x0x1x2x3
x0x8x3 x9xaxbxcxdxexfx10x11
x0 x1 6–1
1,x28x21x22x4bx29x4dx2ax2bx2cx2dx29x2ex4dx2ax2bx2cx2fx30x41x30x4dx3c2,3,4xcx30x54x31x47x2dx29x32x33x31.
x4:
x70x74
x74
x74
x52x53x52x53
x52x53x52x53
x52x74
x74
x74
x70
x52x53x52x53
x52x53x52x53
x52
x2ex2dx2ex2d
x2ex2dx52x53x52x53x52x53
x52x53x52
x2dx2ex2dx2e
x2dx2e
x2ex2dx2ex2d
x2ex2dx74
x74
x74
x70
x47x1b x47x1b x46x1c
x47x1b x59x25 x59x25 x5ax24 x59x25
x59x25 x59x25
x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10x10x0
x0x74
x74
x74
x74
x44x44x44x44x44x44x44x44x44x44x44x44x74
x74
x74
x74
x0x44x44x44x44x44x44x44x44x44x44x44x44x24
x23x24x44x44x44x44x44x44x44x44x44x44x44x44
x24x23x24
x24x23x24
x74
x74
x74
x74
x0
x42x19 x41x19
x52x21 x52x21 x52x21 x52x21 x52x21 x52x21 x52x21 x52x21x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10
x3bx3bx3bx3bx3bx3bx3bx3bx3bx3bx3cx3bx3bx3bx3b
x3bx3bx3bx3bx3bx4dx8
x69x68x68
x69x68x68
x68x69x68
x68x69x68
x68x68x69
x68
x14xexfxe
xfxexf
xexfxe
xfxexf
xexfxe
xfxexf
xe
xfxfxf
xexfxf
xfxfxf
xexfxf
xexexe
xexexd
xex14x1
x25x59x25x59x24x5a
x25x59x25x59x25x59
x24x5ax25x59x25x59
x25x59xb
x37x38x37x37x37x38x37x37x37
x38x37x37x37x38x37x37x4c
x2ax63x68x67
x67x68x67
x67x68x67
x67x67x68
x67x67x68
x67
x11x7x7x7
x7x7x7
x7x6x7
x7x7x7
x7x7x7
x6x7x7
x7x7
x70x70x71
x70x70x71
x70
x13x13x13
x12x13x13
x13x13x13
x13x12x13
x13x15
x21x52x21x52x21x52x21x52x21x52x21x52x21x52x21x52x21x52x21x53x21x52x21x52
x21x52x52
x0 x1 6–2
1,x21x22x34x33x34x5ex5fx58x35x33x54x55:
(1)x42x1x36xcx40x37x26x23A(2;?3;5)x38B(4;1;?3);
(2)x35x39x18x1x1d x?42 = y +8?4 = z?21 xb,x24x3ax26(2;?3;6)x38(6;3;?2);
(3)x3ax26(?1;2;5),x24x193x43x1fx20x3bx33x18x3c;
(4)x3ax26(p2;p2;2),x24x3dx3ex23:
( x2 +y2 = 4;
z = 0:
x4,(1)x35x39x1fx20C
2+4
2 ;
3+1
2 ;
5?3
2
·
= (3;?1;1),x3fx36
R =
p
(3?2)2 +(?1+3)2 +(1?5)2 = p21;
x46x47x35x33x54x55x23(x?3)2 +(y +1)2 +(z?1)2 = 21.
(2)x28x35x39x18x40x3ax1x1dxb,x53x41xcx1fx20x42x23(4+2t;?8?4t;2+t),x45x28x26(2;?3;6)x38(6;3;?2)x18
x35x33xb,x46x47x41xex0x35x39xcx43x44x18x4d,x74
(4+2t?2)2 +(?8?4t+3)2 +(2+t?6)2 = (4+2t?6)2 +(?8?4t?3)2 +(2+t+2)2;
x25x3dt =?2,x66x2dx35x39x1fx20x8(0;0;0),x24x3fx36x4dx3c7,x35x33x54x55x23x2 +y2 +z2 = 49.
