A1A2 =
a
1a2 +b1b2 a1b2 +b1a2
a1b2 +b1a2 a1a2 +b1b2
x79x32x30x35x5cx2bx16xbxc,x26x43x76x8(1)x66x52x53;x1ba = 1,b = 0x1cx31x32x6x1fx1d
x1e,x26x43(2)x79x52x53; A?1 =
a
a2?b2?
b
a2?b2
ba2?b2 aa2?b2
!
x79x32x5cx2bx16xbxc,x26x43(3)x79x52x53.x30x35x5cx2bx6x53
x32x7.
x0 x1 9–5
1,x24x45x46x47x48x2 +2y2?4x?2y?6 = 0xdx46x12(8;0)x16x8xex49x4a,x63x24x1cx27x10xfx8xex16x49x4a.
x4,x3fx45x46x47x48x49x4ax2dx1bx5d
(x?2)2
21
2
+
y? 12
·2
21
4
= 1;
x32x67x68.
x78x1dx1e 8
<
:
x0 =
p42
21 (x?2)
y0 = 2
p21
21
y? 12
·;
x2b:
8<
:
x =
p42
2 x
0 +2
y =
p21
2 y
0 + 1
2,x34x43x1dx1ex10x68x02 +y02 = 1x1dx1ax67x68
(x?2)2
21
2
+
y? 12
·2
21
4
= 1;
x37x10x12
2p42
7 ;?
p21
21
x1dx5d(8;0),x28x29x43x1dx1ex10x68x16x8xey0 =?
p2
12 x
0x1dx1ax67x68x16x46x12(8;0)x16x8
xe,x+12y?8 = 0.
xdx68x8xey0 =?
p2
12 x
0x3bx8x16x8xey0 = 6p2x0x66x43x1dx1ex1dx5dxdx67x68x16x8xex+12y?8 = 0x10xfx16
x8xe12x?2y?23 = 0.
2,x44x77x47x48x16xbx41x49x4ax1a
x2
a2?
y2
b2 = 1;
x38x24x8xey = kba x(jkj < 1)x16x10xfx8xe(x28x7ax8xex16x33x4fx11x16x20x12x12x48).
x4,x6cx4ex72x2ex76,x77x47x48x16x20x75x32x28xexdx46x25x16x11x16x20x12,x26x43x28xex16x8xexbx7xdx46x25x16x20x75,x1ax6
x7x10xfx8xe,x13x70x74x14xbx35x12,x78xbx0x33x4fx11y = kba x+t,x25xdx77x47x48x16x24x12x52x53 x2a2?
kb
a x+t
·2
b2 =
1,x61x62x1ab2(1?k2)x2?2abktx?a2(b2 +t2),x26x43x33x4fx11x16x20x12x16x15xaxbx32 x1 +x22 = aktb(1?k2), x16
￠ 13 ￠
xaxbx1a kba ￠ aktb(1?k2) +t = t1?k2, x60x3dx10xfx8xex16x17x18x1a t1?k2 ￠ b(1?k
2)
akt =
b
ak,x10xfx8xex16x49
x4ax1ay = bak x,x2dx64x77x47x48x16x8xexdx10xfx8xex16x17x18x16x19x40x3ax5bx1ax5a b2a2,
3,x22x23x47x48xy?y2?2x+3y?1 = 0x16x8xexdyx35x33x4f,x24x25x16x49x4a,x63x24x1cx30x8xex16x10xfx8xe.
x4,x26x1aI2 =
flfl
flfl
fl
0 12
1
2?1
flfl
flfl
fl =?
1
4 < 0,I3 =
flfl
flfl
flfl
fl
0 12?1
1
2?1
3
2
1 32?1
flfl
flfl
flfl
fl
6= 0,x28x29x3fx47x48x32x77x47x48.x1ax24x25x16x20
x75,x57x29x14x49x4ax56,( 1
2 y0?1 = 0
1
2 x0?y0 +
3
2 = 0;
x60x0 = 1,y0 = 2,x2ax5bx22x7ax8xexdyx35x33x4f,x25x16x49x4ax32x = 1.
x78xbx0x33x4fx11x = t,x25xdx77x47x48x16x24x12xaxbx73x52x53ty?y2?2t+3y?1 = 0,x26x43x24x12x16x20x12x16
x16xaxbx3ax5b t+32,x25x15xaxbx1at,x40x10xfx8xex16x49x4ax1ay =
3+t
2?2t?1 (x?1)+2,x2bx?2y +3 = 0.
4,x38x2cx2d,x79x7ax48x16x28xex8xex5cx5dxdx72x2ex35x33x4fx16x8x48x1b.
x2x3,x44x43x79x7ax48x16xbx41x49x4ax32y2 = 2px (p > 0),x44x79x7ax48x16x8xex32x2ax33x4fx5by = kxx16x33x4fx11
x16x20x12x5cx5dx16,x44x33x4fx11x16x49x4ax32y = kx+t,x34x25xdx79x7ax48x24x12x16x15xaxbx52x53x49x4a(kx+t)2 = 2px,x2b
k2x2 +2(kt?p)x+t2 = 0,x2ax43x2dx60x20x12x16x15xaxbx1a x1 +x22 = p?ktk2,x16xaxbx1ak￠ p?ktk2 +t = pk,
x32xbx35x1ax5a.x2bx8xex33x4fx5bxx35.
x0 x1 9–6
1,x22x233x35x33x34:
Π1, x+2y?2z +3 = 0;
Π2, 2x+y +2z?1 = 0;
Π3, 2x?2y?z?3 = 0;
x21x55x4bx1aO0x0y0;O0y0z0;O0x0z0x33x34,x24x8x9xaxbx1dx1exfx1f,x63x1bx1cx15x11x12x16x4dxaxbxdx4dx11x12x16x15xaxb.
x4,x44P(x;y;z)x13x15xaxbxcx14x16xaxbx32(x0;y0;z0),x34Px3d3x35xaxbx33x34x16x6ax1cx3ax5bx303x35xaxbx16
x1dx72x6b,x2b 8
>>>
<
>>>:
jx0j = j2x+y +2z?1j3
jy0j = j2x?2y?z?3j3
jz0j = jx+2y?2z +3j3,
x44T =
0
B@
2
3
1
3
2
32
3?
2
3?
1
31
3
2
3?
2
3
1
CA,x2ax5bTx32x31x24x3ax3bx37jTj = 1,x1ex3x2dx29x5dxaxbx1dx1exfx1fx4bx1a:
0
BB
B@
x0
y0
z0
1
1
CC
CA =
0
BB
BB
@
2
3
1
3
2
3?
1
32
3?
2
3?
1
3?11
3
2
3?
2
3 10 0 0 1
1
CC
CC
A
0
BB
B@
x
y
z
1
1
CC
CA:
x30x32x5ex5fxcx44x4ex16x8x9xaxbx1dx1e,x58x59x4dxaxbx11x12Ox16x15xaxbx32
13 ;?1;1
·
.
￠ 14 ￠
x4dxaxbx39x15xaxbx13x14x16xfx1fx1a:
0
BB
B@
x
y
z
1
1
CC
CA =
0
BB
BB
@
2
3
2
3
1
3
5
91
3?
2
3
2
3?
11
92
3?
1
3?
2
3
5
90 0 0 1
1
CC
CC
A
0
BB
B@
x0
y0
z0
1
1
CC
CA;
x26x43x15xaxbx11x12O0x16x4dxaxbx32
5
9 ;?
11
9 ;
5
9
·
.
2,x61x62x45x46x47x34x16x49x4a:
x2 +y2 +5z2?6xy?2xz +2yz?6x+6y?6z +10 = 0;
x63x2x1cx30x32x4x5x47x34.
x4,x2ax3ex44,
A =
0
@
1?3?1
3 1 1
1 1 5
1
A; B =
0
@
3
3
3
1
A;
x3ax3bAx16x69x6ax1fx20x1fx32′A(?) =?3? 7?2 + 36,Ax16x69x6ax6bx1a?1 = 6,?2 = 3,?3 =?2,xdx303
x35x69x6ax6bx72x73x16x41x42x69x6ax18x19(x26x69x6ax6bx28x1f,x25x3x30x23x31x24)x21x55x32?1 =
p6
6 ;
p6
6 ;
p6
3
,?2 =
p
3
3 ;?
p3
3 ;
p3
3
,?3 =
p
2
2 ;
p2
2 ;0
.
x29x303x35x69x6ax18x19x16xaxbx78x1ax50x18x19x5cx21x3ax3b
T =
0
BB
B@
p6
6
p3
3
p2
2p
6
6?
p3
3
p2
2p
6
3
p3
3 0
1
CC
CA;
x34Txbx7x32x31x24x3ax3b,x1dx26jTj = 1,Tx52x53x1ex3x16x71x24,x2bxeTTAT = diag(6;3;?2).
x74x24AX +B = 0x16x57?,x23x1bx5bx57x48x76x49x4ax56
8>
<
>:
x?3y?z = 3
3x+y +z =?3
x+y +5z = 3
x60? = (1?1 1)T,x26x43x78x8x9xaxbx1dx1e
0
BB
B@
x
y
z
1
1
CC
CA =
0
BB
BB
B@
p6
6
p3
3
p2
2 1p
6
6?
p3
3
p2
2?1p
6
3
p3
3 0 10 0 0 1
1
CC
CC
CA
0
BB
B@
x0
y0
z0
1
1
CC
CA
x2dx4cx11x49x4ax61x62x1a:
6x02 +3y02?2z02 +1 = 0:
x30x32xbx35x77x22x77x47x34.