(3)x35x39x19x26(?1;2;5)x18x45x42x46x47x48,x28x29x2ex7x41xcx1fx20x23(?a;a;a),x1bx35x33x54x55x23
(x+a)2 +(y?a)2 +(z?a)2 = a2:
￠ 1 ￠
x49(?1;2;5)xcx1fx20x4ax4b,x3da2?8a+15 = 0,x25x3da = 5x6f3,x74x35x33x54x55x23
(x+5)2 +(y?5)2 +(z?5)2 = 25 x47x2d (x+3)2 +(y?3)2 +(z?3)2 = 9:
(4)x40x3ax23x4cx3cx1fx20x3bx33xOyxb,x23x39x8x37x26,x28x29x35x39x42x6dx18zx4dxb,x7x35x39x1fx20x23(0;0;t),x1b
4+t2 = 2+2+(2?t)2,x25x3dt = 1,x46x47x35x33x54x55x23x2 +y2 +(z?1)2 = 5.
2,x58x33x34x23xcx23x39x2dx3fx36:
(1)
( x2 +y2 +z2 = 4;
x+y +z?3 = 0;
(2)
( x2 +y2 +z2 = 5;
x2 +y2 +z2 +x+2y +3z?7 = 0:
x4,(1) x46x4excx23x2ex47x4fx59x8x7cx42x43x54x55x46x50x6dxcx35x33x19x7cx2ex43x54x55x50x6dxcx3bx33xcx25x1d,x2dx35x39
x8x37x26,x46x47x23x39x42x18x37x26x51x52x43x3bx33x46x53xcx54x1dxcx54x55xb,x29x54x1dxcx54x51x51x56x8(1;1;1),x53x54x1dx54
x55x23
8>
<
>:
x = t;
y = t;
z = t:
x54x1dx19x3bx33xcx25x26x8(1;1;1),x29x74x35x39,x57x15x57x58x6dx6e,x35xcx3fx36x23p4?3 = 1.
(2)x7cx2ex43x54x55x59x5ax7cx42x43x54x55x67x2ex3dx+2y+3z?2 = 0,x5bx4bx19(1)x49x5cxcx54x19,x2ex3ax23x39x34x8
x29x54x55x46x50x6dxcx3bx33x19x1x1d
8>
<
>:
x = t;
y = 2t;
z = 3t
xcx25x26,x25x3dx23x39x1fx20x23
1
7 ;
2
7 ;
3
7
·
,x3fx36
q
5? 27 =
p231
7,
3,x58x13,8
><
>:
x = acos2 t;
y = asin2 t;
z = ap2sintcost;
0 6 t < … (a > 0)
x50x51x42x23,x58x29x23xcx23x39x38x3fx36.
x4,x29x1cx1dxbxcx7x8x42x26(x;y;z)x5dx55x54x55x2 +y2 +z2 = a2x47x2dx+y?a = 0,x53x1cx1dx8x35x33x19
x3bx33xcx25x1d(x23),(
x2 +y2 +z2 = a2;
x+y?a = 0;
x6fx1fx42x5ex21,x23x13x29x1cx1dx50x8x23,x7(x;y;z)x8x23xbx7x8x42x26,x3cx8y = a?x,x2 + (a?x)2 + z2 = a2.
x67x12x2ex5fx232
x? a2
·2
+z2 = a22, x28x29x17x180 6 < 2…x1ax3d
8<
:
x = a2 + a2 cos = acos2 2 ;
z =
p2a
2 sin =
p2asin
2 cos
2 ;
x66x2dy = a?x = asin2 2, x1et = 2,x34x60x3dx0x61x7xcx62x9x54x55,x63x14x5dx55x23x54x55xcx26x1dx8x61x7x1cx1dxb
xcx26,x28x29x40x3ax1cx1dx50x8x23.
x1fx23x39x42x8x1x1d
8>
<
>:
x = t;
y = t;
z = 0
x19x3bx33x+y?a = 0xcx25x26,x74
a
2 ;
a
2 ;0
·
,x3fx36x1bx23
r
a2? a24? a24 =
p2
2 a.
4,x58x13,x40x43x35x33
Si, x2 +y2 +z2 +Aix+Biy +Ciz +Di = 0; (i = 1;2);
￠ 2 ￠
x25x1dx23x46x18x3bx33x23
(A1?A2)x+(B1?B2)y +(C1?C2)z +(D1?D2) = 0:
x2x3,x7x52x40x43x35x33xcx25x1dx23xbxc3x43x2cx45x26Mj(aj;bj;cj) (j = 1;2;3),x1b(
a2j +b2j +c2j +A1aj +B1bj +C1cj +D1 = 0;
a2j +b2j +c2j +A2aj +B2bj +C2cj +D2 = 0; (j = 1;2;3):
x49x40x12x18x59x3d
(A1?A2)aj +(B1?B2)bj +(C1?C2)cj +(D1?D2) = 0; (j = 1;2;3):
x52x63x14x23xbxc3x43x26M1;M2;M3x1dx18x3bx33
(A1?A2)x+(B1?B2)y +(C1?C2)z +(D1?D2) = 0
xb,x28x29x64x43x23x1cx18x29x3bx33xb.
5,x40x3ax40x35x33:
Si, x2 +y2 +z2 +2Uix+2Viy +2Wiz +di = 0; (i = 1;2)
x58x13x52x40x35x33x29x25(x18x25x26x65xcx3cx3bx33x54x1)xcx5ex5fx8:
2(U1U2 +V1V2 +W1W2) = d1 +d2:
x2x3,x7x26M(x0;y0;z0)x23x40x43x35x33xcx7x8x42x43x25x26,x1bx3aMx26xcx3cx3bx33xcx19x51x56x34x8x3ax52x43x26
xcx35x3fx36,x4bx66x54x19x2cx67x4fx30x52x40x43x35x33xcx35x39x8(?Ui;?Vi;?Wi) (i = 1;2),x28x29x35x3fx36xcx54x51x51x56
x8(x0 +Ui;y0 +Vi;z0 +Wi) (i = 1;2),x52x40x43x35x33x29x25x4dx68x3cx52x40x43x35x3fx36x29x25,x74
(x0 +U1)(x0 +U2)+(y0 +V1)(y0 +V2)+(z0 +W1)(z0 +W2) = 0;
x5x6x3d
x20 +y20 +z20 +(U1 +U2)x0 +(V1 +V2)y0 +(W1 +W2)z0 +U1U2 +V1V2 +W1W2 = 0:
x3bx3cMx18x35x33xcx25x1dxb,x28x29x41xcx1fx20x45x2bx5dx55x40x43x35x33xcx54x55,x49x29x40x43x54x55x18x7fx67x14x472x2ex3d
x20 +y20 +z20 +(U1 +U2)x0 +(V1 +V2)y0 +(W1 +W2)z0 + d1 +d22 = 0:
x4ax4bx2x33x4dx12x67x74x3d
2(U1U2 +V1V2 +W1W2) = d1 +d2:
x52x43x5ex5fx8x5bx21x24x5cx5dxc.
6,x58x13x40x23
S1,
( x2 +y2 = 4;
z = 0 x19S2,
( x2 +y2 +z2?2x?2y?2 = 0;
x+y +z = 1
x18x45x42x35x33xb.
x2x3,x69x3aS1xcx35x33xcx35x39x42x6dx18zx4dxb(x62x6ax6bx611(4)),x28x29x1fx1fx20x23(0;0;a),x3x58S2xcx23
x39,x41x8x1x1d
8>
<
>:
x = 1+t;
y = 1+t;
z = t
x19x3bx33x + y + z = 1xcx25x26
2
3 ;
2
3 ;?