3,x61x62x45x46x47x34x16x49x4a:
2x2 +2y2 +3z2 +4xy +2xz +2yz?4x+6y?2z +3 = 0;
x63x2x1cx30x32x4x5x47x34.
￠ 15 ￠
x4,x2ax3ex44,
A =
0
@
2 2 1
2 2 1
1 1 3
1
A; B =
0
@
2
3
1
1
A;
x3ax3bAx16x69x6ax1fx20x1fx32′A(?) =?3? 7?2 + 10?,Ax16x69x6ax6bx1a?1 = 5,?2 = 2,?3 = 0,xdx303
x35x69x6ax6bx72x73x16x41x42x69x6ax18x19(x26x69x6ax6bx28x1f,x25x3x30x23x31x24)x21x55x32?1 =
p
3
3 ;
p3
3 ;
p3
3
,?2 =
p
6
6 ;
p6
6 ;?
p6
3
,?3 =
p
2
2 ;?
p2
2 ;0
.
x29x303x35x69x6ax18x19x16xaxbx78x1ax50x18x19x5cx21x3ax3b
T =
0
BB
B@
p3
3
p6
6
p2
2p
3
3
p6
6?
p2
2p
3
3?
p6
3 0
1
CC
CA;
x34Txbx7x32x31x24x3ax3b,x2dx32jTj =?1,x26x43x3dx23x4cx27x20x24xbx50x1dx20,x2ax4b
T =
0
BB
B@
p3
3
p6
6?
p2
2p
3
3
p6
6
p2
2p
3
3?
p6
3 0
1
CC
CA;
Tx52x53x1ex3x16x71x24,x2bxeTTAT = diag(5;2;0).
x2ax5bAx25x61,x63x372 = rankA < rank(A B) = 3,x1ex3x7bx78x7cx7dxaxbx1dx1e
X
1
T 0
0 1
X0
1
x60
x3d:
TT 0
0 1
A B
BT 3
T 0
0 1
=
0
BB
BB
B@
5 0 0 0
0 2 0
p6
2
0 0 0 5
p2
2
0
p6
2
5p2
2 3
1
CC
CC
CA:
x2b5x02 +2y02 +p6y0 +5p2z0 +3 = 0,x26x49x1a5x02 +2
y0 +
p6
4
2
+5p2
z0 + 9
p2
40
= 0,x26x43x74x78
x33x7exaxbx1dx1e 0
BB
B@
x0
y0
z0
1
1
CC
CA =
0
BB
BB
@
1 0 0 0
0 1 0?
p6
4
0 0 1? 9
p2
400 0 0 1
1
CC
CC
A
0
BB
B@
x00
y00
z00
1
1
CC
CA
x2dx5dx49x4ax61x62x1a
5x002 +2y002 +5p2z00 = 0:
x30x32xbx35x67x68x79x7ax34.
x23x73x16xaxbx1dx1exfx1fx1a
0
BB
B@
x
y
z
1
1
CC
CA =
0
BB
BB
B@
p3
3
p6
6?
p2
2?
1
40p
3
3
p6
6
p2
2?
19
40p
3
3?
p6
3 0
1
20 0 0 1
1
CC
CC
CA
0
BB
B@
x00
y00
z00
1
1
CC
CA:
￠ 16 ￠
x0x1x2 x3x4x5x6x7x8x5x6
x0 x1 9–1
1,x7x8x9xaxbxc[O;·1;·2]xd[O;·01;·02]xexfx10x11x12,x13x11xaxbxc[O;·1;·2]x14,x15xaxbxcx16x17x18x19
x1a:
·01 =
p2
2p
2
2
!; ·02 =
p2
2p
2
2
!
:
(1)x1bx1cxaxbx1dx1exfx1f;
(2)x1bx1cx11xaxbxcx20x16x17x18x19·1 =
1
0
,·2 =
0
1
x13x15xaxbxcx14x16xaxbx21x19;
(3)x22x23x18x19?!v x13[O;·1;·2]x16x21x19x1a
1
1
,x24x25x13x15xaxbxc[O;·01;·02]x14x16x21x19.
x4,(1)x26x1a(·01;·02) = (·1;·2)T,x27x20T =
p2
2?
p2
2p
2
2
p2
2
!
,x28x29xaxbx1dx1exfx1fx1a
x
y
= T
x0
y0
=
p2
2?
p2
2p
2
2
p2
2
!
x0
y0
:
(2) x2a(1)x23,(·1;·2) = (·01;·02)T?1,x27x20T?1 =
p2
2
p2
2
p2
2
p2
2
!
,x28x29 ·1 =
p2
2 ·
01?
p2
2 ·
02,
·2 =
p2
2 ·
01 +
p2
2 ·
02,x2b,·1 =
p2
2
p2
2
!
,·2 =
p2
2p
2
2
!
.
(3)x2c
x
y
= T
x0
y0
x2dx2ex23
x0
y0
= T?1
x
y
,x2fx13?!v = ·1?·2,x28x29
x0
y0
=
p2
2
p2
2
p2
2
p2
2
!
1
1
=
0
p2
;
x30x31x32?!v x13x15xaxbxcx14x16x21x19.
2,x13x33x34x8x9xaxbxc[O;·1;·2]x20,x22x23x15x16x8x9xaxbxc[O0;·01;·02]x16x11x12O0x16xaxbx1a(3;2),x12
M(5;3)x13x15xaxbxcx16x0x35x36,x37x12Mx16x15xaxbx0 > 0,x38x39x3ax3bx3cx1fx1bx1cx2c[O;·1;·2]x3d[O0;·01;·02]x16
xaxbx1dx1exfx1f.
x4,x26x1aX0 =
3
2
,x37x2ax3ex3fx23!O0M =
5
3
3
2
=
2
1
,x40!O0Mx16x41x42x18x19x32
2p5
5p
5
5
!
,
x2bcos = 2
p5
5,sin =
p5
5, x28x29T =
2p5
5?
p5
5p
5
5
2p5
5
!
,x26x43x1dx1exfx1fx1a
0
@
x
y
1
1
A =
0
BB
@
2p5
5?
p5
5 3p
5
5
2p5
5 20 0 1
1
CC
A
0
@
x0
y0
1
1
A:
￠ 1 ￠
3,x44x45x46x47x48Cx13x8x9xaxbxc[O;·1;·2]x20x16x49x4ax32:
x2?xy +y2 +2x?4y +1 = 0:
(1)x4bx15x16x8x9xaxbxc[O0;·01;·02],x4cO0x13x4dxaxbxcx14x16xaxbx1a(0;2),x37xe8
<
:
·01 =
p2
2 ·1 +
p2
2 ·2
·02 =?
p2
2 ·1 +
p2
2 ·2;
x38x39x3ax3bx3cx1fx1bx1cxaxbx1dx1exfx1f;
(2)x24x47x48Cx13x15xaxbxcx14x16x49x4a.
x4,(1)x4ex3ex44,X0 =
0
2
,T =
p2
2?
p2
2p
2
2
p2
2
!
,x37x4fx50x1fjTj = 1,x28x29xaxbx1dx1exfx1fx1a:
0
@
x
y
1
1
A =
0
BB
@
p2
2?
p2
2 0p
2
2
p2
2 20 0 1
1
CC
A
0
@
x0
y0
1
1
A:
(2) Cx16x49x4ax1a x026 + y022 = 1.
4,x44xex33x34x8x9xaxbxc[O;·1;·2],x51x15x16x8x9xaxbxc[O0;·01;·02]x52x53,x0x35x54y0x35x13x4dxaxbxcx20
x16x49x4ax21x55x32x?2y +2 = 0x542x+y +4 = 0.
(1)x24x2cx4dxaxbxcx3dx15xaxbxcx16x1dx1exfx1f;
(2)x24x8x48x?y +2 = 0x13x15xaxbxcx20x16x49x4a;
(3)x24x8x483x0 +y0 +1 = 0x13x4dxaxbxcx20x16x49x4a.
x4,(1)x26O0x12x16xaxb(x0;y0)x32x49x4ax56
( x?2y +2 = 0
2x+y +4 = 0 x16x57,x2bX0 =
x
0
y0
=
2
0
.
x58x59x?2y+2 = 0x16x49x18xcx5ax1a2, 1,2x+y+4 = 0x16x49x18xcx5ax1a?1, 2,x28x29·01 = 2
p5
5 ·1+
p5
5 ·2,
·02 =?
p5
5 ·1 +
2p5
5 ·2,x2b(·
01;·02) = (·1;·2)
2p5
5?
p5
5p
5
5
2p5
5
!
,x2ax5b
flfl
flfl
fl
2p5
5?
p5
5p
5
5
2p5
5
flfl
flfl
fl = 1,·
01;·02x5cx5d
x5ex5fxc,x28x29xaxbx1dx1exfx1fx1a
0
@
x
y
1
1
A =
0
BB
@
2p5
5?
p5
5?2p
5
5
2p5
5 00 0 1
1
CC
A
0
@
x0
y0
1
1
A:
(2) x?y +2 = 0x13x15xaxbxcx20x16x49x4ax1a:
2p5
5 x
0?
p5
5 y
0?2
!
p
5
5 x
0 + 2
p5
5 y
0
!
+2 = 0;
x2bx0?3y0 = 0.
(3)x2a(1)x16xaxbx1dx1exfx1fx2dx29x60x3d
0
@
x0
y0
1
1
A =
0
BB
@
2p5
5
p5
5
4p5
5
p5
5
2p5
5?
2p5
50 0 1
1
CC
A
0
@
x
y
1
1
A;
￠ 2 ￠
x403x0 +y0 +1 = 0x13x4dxaxbxcx14x16x49x4ax1a:
3
2p5
5 x+
p5
5 y +
4p5
5
!
+
p5
5 x+
2p5
5 y?
2p5
5
!
+1 = 0;
x2b5x+5y +10+p5 = 0.
x0 x1 9–2
1,x61x62x45x46x47x48x16x49x4a
5x2 +4xy +2y2?24x?12y +12 = 0;
x63x64x1cx25x16x65x3cx29x66x15x16xaxbx35.
x4,x3ax3b A =
5 2
2 2
,B =
12
6
,x26x43 I1 = Tr(A) = 7 > 0,I2 = jAj = 6 > 0,I3 =
flfl
flfl
flfl
5 2?12
2 2?6
12?6 12
flfl
flfl
flfl =?108 < 0,x43x47x48x32x67x68,Ax16x69x6ax6bx32x49x4a?2?7? + 6 = 0x16x6c,x57x60?1 = 6,
2 = 1,x40x62x61x6dx16x49x4ax1a6x02 +y02? 1086 = 0,x2b x023 + y0218 = 1.
x1ax64x1cx27x6ex6fx65x3c,x70x71x24x1cxaxbx1dx1exfx1f,x72x73x5bx69x6ax6c 6 xd 1 x16x41x42x69x6ax18x19x21x55x32
2p5
5 ;
p5
5
xd
p5
5 ;
2p5
5
,x28x29T =
2p5
5?
p5
5p
5
5
2p5
5
!
,x37jTj = 1,x74x24x47x48x16x20x75(x2bx15
xaxbxcx16x11x12)O0(x0;y0),x57x48x76x49x4ax56(
5x0 +2y0?12 = 0
2x0 +2y0?6 = 0
x60X0 =
x
0
y0
=
2
1
,x26x43xaxbx1dx1exfx1fx32
0
@
x
y
1
1
A =
0
BB
@
2p5
5?
p5
5 2p
5
5
2p5
5 10 0 1
1
CC
A
0
@
x0
y0
1
1
A:
x24x1cx15xaxbxcx20x0x35xdy0x35x13x4dxaxbxcx20x16x49x4ax21x55x32x?2y = 0 (x2by0 = 0)xd2x + y?5 = 0 (x2b
x0 = 0).
x30x10x1x0x11x31x32x12
x11x34x12x13x12x13x38x14
x15x3cx18x43x1bx48x4cx4dx50x51x52x4ax56x50x54x5bx5ax5cx5fx61
x62x62x66
x67x69x6b
x6bx6dx6d
x6fx6fx70
x71x72x73
x74x74x75
x76x77x78
x79x79x6e
x2fx2ex6ex0x6ex2fx10xf
x10x2fx0x30xfx1x0
x10x30x2fx6fx10x7cx6e
x2fx6dx7ax79
x78x77x76
x76x74x74
x73x72x71
x71x6fx6f
x6ex6cx6c
x6bx69x68
x66x64x62
x5ex5cx58
x5bx5ax50x57x56x55x53x52x50x4ex4dx1dx48x45x19x17x16x3x15x39x14x13x1x13x35x12x11
x12x1x32x11x0x32x30x10
x11
x75x75x75x75x75x75x75x40x50x20x8
x74
x74
x74
x74
x74
x74
x74
x74
x74
x74
x70x0
x10x0
x30x30x30x30x30
x30x30x30x30x30
x30x30x30x30x30
x30x30x30x30x30
x30x30x30x30x30
x30x30x30x30x48
x1cx6
xex2cx70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70x70
x70x70
x7x1fx3ax9x21x46
x7ax2fxfx1x11x6
x5x28x7x1dx37
x68xdx77x2dxf
x
y
x0
y0
O
O0
x01x1x2
￠ 3 ￠
2,x61x62x45x46x47x48x49x4a
x2?4xy?2y2 +10x+4y = 0;
x63x64x1cx25x16x65x3cx29x66x15x16xaxbx35.
x4,x3ax3bA =
1?2
2?2
,B =
5
2
,x26x43I1 = Tr(A) =?1 < 0,I2 = jAj =?6 < 0,I3 =
flfl
flfl
flfl
1?2 5
2?2 2
5 2 0
flfl
flfl
flfl = 6 > 0,x43x47x48x32x77x47x48,Ax16x69x6ax6bx32x49x4a?2 +6 = 0x16x6c,x57x60?1 = 2,?2 =?3.
x40x62x61x6dx16x49x4ax1a2x02?3y02?1 = 0,x2b x021
2
y021
3
= 1.
x1ax64x1cx27x6ex6fx65x3c,x70x71x24x1cxaxbx1dx1exfx1f,x72x73x5bx69x6ax6c 2 xd?3 x16x41x42x69x6ax18x19x21x55x32
2p5
5 ;?
p5
5
xd
p
5
5 ;
2p5
5
,x28x29T =
2p5
5
p5
5
p5
5
2p5
5
!
,x37jTj = 1,x74x24x47x48x16x20x75(x2bx15
xaxbxcx16x11x12)O0(x0;y0),x57x48x76x49x4ax56(
x0?2y0 +5 = 0
2x0?2y0 +2 = 0
x60X0 =
x
0
y0
=
1
2
,x26x43xaxbx1dx1exfx1fx32
0
@
x
y
1
1
A =
0
BB
@
2p5
5
p5
5?1
p5
5
2p5
5 20 0 1
1
CC
A
0
@
x0
y0
1
1
A:
x24x1cx15xaxbxcx20x0x35xdy0x35x13x4dxaxbxcx20x16x49x4ax21x55x32x+2y?3 = 0xd2x?y +4 = 0.
x72x73x72
x73x73x73
x74x75x76
x2dx6cxfx6dx2fx6f
x35x32x48x45x19x18x3x14x2
x34x0x11
x10x12x0x1x38x16x18x1ax45
x48x49x4ax35x36x37x37x38x7
x10xfx0xfx2fx6dx2ex6bx76
x75x74x73
x73x73x72
x73x72x72
x72x72x72
x72x72x72
x72x72x69
x2c
x75x75x75x75x40x8
x74
x74
x74
x74
x74
x74
x74
x74
x74
x70x10x0
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x38x14
x50x50x50x50x50
x50x50x50x50x50
x50x50x50x50x50
x50x50xa
x7x1fx3ax9x21x46
x7ax2fxfx1x11x6 x1x13x9x3x18
x9x23x49xbx58
x
y
x0
y0
O
O0
x02x1x2
3,x61x62x45x46x47x48x49x4a
x2?3xy +y2 +10x?10y +21 = 0;
x63x78x1cx25x16x65x3cx29x66x15x16xaxbx35.
x4,x3ax3bA =
1? 3
2
32 1
!