1
3
·
,x2dx35x39x19x23x39xcx6cx1dx42x6dx19x3b
x33x + y + z = 1xcx19x51x56x3bx35,x74 23, 23,
13?a
·
= 1, 1, 1,x46x47a =?1,x74x35x39x23(0;0;?1),x45
x28(2;0;0)x18S1xb,x66x2dx18x35x33xb,x58x3dx35x3fx36x4dx3cp22 +12 = p5,x28x29S1x18x35x33
x2 +y2 +(z +1)2 = 5
￠ 3 ￠
xb,x29x54x55x59x5ax54x55x+y +z = 1xc2x6ex67x3dx0
x2 +y2 +z2?2x?2y?2 = 0;
x63x14x29x35x33x19x3bx33x+y +z = 1xcx25x1dx34x8S2,x28x29S1x19S2x18x45x42x43x35x33xb.
7,x13x14,x3ax23:
( S = x2 +y2 +z2 +2ux+2vy +2wz +d = 0; (u2 +v2 +w2?d > 0);
E = Ax+By +Cz +D = 0; (A2 +B2 +C2 6= 0)
xcx35x33x6fx54x55x2ex50x51x23,S +2?E = 0 (?x23x62x9).
x2x3,x4ex3cx62x9?,x63x54x55S +2?E = 0x50x70x50x51x42x43x35x33,x1bx23
( S = 0;
E = 0 x42x6dx18x29x35x33xb.
x4ex7x8x42x43x3ax23
( S = 0;
E = 0 xcx35x33
x2 +y2 +z2 +2px+2qy +2tz +r = 0;
x40x3ax23x42x18x40x43x35xcx25x1d (
x2 +y2 +z2 +2ux+2vy +2wz +d = 0;
x2 +y2 +z2 +2px+2qy +2tz +r = 0
xb,x3bx7c4x61x3a,x29x23x42x18x3bx33
(p?u)x+(q?v)y +(t?w)z + r?d2 = 0
xb,x71E = 0x1cx3ax29x23,x28x29x40x43x3bx33x21x72,x74x17x18x70x9?x1ax3d
p?u =?A; q?v =?B; t?w =?C; r?d2 =?D:
x25x30p;q;t;r,x4ax4bx35x33x54x55x74x3d
x2 +y2 +z2 +2ux+2vy +2wz +d+2?(Ax+By +Cz +D) = 0:
x0 x1 6–3
1,x58x33x34x73x17x1cx33xcx54x55:
(1)x1x1d x2 = y1 = z?10 x74x1x1dx = y = zx73x17;
(2)x1x1d x?11 = y?3 = z3 x74zx4dx73x17;
(3)x75x76x1d
( y2 = 2px;
z = 0 x74x41xcx77x1dx73x17;
(4)x1cx1d
( x2 = y;
x+z = 0 x74x1x1d
x
1 =
y
2 =
z
1 x73x17.
x4,(1)x78x79x37x26Ox18x73x17x4dxb,x24x4dxcx54x51x51x56x8? = (1;1;1),x62x78(3.1),x2ex47x3dx0x54x55x22
8>
<
>:
(x?x0)+(y?y0)+(z?z0) = 0;
x2 +y2 +z2 = x02 +y02 +z02;
x0
2 =
y0
1 =
z0?1
0 ;
x18x54x55x22x20x79x5ax62x9x0;y0;z0x67x2ex3d
x2 +y2 +z2?1 = 59 (x+y +z?1)2;
x28x29x46x58x73x17x1cx33xcx54x55x23
2(x2 +y2 +z2)?5(xy +xz +yz)+5(x+y +z)?7 = 0:
￠ 4 ￠
(2)x78x79x37x26Ox18x73x17x4dxb,x24x4dxcx54x51x51x56x8? = (0;0;1),x45xbx61,x2ex47x3dx0x54x55x22
8>
<
>:
z?z0 = 0;
x2 +y2 +z2 = x02 +y02 +z02;
x0?1
1 =
y0
3 =
z0
3 ;
x28x29z0 = z,y0 =?z,x0 = 1+ z3,x4ax4bx54x55x22x79x5ax62x9x0;y0;z0x67x2ex3d
x2 +y2 +z2 =
1+ z3
·2
+z2 +z2;
x28x29x46x58x73x17x1cx33xcx54x55x23
9x2 +9y2?10z2?6z?9 = 0:
(3)x75x76x1dxcx77x1dxcx42x7ax54x55x23
( x =?p
2 ;
z = 0; x1bx1fx20x77x54x55x23
x+ p2
0 =
y
1 =
z
0,
x52x4dxbxcx42x26(x0;y0;z0) =
p2 ;0;0
·
,x34x2ex7bx30x47x33x54x55x22
8>
>>>>
<
>>>>
>:
y?y0 = 0;?