,B =
5
5
,x26x43I1 = Tr(A) = 2 > 0,I2 = jAj =? 54 < 0,
I3 =
flfl
flfl
flfl
fl
1? 32 5
32 1?5
5?5 21
flfl
flfl
flfl
fl
=? 54 < 0,x43x47x48x32x77x47x48,Ax16x69x6ax6bx32x49x4a?2?2 54 = 0x16x6c,x57x60
￠ 4 ￠
1 = 52,?2 =? 12, x40x62x61x6dx16x49x4ax1a 52 x02? 12 y02 +1 = 0,x2b y022? x022
5
= 1.
x1ax64x1cx27x6ex6fx65x3c,x70x71x24x1cxaxbx1dx1exfx1f,x72x73x5bx69x6ax6c 52 xd? 12 x16x41x42x69x6ax18x19x21x55x32
p
2
2 ;?
p2
2
xd
p
2
2 ;
p2
2
,x28x29T =
p2
2
p2
2
p2
2
p2
2
!
,x37jTj = 1,x74x24x47x48x16x20x75(x2bx15xaxbxcx16
x11x12)O0(x0;y0),x57x48x76x49x4ax56 (
x0? 32 y0 +5 = 0
32 x0 +y0?5 = 0
x60X0 =
x
0
y0
=
2
2
,x26x43xaxbx1dx1exfx1fx32
0
@
x
y
1
1
A =
0
BB
@
p2
2
p2
2?2
p2
2
p2
2 20 0 1
1
CC
A
0
@
x0
y0
1
1
A:
x24x1cx15xaxbxcx20x0x35xdy0x35x13x4dxaxbxcx20x16x49x4ax21x55x32x+y = 0xdx?y +4 = 0.
x39x3ax4fx50x52x55x26x2axex2dxex0x2fxfx10
xcxbxc
xbxbxb
xbxbxb
xaxax9
x8x34x11x12x1x4
x31
x10x0x1x0x11x33x11x34x7
x9xaxa
xaxbxb
xbxbxb
xcxbxc
xcxbx36
x13
x10xfx0x2exfx2cxexdx26xbx23x53x50x4fx4ex3ax38x38x36x1e
x75x75x75x75x75
x74
x74
x74
x74
x74
x74
x74
x74
x74
x74
x74
x30x10x0
x73x73
x73x73
x73x73
x73x73
x73x73x71x71
x71x71
x71x71
x71x71
x71x71
x7x1fx3ax9x21x46
x7ax2fxfx1x11x6
xbx56xdx6a
x3x16x5x2a
x
y
x0
y0
O
O0
x03x1x2
4,x61x62x45x46x47x48x49x4a
4x2?4xy +y2 +6x?8y +3 = 0;
x63x64x1cx25x16x65x3cx29x66x15x16xaxbx35.
x4,x3ax3b A =
4?2
2 1
,B =
3
4
,x26x43 I1 = Tr(A) = 5 > 0,I2 = jAj = 0,I3 =
flfl
flfl
flfl
4?2 3
2 1?4
3?4 3
flfl
flfl
flfl =?25 < 0,x43x47x48x32x79x7ax48.Ax16x69x6ax6bx32x49x4a?2?5? = 0x16x6c,x57x60?1 = 0,?2 = 5,x72
x73x5bx69x6ax6c0xd5x16x41x42x69x6ax18x19x21x55x32
p
5
5 ;
2p5
5
xd
2
p5
5 ;
p5
5
,x28x29T =
p5
5?
2p5
5
2p5
5
p5
5
!
,
x37jTj = 1.
￠ 5 ￠
x7bx39Tx78x7cx7dxaxbx1dx1e,x2dx60
TT 0
0 1
A B
BT 3
T 0
0 1
=
0
@
0 0?p5
0 5?2p5
p5?2p5 3
1
A:
x2bb01 =?p5,b02 =?2p5,x4b
X0 =
0
@
b0222c
2b01?2
b
02
2
1
A =
p5
10
2p5
5
!;
x39X0x78x33x7exaxbx1dx1e
X0
1
=
E X
0
0 1
X00
1
;
x2dx60
E 0
XT0 1
0
@
0 0?p5
0 5?2p5
p5?2p5?2
1
A
E X
0
0 1
=
0
@
0 0?p5
0 5 0
p5 0 0
1
A;
x2bx49x4ax61x62x1a5y002?2p5x00 = 0,x27x62x61x49x4ax1ay002 = 2
p5
5 x
00,x7fx16xaxbx1dx1exfx1fx1a
X
1
=
T TX
0
0 1
X00
1
=
0
BB
@
p5
5?
2p5
5?
9
10
2p5
5
p5
5
1
50 0 1
1
CC
A
0
@
x00
y00
1
1
A:
x30x0x79x7ax48x16x1x12xaxbx32TX0 =
910 ; 15
·T
,x15xaxbxcx20x0x35xdy0x35x13x4dxaxbxcx20x16x49x4ax21x55x32
2x?y +2 = 0xd2x+4y +1 = 0.
x30x10x11x31x10x11x10x32
x10x32x4x5
x5x5x6
x6x7x7
x7x7x8
x8x8x8
x8x9x8
x9x9x9
x9xax9
xax9xa
xax9x35
x2
x10x30xfx0x10x7cx2fx6cx2e
x2dxex2dx67x29x28x25x24x54x50x4ex4bx4ax48x47x46x2ax26x24x23
x75x75x75x72x40x74
x74
x74
x74
x74
x74
x74
x74
x74
x70x0
x0
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x10x10x10
x10x10x38
x14
x50x50x50x50x50
x50x50x50x50x50 x7x1fx3ax9x21x46
x7ax2fxfx1x11x6
x1x13x9x3x18
x49x23x9x58xb
x
y
x0
y0
OO0
x04x1x2
5,x61x62x14x50x45x46x47x48x16x49x4a,x63x2x1cx25x3x32x4x5x47x48:
(1) 4x2?4xy +y2 +4x?2y = 0;
(2) x2?2xy +y2 +2x?2y?3 = 0.
x4,(1) x3ax3bA =
4?2
2 1
,B =
2
1
,x26x43I1 = Tr(A) = 5 > 0,I2 = jAj = 0,I3 =
flfl
flfl
flfl
4?2 2
2 1?1
2?1 0
flfl
flfl
flfl = 0,x1ax6x7x47x48x16x8x9x70x71xaxbxcxdxe,Ax16x69x6ax6bx32x49x4a?2? 5? = 0x16x6c,x57x60
1 = 0,?2 = 5,x72x73x5bx69x6ax6c0xd5x16x41x42x69x6ax18x19x21x55x32
p
5
5 ;
2p5
5
xd
2
p5
5 ;
p5
5
,x28x29
￠ 6 ￠
T =
p5
5?
2p5
5
2p5
5
p5
5
!
,x37jTj = 1.
x7bx39Tx78x7cx7dxaxbx1dx1e,x2dx60
TT 0
0 1
A B
BT 0
T 0
0 1
=
0
@
0 0 0
0 5?p5
0?p5 0
1
A:
x2bb01 = 0,b02 =?p5,x4bX0 =
0
b
02
2
=
0p
5
5
,x39X0x78x33x7exaxbx1dx1e,x2dx60
E 0
XT0 1
0
@
0 0 0
0 5?p5
0?p5 0
1
A
E X
0
0 1
=
0
@
0 0 0
0 5 0
0 0?1
1
A;
x2bx49x4ax61x62x1a5y002?1 = 0,x2by00 = §
p5
5,x30x32xbx72x33x4fx8x48.x7fx16xaxbx1dx1exfx1fx1a
X
1
=
T TX
0
0 1
X00
1
=
0
BB
@
p5
5?
2p5
5?
2
5
2p5
5
p5
5
1
50 0 1
1
CC
A
0
@
x00
y00
1
1
A:
(2) x3ax3b A =
1?1
1 1
,B =
1
1
,x26x43 I1 = Tr(A) = 2 > 0,I2 = jAj = 0,I3 =
flfl
flfl
flfl
1?1 1
1 1?1
1?1?3
flfl
flfl
flfl = 0,x1ax6x7x47x48x16x8x9x70x71xaxbxcxdxe,Ax16x69x6ax6bx32x49x4a?2? 2? = 0x16x6c,x57
x60?1 = 0,?2 = 2,x72x73x5bx69x6ax6c0xd2x16x41x42x69x6ax18x19x21x55x32
p
2
2 ;
p2
2
xd
p2
2 ;
p2
2
,x28x29
T =
p2
2?
p2
2p
2
2
p2
2
!
,x37jTj = 1.
x7bx39Tx78x7cx7dxaxbx1dx1e,x2dx60
TT 0
0 1
A B
BT?3
T 0
0 1
=
0
@
0 0 0
0 2?p2
0?p2?3
1
A:
x2bb01 = 0,b02 =?p2,x4bX0 =
0
b
02
2
=
0p
2
2
,x39X0x78x33x7exaxbx1dx1e,x2dx60
E 0
XT0 1
0
@
0 0 0
0 2?p2
0?p2?3
1
A
E X
0
0 1
=
0
@
0 0 0
0 2 0
0 0?4
1
A;
x49x4ax61x62x1a2y002?4 = 0,x2by00 = §p2,x30x32xbx72x33x4fx8x48.x7fx16xaxbx1dx1exfx1fx1a
X
1
=
T TX
0
0 1
X00
1
=
0
BB
@
p2
2?
p2
2?
1
2p
2
2
p2
2
1
20 0 1
1
CC
A
0
@
x00
y00
1
1
A:
6,x24x14x50x45x46x47x48x16xfx10x48:
(1) 6x2?xy?y2 +3x+y?1 = 0;
(2) 2xy?4x?2y +3 = 0.
￠ 7 ￠
x4,(1)x3ax3bA =
6? 1
2
12?1
!
,B =
3
21
2
!
,x26x43I2 = jAj =? 254 < 0,I3 =
flfl
flfl
flfl
fl
6? 12 32
12?1 12
3
2
1
2?1
flfl
flfl
flfl
fl
=
25
4 > 0,x43x47x48x32x77x47x48.x47x48x16x20x75(x0;y0)x52x53x48x76x49x4ax56(
6x0? 12 y0 + 32 = 0
12 x0?y0 + 12 = 0
x60X0 =
x
0
y0
=
1
53
5
!
,xfx10x48x49x4ax1a
6
x+ 15
2
x+ 15
y? 35
y? 35
2
= 0:
x36x1fx2dx21x57x1a
3
x+ 15
+
y? 35
2
x+ 15
y? 35
= 0;
x28x29xfx10x48x49x4ax1a3x+y = 0x542x?y +1 = 0.
(2)x3ax3bA =
0 1
1 0
,B =
2
1
,x26x43I2 = jAj =?1 < 0,I3 =
flfl
flfl
flfl
0 1?2
1 0?1
2?1 3
flfl
flfl
flfl = 1 > 0,x43x47
x48x32x77x47x48.x47x48x16x20x75(x0;y0)x52x53x48x76x49x4ax56
( y
0?2 = 0
x0?1 = 0
x60X0 =
x
0
y0
=
1
2
,xfx10x48x49x4ax1a2(x?1)(y?2) = 0,x2bx = 1x54y = 2.
7,x31?x16x6bx11x12x49x4a
x2?2xy +?y2?2x+2y +5 = 0
x28x13x14x16x47x48x3cx15.
x4,x3ax3bA =
1
1?
,B =
1
1
,x26x43I1 = Tr(A) = 2?,I2 = jAj =?2?1 = (1)(?+1),
I3 =
flfl
flfl
flfl
1?1
1? 1
1 1 5
flfl
flfl
flfl = (5?+3)(1),x21x16x17x18x19x1ax11x12.
(i)x1b? 6= §1x1c,I2 6= 0,x1dx2dx21x1ax7x17x18x19,(a)? > 1x1e? <?1,x43x1cI2 > 0,x1b? > 1x1c,I1xdI3
x1fx20,x47x48x32x21x67x68;x1b? <?1x1c,I1xdI3x22x20,x47x48x32x67x68,(b)?1 <? < 1,x43x1cI2 < 0,x1b? =? 35
x1c,I3 = 0,x47x48x1axbx72x23x24x8x48;x25x1b? 6=? 35 x1c,x47x48x7fx32x77x47x48.
(ii)x1b? =?1x1c,I2 = 0,I3 6= 0,x47x48x32x79x7ax48.
(iii)x1b? = 1x1c,I2 = I3 = 0,x26x39x27x28x1dx19K1 =
flfl
flfl1
1 5
flfl
flfl+
flfl
flfl? 1
1 5
flfl
flfl = 102 = 8 > 0,x2dx23x47
x48x32xbx72x21x33x4fx8x48.
8,x31?x16x6bx11x12x49x4a
x2 +2?xy +y2?2x?2?y +? = 0
x28x13x14x47x48x16x3cx15.
x4,x3ax3bA =
1
,B =
1
,x26x43I1 = Tr(A) = 1 +?,I2 = jAj =?(1),I3 =
￠ 8 ￠
flfl
flfl
flfl
1
1
1
flfl
flfl
flfl =?(1)2(2?+1),x21x16x17x18x19x1ax11x12.
(i)x1b? 6= 1;0x1c,I2 6= 0,x1dx2dx21x1ax7x17x18x19,(a) 0 <? < 1,x43x1cI2 > 0,I1 > 0,I3 < 0,x47x48x32x67
x68; (b)? < 0x1e? > 1,x43x1cI2 < 0,x29x1b? =? 12 x1c,I3 = 0,x47x48x1axbx72x23x24x8x48;x25x1b? 6=? 12 x1c,x47
x48x7fx32x77x47x48.
(ii)x1b? = 0x1c,I2 = 0,I3 6= 0,x47x48x32x79x7ax48.
(iii)x1b? = 1x1c,I2 = I3 = 0,x26x39x27x28x1dx19K1 =
flfl
flfl 1?1
1 1
flfl
flfl+
flfl
flfl 1?1
1 1
flfl
flfl = 0,x2dx23x47x48x32xbx72x2a
x2bx8x48.
9,x22x23x49x4a(A1x+B1y+C1)2+2(A2x+B2y+C2) = 0,x27x20A1B2?A2B1 6= 0,A1A2+B1B2 = 0.
(1)x2cx2dx43x49x4ax13x14xbx0x79x7ax48;
(2)x24x1cx72x2ex35x16x49x4a.
x4,(1) x44x43x47x48x49x4ax32x2fx5bx8x9xaxbxc[O;·1;·2]x16,x2ax5bA1A2 + B1B2 = 0,x26x43x8x48L1,
A1x + B1y + C1 = 0 xdx8x48 L2, A2x + B2y + C2 = 0 x30x23x31x24,x28x32x44 A1B2? A2B1 > 0,x33
1 = pA21 +B21,?2 = pA22 +B22,x34L1;L2x16x41x42x49x18x18x19·01 =
A
1?
1; B1?
1
·
xd·02 =
A
2?
2; B2?
2
·
x5c
x5dxbx35x36x37x31x24x56.x33T =
A1
1
A2
2
B1
1
B2
2
!
,x34xe(·01;·02) = (·1;·2)T,Tx32xbx35x31x24x3ax3b,x37jTj = 1,x26
x43·01;·02x5cx5dxbx35x5ex5fxc.
x38x39x33 8
<
:
x0 = A1?
1
x+ B1?
1
y + C1?
1
y0 = A2?
2
x+ B2?
2
y + C2?
2;
x31xe
0
@
x0
y0
1
1
A =
0
B@
A1
1
B1
1
C1
1
A2
2
B2
2
C2
20 0 1
1
CA
0
@
x
y
1
1
A;
x30x32xbx35x8x9xaxbx1dx1exfx1f,x13x15x16xaxbxc[O0;·01;·02]x20,x47x48x16x49x4ax61x62x1a?21x02 +2?2Y0 = 0,x58x59
1 6= 0,x26x43x02 + 2?2?2
1
y0 = 0,x30x32xbx0x79x7ax48.
(2)x43x79x7ax48x16x72x2ex35x32y0x35,x49x4ax1ax0 = 0,x2bA1x+B1y +C1 = 0.
10,x44x45x46x47x48x49x4ax1a
a11x2 +2a12xy +a22y2 +2b1x+2b2y +c = 0:
x2cx2d:
(1)x45x46x47x48x1axbx0x3ax35x77x47x48x1ex7x0x23x30x3bx8x16x8x48x16x3cx21x3dx71x0x3ex32I1 = 0;
(2)x45x46x47x48x1ax68x16x3cx21x3dx71x0x3ex32I21 = 4I2,I1I3 < 0;
(3)x45x46x47x48x51x13x14xbx35x67x68,x38x24x3fx67x68x34x40.
x2x3,(1) x51x43x45x46x47x48x1ax3ax35x77x47x48,x34x2ax5bI1x32x31x24x28x1dx19,x2cx3ax35x77x47x48x16xbx41x49x4ax42x23
I1 = 0;x51x32x7x0x30x23x3bx8x16x8x48,x34x27xbx41x49x4ax1a?1x02 +?2y02 = 0,x37?1?2 < 0,x2by0 = §
r
1
2 x
0
x13x14x7x0x30x23x3bx8x16x8x48,x26x43
r
1
2 ￠
r
1
2
=?1,x2ex601?
2
= 1,x2bI1 = 0.