x+ p2
·2
+y2 +z2 =
x0 + p2
·2
+y02 +z02;
y02 = 2px0;
z0 = 0:
x66x54x55x22x79x5ax62x9x0;y0;z0,x34x60x3dx0x73x17x1cx33xcx54x55
y4?4p2x2 +2p2y2?4p2z2?4p3x = 0:
(4)x78x79x37x26Ox18x73x17x4dxb,x28x29x2ex3dx54x55x22:8
>>>
><
>>>
>:
(x?x0)+2(y?y0)+(z?z0) = 0;
x2 +y2 +z2 = x02 +y02 +z02;
x02 = y0;
x0 +z0 = 0:
x66x54x55x22x79x5ax62x9x0;y0;z0,x34x60x3dx0x73x17x1cx33xcx54x55
3x2 +3z2?4xy?2xz?4yz?4x?8y?4z = 0:
2,x57x15k;lxcx2cx45x52x1b(x7cx6fx7dx7c)x7ex10x1x1d
L, x1 = yk = z?l0
x74xx4dx73x17x46x59x1cx33Sx8x1ax7fx1cx33.
x4,x21x47x33x0x7fx1x2x7ex10.
(i) k = l = 0x2b,Lxcx54x55x59x23 x1 = y0 = z0,Lx34x8xx4d,x28x29x74xx4dx73x17x3x79x8xx4dx6cx4;
(ii) k = 0,l 6= 0x2b,Lxcx54x55x23
( z = l;
y = 0; Lx8x1fx20x3bx33xOzxbxcx1cx1d,x57x15x53.1xcx7ex10,xOzx1f
x20x3bx33xbxcx1cx1dx74xx4dx73x17x3dx0xcx73x17x1cx33xcx54x55x2ex47x4bpy2 +z2x4ax6x54x55x20xczx2dx3dx0,x28x29x73x17
x1cx1dxcx54x55x23y2 +z2 = l2,x8x42x43x23x7x33;
(iii) k 6= 0,l = 0x2b,Lxcx54x55x8
( y = kx;
z = 0; Lx8x1fx20x3bx33xOyxbxcx1cx1d,x45x6e,x73x17x1cx33xcx54x55x2e
x47x4bpy2 +z2x4ax6x54x55x20xcyx2dx3dx0,x74x23y2 +z2 = k2x2,x52x8x23x24x33;
￠ 5 ￠
(iv) k 6= 0,l 6= 0x2b,x28x37x26x18x73x17x4dxb,x2ex3dx47x33x54x55x22
8>
<
>:
x?x0 = 0;
x2 +y2 +z2 = x02 +y02 +z02;
x0
1 =
y0
k =
z0?l
0 ;
x79x5ax62x9x67x3dx0x1cx33x54x55
y2 +z2
l2?
k2x2
l2 = 1;
x52x8x8x9x27x1cx33.