￠ 9 ￠
x43x44,x51I1 = 0,x34x26I1 =?1 +?2 = 0,x2dx23?2 =1(6= 0),x63x37I2 =21 < 0,x2ax62x61x49x4a
1(x02?y02)+ I3I
2
= 0x2dx23,x1bI3 6= 0x1cx47x48x32x3ax35x77x47x48;x1bI3 = 0x1cx47x48x32x7x0x30x23x3bx8x16x8x48.
(2)x51x43x45x46x47x48x32x68,x34x3dxeI2 > 0,I1 ￠I3 < 0(x67x68x9),x45x46x47x1bx48x49x8x9xaxbxcx23x62x61x49x4a
x1a?1x02 +?2y02 + I3I
2
= 0,x26x1ax32x68,x28x29?1 =?2,x5bx32I1 = 2?1,I2 =?21,x28x29I21 = 4I2.
x43x44,x51I1 ￠I3 < 0,I21 = 4I2,x2dx23I2 > 0,x47x48x32x67x68,x1dx26I1 =?1 +?2,I2 =?1?2,x2aI21 = 4I2
x2dx60?1 =?2,x40x43x47x48x32x68.
(3)x13xbx41x67x68x49x4a x2a2 + y2b2 = 1 (a > 0,b > 0)x20,x67x68x16x34x40S = …ab,x51x47x48x13x14xbx35x67x68,
x34x47x1bx48x49xaxbxcx44x6dx2dx60x25x16x62x61x49x4ax1a?1x02 +?2y02 + I3I
2
= 0,x2b x02?I
3I
2
￠ 1?
1
+ y02?I
3I
2
￠ 1?
2
= 1.
x28x29x34x40S = …
rI2
3
I22 ￠
1
1?2 =
…jI3jpI2
I22,
11,x22x23x67x68x4ax35x54x4bx35x21x55x13x8x48x+y?1 = 0x54x?y +1 = 0x36,x37x4ax4bx35x4ax21x55x1a4xd2.
x24x43x67x68x16x49x4a.
x4,x28x32x44x4ax35x13x0x35(x2by0 = 0)x36,x4bx35x13y0x35(x2bx0 = 0)x36,x78x1dx1e8
<
:
x0 =
p2
2 x?
p2
2 y +
p2
2
y0 =
p2
2 x+
p2
2 y?
p2
2 ;
x1bx5dx3ax3bx3cx1fx1a
0
@
x0
y0
1
1
A =
0
BB
@
p2
2?
p2
2
p2
2p
2
2
p2
2?
p2
20 0 1
1
CC
A
0
@
x
y
1
1
A;
x27x4cx36x9x16x4dx3ax3b
p2
2?
p2
2p
2
2
p2
2
!
x32x4fx50x1fx3ax5b1x16x31x24x3ax3b,x26x43x30x32x5ex5fx8x9xaxbxcx4ex16xaxbx1d
x1e,x13x15x16xaxbxcx4fx47x48x16x49x4ax73x1a x024 + y021 = 1,x26x39xaxbx1dx1exfx1f,x2dx60x13x4dxaxbxcx14x16x49x4a:
1
4
p
2
2 x?
p2
2 y +
p2
2
2
+
p
2
2 x+
p2
2 y?
p2
2
2
= 1,x2bx1a5x2 +6xy +5y2?6x?10y?3 = 0.
x0 x1 9–3
1,x50x55x14x50x72x73x51x34x32x52x1ax53x5ax54Rx3dx55x56x16x57x58,x63x2x1cx59x5ax32x41x58?x52x58?
(1) x 7! x2; (2) x 7! x3; (3) x 7!jxj;
(4) x 7! 2x; (5) x 7! sin(x2); (6) x 7! tanx.
x4,(1)–(5)x5bx32Rx3dx55x56x16x57x58,x27x20(2),(4)x32x41x58,(2)x32x52x58,(6)x28x32x57x58.
2,x44Sx13x14x33x34x36x28xex12x56x5dx16x5cx2b,Lx32xbx0x8x48,x5dx33x34x36x5ex35x12P(x;y)x72x73x3dx25x2fx5bLx16
x72x2ex12P0(x0;y0),x30x32Sx3dx55x56x16xbx35x1dx1e,x2ex1ax2fx5bx8x48Lx16x43x58,x2eLx32x43x58x35.
(1)x24x1cx33x34x2fx5bx8x48y = xx16x43x58xfx1f;
(2)x44x43x58x35x1aAx+By +C = 0,x24x1cx30x1cx16x43x58xfx1f;
(3)x44x53 1;x53 2x32x2fx5bx33x34x36x7x0x33x4fx8x48L1;L2x16x43x58.x2cx2dx53 2x53 1x32xbx35x33x7e.
x4,(1)x22x23P(x;y)x2fx5bx8x48y = xx16x72x2ex12x1aP0(x0;y0),x348
<
:
x+x0
2 =
y +y0
2
y?y0
x?x0 =?1;
￠ 10 ￠
x57x60x43x58xfx1fx1a (
x0 = y
y0 = x:
(2)x12P(x;y)x2fx5bx8x48Ax+By +C = 0x16x72x2ex12x1aP0(x0y0),x348
<
:
A
x+x0
2
·
+B
y +y0
2
+C = 0
A(y?y0) = B(x?x0);
x57x60x43x58xfx1fx1a (
x0 = 1A2 +B2?(B2?A2)x?2ABy?2AC￠
y0 = 1A2 +B22ABx+(A2?B2)y?2BC￠:
(3) x29L1 x78x1axx35x5fx60xaxbxc,x34L1;L2 x16x49x4ax21x55x1ay = 0xdy + C = 0,x27x20C 6= 0,x61
x53 1(P) = P0(x0;y0),x53 2(P0) = P00(x00;y00),x2a(2)x23,(
x0 = x
y0 =?y;
( x00 = x0
y00 =?y0?2C:
x62x63x6dxex60 (
x00 = x
y00 = y?2C:
x2dx64x53 2x53 1x32xbx35x33x7e.
3,x44x4d x32x1dx1e(x;y) 7! (x0;y0),(
x0 = 2x+3y +2
y0 =?x+2y?3
x65:
(1)x12(?1;1)x66x1dx5dx67x4x5x12?
(2)x8x48y = 2x66x1dx5dx67x4x5x65x3c?
(3)x12(9;?3)x32x2ax59x35x12x1dx46x1ax16?
x4,(1)x12(?1;1)x66x1dx1a(3;0).
(2)x8x48y = 2x36x16x12x32(t;2) (t 2R),x25x4d((t;2)) = (2t+8;?t+1),x26x43x1dx1ex6dx16x12x5dxbx0x8x48,
x27x68x5ax49x4ax1a
( x = 8+2t
y = 1?t;,x1ex+2y?10 = 0.
(3)x57x49x4ax56
( 2x+3y +2 = 9
x+2y?3 =?3 x60x = 2,y = 1,x40x4d((2;1)) = (9;?3).
4,x13x8x9xaxbxc[O;·1;·2]x20,x24x1cx33x34x69x12M0(x0;y0)x7cx7d 0x9x16x1dx1exfx1f.
x4,x29M0(x0;y0)x1ax11x12x5fx60x15x8x9xaxbxc[M0;·1;·2],x34x69x12M0x7cx7d 0x9x16x1dx1ex13x15xaxbx14x16
x1dx1exfx1fx1a (
x0 = cos 0?x?sin 0?y
y0 = sin 0?x+cos 0?y:
x44x12P(x;y)x45x7cx7dx1dx1aP0(x0;y0),x6cx4ex33x7exaxbx1dx1ex16xfx1f,P;P0x12x16x15xaxbx73x1a(x? x0;y? y0),
(x0?x0;y0?y0),x62x63x36x34x16xfx1fx2bx60(
x0 = cos 0(x?x0)?sin 0(y?y0)+x0
y0 = sin 0(x?x0)+cos 0(y?y0)+y0:
5,x51x5dx47x482xy = a2x69x11x12x7cx7d …4,x24x15x16x47x48x49x4a.
￠ 11 ￠
x4,x51P(x;y)x12x7cx7dx3dx67P0(x0;y0)x12,x34Px12x2dx2aP0x12x45x7cx7d?…4 x60x3d,x26x43
8<
:
x = cos
…4
·
x0?sin
…4
·
y0 =
p2
2 (x
0 +y0)
y = sin
…4
·
x0 +cos
…4
·
y0 =
p2
2 (?x
0 +y0);
x62x63x11x49x4a2xy = a2x2bx60y02?x02 = a2.
6,x33x34x16x3ax6ax1dx1ex4d x51xex7x35x28x6bx12A;B,x34x8x48ABx36x5ex35x12x5bx32x28x6bx12.
x4,x44C x12x13x8x48AB x36,x34x6cx4eC x32x52x42x5bx48x6cAB x36,xed(A;C) + d(C;B) = d(A;B)x1e
jd(C;A)?d(C;B)j = d(A;B),x33x4d(C) = C0,x34x26A;Bx32x4dx16x28x6bx12,xed(A;C0)+d(C0;B) = d(A;B)
x1ejd(C0;A)?d(C0;B)j = d(A;B),x38x39A;B;C0x28x10x48,x34x25x3x5cx5dxbx35x6dx9x3c,x25x6dx9x3cx7x6ex44x54x6e
x5bx6fx6dx6e,x7x6ex44x70x71x5bx6fx6dx6e,xdx36x72x3ax1fx73x74.
7,x24x14x72x75x58x1dx1ex16x28x6bx12,(
x0 = 3x?y?5
y0 = 2x+y +1:
x4,x57x49x4ax56
( x = 3x?y?5
y = 2x+y +1; x60x =?
1
2,y =?6,x2bx28x6bx12x29xe
12 ;?6
·
xbx35.
8,x51x13x75x58x1dx1ex41x14xbx0x8x48x16x76xdx27x55x56x2ax2b,x34x2ex30x0x8x48x1ax41x16x28x1dx8x48.x24x14x72x75x58x1d
x1ex16x28x1dx8x48,(
x0 = 7x?y +1
y0 = 4x+2y +4:
x4,x75x58x1dx1ex5dx8x48x1dx5dx8x48.x44x28x1dx8x48x1aL, Ax+By +C = 0,x51P(x;y) 2 L,x34P0(x0;y0) =
x41(P) 2 L,x2bAx0 +By0 +C = 0,x62x63x61x62x6dx60
(7A+4B)x+(2B?A)y +(A+4B +C) = 0:
x26x1aLx36x16x77x3fx12x5bx52x53x43x49x4a,x78x2dx30x35x49x4ax79x32Lx16x49x4a.x26x43xe
7A+4B
A =
2B?A
B =
A+4B +C
C = k:
x57x60k = 6x1ek = 3,x28x29 (
A =?4B
C = 0 ·
( A =?B
C = 32 B;
x40x28x1dx8x48xex7x0,?4x+y = 0x54?2x+2y +3 = 0.
9,x44x13x33x34x36x7ax1cx67x7x35x6dx9x3cABCx54DEF,x65xex16x35x75x58x1dx1ex5d4ABCx1dx5d4DEF?
x4,x2ax7bx3e3.3(9)x2dx23xe3! = 6x35x28x1fx16x75x58x1dx1e.
10,x2cx2d,x33x34x36x77x7ax7x35x8x9xbx7c(I)x54(II),x7fx7dx13x7exbx16x3ax6ax1dx1ex5d(I)x1dx5d(II).
x2x3,x7bx3e3.2(6)x22x45x7fx0x67x52x53x0x3ex16x3ax6ax1dx1ex16x7exbx76,x44xex8x9xbx7c[O;·1;·2]xd[O0;·01;·02],
x34x36x37x31x24x17·1;·2xd·01;·02x2dx29x6x7x7exbx16x31x24x1dx1ex41 x4cx60x41(·i) = ·0i,i = 1;2,x44x18x19– =!OO0,x1d
x2dx7x1xbx35x33x7ex1dx1ex54 –,x5bx32x2x2bx1dx1ex54 –x41x32xbx35x3ax6ax1dx1e,x25x5dx8x9xbx7c[O;·1;·2]x57x3d[O0;·01;·02].
x7dx13x76x3x2c.
x0 x1 9–4
1,x2cx2d,x33x34x36x69xbx35x4x7x12x7d90–x5 180–x5 270–x16x6dx35x7cx7dx521x5 x522x5 x523x54x6x1fx1dx1ex45x56x5dxb
x35x1dx1ex7.
￠ 12 ￠
x2x3,x61x520 = E,x34x26x52ix52j =
( x52
i+j x51i+j 6 3,
x52i+j?4 x51i+j > 4,x7x16x76x8(1)x66x52x53;x76x8(2)x34x32x58x59
x16;x1dx26x52?1i = x524?i,x76x8(3)x79x66x52x53,x28x29x30x32xbx35x7.
2,x1ba;bx4bx1ax77x3fx16x28x9x1axax16x5ax1c,x14x50x28xex16x75x58x1dx1ex56x5dx16x5cx2bx32x52x1axbx35x7?(
x0 = ax+by
y0 = bx+ay:
x4,x30x35x75x58x1dx1ex72x73x16x3ax3bx32A =
a b
b a
,x44A1 =
a
1 b1
b1 a1
,A2 =
a
2 b2
b2 a2
,x34A1A2 =
a
1a2 +b1b2 a1b2 +b1a2
a1b2 +b1a2 a1a2 +b1b2
x79x32x30x35x5cx2bx16xbxc,x26x43x76x8(1)x66x52x53;x1ba = 1,b = 0x1cx31x32x6x1fx1d
x1e,x26x43(2)x79x52x53; A?1 =
a
a2?b2?
b
a2?b2
ba2?b2 aa2?b2
!
x79x32x5cx2bx16xbxc,x26x43(3)x79x52x53.x30x35x5cx2bx6x53
x32x7.
x0 x1 9–5
1,x24x45x46x47x48x2 +2y2?4x?2y?6 = 0xdx46x12(8;0)x16x8xex49x4a,x63x24x1cx27x10xfx8xex16x49x4a.
x4,x3fx45x46x47x48x49x4ax2dx1bx5d
(x?2)2
21
2
+
y? 12
·2
21
4
= 1;
x32x67x68.
x78x1dx1e 8
<
:
x0 =
p42
21 (x?2)
y0 = 2
p21
21
y? 12
·;
x2b:
8<
:
x =
p42
2 x
0 +2
y =
p21
2 y
0 + 1
2,x34x43x1dx1ex10x68x02 +y02 = 1x1dx1ax67x68
(x?2)2
21
2
+
y? 12
·2
21
4
= 1;
x37x10x12
2p42
7 ;?
p21
21
x1dx5d(8;0),x28x29x43x1dx1ex10x68x16x8xey0 =?
p2
12 x
0x1dx1ax67x68x16x46x12(8;0)x16x8
xe,x+12y?8 = 0.
xdx68x8xey0 =?
p2
12 x
0x3bx8x16x8xey0 = 6p2x0x66x43x1dx1ex1dx5dxdx67x68x16x8xex+12y?8 = 0x10xfx16
x8xe12x?2y?23 = 0.
2,x44x77x47x48x16xbx41x49x4ax1a
x2
a2?
y2
b2 = 1;
x38x24x8xey = kba x(jkj < 1)x16x10xfx8xe(x28x7ax8xex16x33x4fx11x16x20x12x12x48).
x4,x6cx4ex72x2ex76,x77x47x48x16x20x75x32x28xexdx46x25x16x11x16x20x12,x26x43x28xex16x8xexbx7xdx46x25x16x20x75,x1ax6
x7x10xfx8xe,x13x70x74x14xbx35x12,x78xbx0x33x4fx11y = kba x+t,x25xdx77x47x48x16x24x12x52x53 x2a2?
kb
a x+t
·2
b2 =
1,x61x62x1ab2(1?k2)x2?2abktx?a2(b2 +t2),x26x43x33x4fx11x16x20x12x16x15xaxbx32 x1 +x22 = aktb(1?k2), x16
￠ 13 ￠
xaxbx1a kba ￠ aktb(1?k2) +t = t1?k2, x60x3dx10xfx8xex16x17x18x1a t1?k2 ￠ b(1?k
2)
akt =
b
ak,x10xfx8xex16x49
x4ax1ay = bak x,x2dx64x77x47x48x16x8xexdx10xfx8xex16x17x18x16x19x40x3ax5bx1ax5a b2a2,
3,x22x23x47x48xy?y2?2x+3y?1 = 0x16x8xexdyx35x33x4f,x24x25x16x49x4a,x63x24x1cx30x8xex16x10xfx8xe.
x4,x26x1aI2 =
flfl
flfl
fl
0 12
1
2?1
flfl
flfl
fl =?
1
4 < 0,I3 =
flfl
flfl
flfl
fl
0 12?1
1
2?1
3
2
1 32?1
flfl
flfl
flfl
fl
6= 0,x28x29x3fx47x48x32x77x47x48.x1ax24x25x16x20
x75,x57x29x14x49x4ax56,( 1
2 y0?1 = 0
1
2 x0?y0 +
3
2 = 0;
x60x0 = 1,y0 = 2,x2ax5bx22x7ax8xexdyx35x33x4f,x25x16x49x4ax32x = 1.
x78xbx0x33x4fx11x = t,x25xdx77x47x48x16x24x12xaxbx73x52x53ty?y2?2t+3y?1 = 0,x26x43x24x12x16x20x12x16
x16xaxbx3ax5b t+32,x25x15xaxbx1at,x40x10xfx8xex16x49x4ax1ay =
3+t
2?2t?1 (x?1)+2,x2bx?2y +3 = 0.