3,x13x14,x0x6dx1x1dx2dx6dx1x1dxbx42x6dx26xcx43x44x3bx54x38x8x44x9xcxax26xbxcx8x42x73x17x1cx33.
x2x3,x7x6dx1x1dx23zx4d,x6dx26x23x37x26O,x7P(x;y;z)x8x5dx55x5ex5fxcx26,x1bPxcx1fx20x5dx55x47x33x54x55:
(x2 +y2)+(x2 +y2 +z2) = k2;
x78x79x52x8x1cx1d (
2x2 +z2?k2 = 0;
y = 0
x74zx4dx73x17x2dx3dxcx73x17x1cx33x54x55.
4,x58x13,yz +zx+xy = a2x8x73x17x1cx33,x24x58x73x17x4d.
x2x3,x28x23
(x2 +y2 +z2)+2(yz +zx+xy)?(x2 +y2 +z2) = 2a2;
x2ex3d
(x+y +z)2?(x2 +y2 +z2) = 2a2:
x4ex7x8x70x9p (jpj > p2jaj),x1cx1d (
x2 +y2 +z2 = p2?2a2;
x+y +z?p = 0
x8x42x43x23,x23x39x18x1x1dx = y = zxb,x28x29x52x8x42x43x73x17x1cx33,x73x17x4dx8x = y = z.
x1cx2ex47x1ax1cx1d (
yz +zx+xy = a2;
2x?y?z = 0
x74x1x1dx = y = zx73x17x2dx3dx0x1cx33yz +zx+xy = a2.
5,x58x13,8
><
>:
x = a(cosu+cosv);
y = a(sinu+sinv);
z = b(u?v)
x8x73x17x1cx33,x52xda;b 6= 0x24a;bx8x44x9.
x2x3,x28x23
x2 +y2 = a2 +2a2(cosucosv +sinusinv) = a2 +2a2 cos(u?v)
= a2 +2a2 cos zb ;
x78x79x41x8x1cx1d (
x2 = a2 +2a2 cos zb ;
y = 0
x74zx4dx73x17x2dx3d.
￠ 6 ￠
6,x58x1cx1d
( x = z2;
x2 +y2 = 1 x74x1x1d
8>
<
>:
x = 2t;
y = 0;
z = 3t
x73x17xex59xcx73x17x1cx33xcx54x55.
x4,x73x17x4dx69x3ax37x26O,x28x29x2ex3dx54x55x22
8>
>>>
<
>>>
>:
2(x?x0)+3(z?z0) = 0;
x2 +y2 +z2 = x02 +y02 +z02;
x0 = z02;
x02 +y02 = 1:
x79x5ax62x9x67x2ex3dx73x17x1cx33x54x55
2x+3z +2?2x2?2y2?2z2 = §3
p
x2 +y2 +z2?1:
7,x13x14x1cx33F(x;y;z) = (x2 +y2 +z2)2?16(x2 +z2) = 0x8x42x43x73x17x1cx33.
x2x3,x52x8x1cx1d (
(x2 +y2)2?16x2 = 0;
z = 0
x74yx4dx73x17x2dx3dx0xcx1cx33.
x0 x1 6–4
1,x40x3ax7x33x77x1dx54x55x23 (
x2 +y2 +z2 = 4;
x2 +(y?3)2 +z2 = 4;
xfx1dx54x51x231, 1, (?1),x28x58x1fx54x55.
x4,x7x52x26M(x0;y0;z0)x18x77x1dxb,P(x;y;z)x23x7x33x3aMxcxfx1dxbxcx26,x1bx62
8
>>>>
>>><
>>>
>>>>
:
x = x0 +u;
y = y0 +u;
z = z0?u;
x02 +y02 +z02 = 4;
x02 +(y0?3)2 +z02 = 4;
x25x3d 8
><
>:
x0 = x?y +y0;
z0 = z +y?y0;
y0 = 32 ;
x2fx30x7x33x54x55
(2x?2y +3)2 +(2z +2y?3)2 = 7:
2,x40x3ax7x33x77x1dx54x55x23 (
y = x2 +z2;
y = 2z;
xfx1dx54x1x3cx77x1dx46x18x3bx33,x28x58x29x7x33x54x55.