4,x38x2cx2d,x79x7ax48x16x28xex8xex5cx5dxdx72x2ex35x33x4fx16x8x48x1b.
x2x3,x44x43x79x7ax48x16xbx41x49x4ax32y2 = 2px (p > 0),x44x79x7ax48x16x8xex32x2ax33x4fx5by = kxx16x33x4fx11
x16x20x12x5cx5dx16,x44x33x4fx11x16x49x4ax32y = kx+t,x34x25xdx79x7ax48x24x12x16x15xaxbx52x53x49x4a(kx+t)2 = 2px,x2b
k2x2 +2(kt?p)x+t2 = 0,x2ax43x2dx60x20x12x16x15xaxbx1a x1 +x22 = p?ktk2,x16xaxbx1ak￠ p?ktk2 +t = pk,
x32xbx35x1ax5a.x2bx8xex33x4fx5bxx35.
x0 x1 9–6
1,x22x233x35x33x34:
Π1, x+2y?2z +3 = 0;
Π2, 2x+y +2z?1 = 0;
Π3, 2x?2y?z?3 = 0;
x21x55x4bx1aO0x0y0;O0y0z0;O0x0z0x33x34,x24x8x9xaxbx1dx1exfx1f,x63x1bx1cx15x11x12x16x4dxaxbxdx4dx11x12x16x15xaxb.
x4,x44P(x;y;z)x13x15xaxbxcx14x16xaxbx32(x0;y0;z0),x34Px3d3x35xaxbx33x34x16x6ax1cx3ax5bx303x35xaxbx16
x1dx72x6b,x2b 8
>>>
<
>>>:
jx0j = j2x+y +2z?1j3
jy0j = j2x?2y?z?3j3
jz0j = jx+2y?2z +3j3,
x44T =
0
B@
2
3
1
3
2
32
3?
2
3?
1
31
3
2
3?
2
3
1
CA,x2ax5bTx32x31x24x3ax3bx37jTj = 1,x1ex3x2dx29x5dxaxbx1dx1exfx1fx4bx1a:
0
BB
B@
x0
y0
z0
1
1
CC
CA =
0
BB
BB
@
2
3
1
3
2
3?
1
32
3?
2
3?
1
3?11
3
2
3?
2
3 10 0 0 1
1
CC
CC
A
0
BB
B@
x
y
z
1
1
CC
CA:
x30x32x5ex5fxcx44x4ex16x8x9xaxbx1dx1e,x58x59x4dxaxbx11x12Ox16x15xaxbx32
13 ;?1;1
·
.
￠ 14 ￠
x4dxaxbx39x15xaxbx13x14x16xfx1fx1a:
0
BB
B@
x
y
z
1
1
CC
CA =
0
BB
BB
@
2
3
2
3
1
3
5
91
3?
2
3
2
3?
11
92
3?
1
3?
2
3
5
90 0 0 1
1
CC
CC
A
0
BB
B@
x0
y0
z0
1
1
CC
CA;
x26x43x15xaxbx11x12O0x16x4dxaxbx32
5
9 ;?
11
9 ;
5
9
·
.
2,x61x62x45x46x47x34x16x49x4a:
x2 +y2 +5z2?6xy?2xz +2yz?6x+6y?6z +10 = 0;
x63x2x1cx30x32x4x5x47x34.
x4,x2ax3ex44,
A =
0
@
1?3?1
3 1 1
1 1 5
1
A; B =
0
@
3
3
3
1
A;
x3ax3bAx16x69x6ax1fx20x1fx32′A(?) =?3? 7?2 + 36,Ax16x69x6ax6bx1a?1 = 6,?2 = 3,?3 =?2,xdx303
x35x69x6ax6bx72x73x16x41x42x69x6ax18x19(x26x69x6ax6bx28x1f,x25x3x30x23x31x24)x21x55x32?1 =
p6
6 ;
p6
6 ;
p6
3
,?2 =
p
3
3 ;?
p3
3 ;
p3
3
,?3 =
p
2
2 ;
p2
2 ;0
.
x29x303x35x69x6ax18x19x16xaxbx78x1ax50x18x19x5cx21x3ax3b
T =
0
BB
B@
p6
6
p3
3
p2
2p
6
6?
p3
3
p2
2p
6
3
p3
3 0
1
CC
CA;
x34Txbx7x32x31x24x3ax3b,x1dx26jTj = 1,Tx52x53x1ex3x16x71x24,x2bxeTTAT = diag(6;3;?2).
x74x24AX +B = 0x16x57?,x23x1bx5bx57x48x76x49x4ax56
8>
<
>:
x?3y?z = 3
3x+y +z =?3
x+y +5z = 3
x60? = (1?1 1)T,x26x43x78x8x9xaxbx1dx1e
0
BB
B@
x
y
z
1
1
CC
CA =
0
BB
BB
B@
p6
6
p3
3
p2
2 1p
6
6?
p3
3
p2
2?1p
6
3
p3
3 0 10 0 0 1
1
CC
CC
CA
0
BB
B@
x0
y0
z0
1
1
CC
CA
x2dx4cx11x49x4ax61x62x1a:
6x02 +3y02?2z02 +1 = 0:
x30x32xbx35x77x22x77x47x34.
3,x61x62x45x46x47x34x16x49x4a:
2x2 +2y2 +3z2 +4xy +2xz +2yz?4x+6y?2z +3 = 0;
x63x2x1cx30x32x4x5x47x34.
￠ 15 ￠
x4,x2ax3ex44,
A =
0
@
2 2 1
2 2 1
1 1 3
1
A; B =
0
@
2
3
1
1
A;
x3ax3bAx16x69x6ax1fx20x1fx32′A(?) =?3? 7?2 + 10?,Ax16x69x6ax6bx1a?1 = 5,?2 = 2,?3 = 0,xdx303
x35x69x6ax6bx72x73x16x41x42x69x6ax18x19(x26x69x6ax6bx28x1f,x25x3x30x23x31x24)x21x55x32?1 =
p
3
3 ;
p3
3 ;
p3
3
,?2 =
p
6
6 ;
p6
6 ;?
p6
3
,?3 =
p
2
2 ;?
p2
2 ;0
.
x29x303x35x69x6ax18x19x16xaxbx78x1ax50x18x19x5cx21x3ax3b
T =
0
BB
B@
p3
3
p6
6
p2
2p
3
3
p6
6?
p2
2p
3
3?
p6
3 0
1
CC
CA;
x34Txbx7x32x31x24x3ax3b,x2dx32jTj =?1,x26x43x3dx23x4cx27x20x24xbx50x1dx20,x2ax4b
T =
0
BB
B@
p3
3
p6
6?
p2
2p
3
3
p6
6
p2
2p
3
3?
p6
3 0
1
CC
CA;
Tx52x53x1ex3x16x71x24,x2bxeTTAT = diag(5;2;0).
x2ax5bAx25x61,x63x372 = rankA < rank(A B) = 3,x1ex3x7bx78x7cx7dxaxbx1dx1e
X
1
T 0
0 1
X0
1
x60
x3d:
TT 0
0 1
A B
BT 3
T 0
0 1
=
0
BB
BB
B@
5 0 0 0
0 2 0
p6
2
0 0 0 5
p2
2
0
p6
2
5p2
2 3
1
CC
CC
CA:
x2b5x02 +2y02 +p6y0 +5p2z0 +3 = 0,x26x49x1a5x02 +2
y0 +
p6
4
2
+5p2
z0 + 9
p2
40
= 0,x26x43x74x78
x33x7exaxbx1dx1e 0
BB
B@
x0
y0
z0
1
1
CC
CA =
0
BB
BB
@
1 0 0 0
0 1 0?
p6
4
0 0 1? 9
p2
400 0 0 1
1
CC
CC
A
0
BB
B@
x00
y00
z00
1
1
CC
CA
x2dx5dx49x4ax61x62x1a
5x002 +2y002 +5p2z00 = 0:
x30x32xbx35x67x68x79x7ax34.
x23x73x16xaxbx1dx1exfx1fx1a
0
BB
B@
x
y
z
1
1
CC
CA =
0
BB
BB
B@
p3
3
p6
6?
p2
2?
1
40p
3
3
p6
6
p2
2?
19
40p
3
3?
p6
3 0
1
20 0 0 1
1
CC
CC
CA
0
BB
B@
x00
y00
z00
1
1
CC
CA:
￠ 16 ￠