x4,x28x23xfx1dx54x1x3cx77x1dx46x18x3bx33y?2z = 0,x46x47xfx1dx54x51x23(0;1;?2),x19xbx61x49x5c,x2ex3dx54x55
￠ 7 ￠
x22 8
>>>>
>>><
>>>
>>>>
:
x = x0;
y = y0 +u;
z = z0?2u;
y0 = x02 +z02;
y0 = 2z0;
x25x3d 8
>><
>>:
x0 = x;
y0 = 25 (2y +z);
z0 = 15 (2y +z);
x2fx30x7x33x54x55
2
5 (2y +z) = x
2 + 1
25 (2y +z)
2;
x5x6x67x3d
25x2 +4y2 +z2 +4yz?20y?10z = 0:
3,x58x1cx1d (
x2 +2y2 +z2 = 1;
x2 +z2 = y
x4exOyx3bx33xcx10x2cx7x33x54x55.
x4,xfx1dxcx54x51x51x56x8(0;0;1),x2ex3dx54x55x228
>>>>
>>><
>>>
>>>>
:
x = x0;
y = y0;
z = z0 +u;
x02 +2y02 +z02 = 1;
x02 +z02 = y0;
x25x3d 8
><
>:
x0 = x;
y0 = y;
2y02 +y0?1 = 0;
x3bx3cy =?1x2cx72x61x8,x28x29x7x33x54x55x23
y = 12 ;
p2
2 6 x 6
p2
2
!
:
x11x25,x49x1cx1dx54x55x22xcx7cx42x43x54x55x59x5ax7cx2ex43x54x55,x2ex3d2y2?y?1 = 0,x52x8xfx1dx19zx4dx3bx35xc
x7x33,x2dx24x69x3ax40x3ax1cx1d,x74x23x46x58xcx10x2cx7x33.
4,x28x63x14x33x34x54x55x46x50x51xcx1cx33x8x7x33:
(1) (x+y)(y +z) = a2; (2) (x+y)(y +z) = x+2y +z;
(3) y2 +2yz +z2 = 1?x2; (4) (x+y +z)2 = (x?y?z)2.
x4,(1) x28x23x1x1d
( x+y = a;
y +z = a x18x29x1cx33xb,x41xcx54x51x231, (?1), 1,x24x26
a
2 ;
a
2 ;
a
2
·
x18x29x1x1d
xb,x2dx3bx33x?y +z? a2 = 0x19x1cx33(x+y)(y +z) = a2xcx25x1dx23
( (x+y)(y +z) = a2;
x?y +z = a2,
￠ 8 ￠
xdxex47x52x5ex1cx1dx23x77x1d,x471, (?1), 1x23xfx1dx54x51,x2ex58x3dx54x55x22
8
>>>>
>>><
>>>>
>>>
:
x = x0 +u;
y = y0?u;
z = z0 +u;
(x0 +y0)(y0 +z0) = a2;
x0?y0 +z0 = a2 ;
x79x5ax62x9x67,x3dx0x7x33x54x55(x+y)(y +z) = a2,x46x47x37x12xcx1cx33x8x7x33.
(x13,x47x33x0x43x14x61xdxex49x15x50x6dxfx1dx54x51,x79x67x13x14x69x3ax1cx33xbx7x8x42x26xcx19xfx1dx54x51x3bx35xcx1
x1dx1dx18x1cx33xb,x4bx52x16xcx54x19x12x13x14x1cx33x8x7x33.)
(2) x28x23(x + y)(y + z) = (x + y) + (y + z),x46x47x1x1d
( x+y = 0;
y +z = 0 x18x29x1cx33xb,x1fx54x51x51x56x8
(1;?1;1).
x7M(x0;y0;z0)x8x1cx33xbxcx7x8x26,P(x;y;z)x8x3aMxcx1x1d
x?x0
1 =
y?y0
1 =
z?z0
1
xbxcx42x26,xdxex5dx17x13Px18x1cx33xb,x23x29,x25x3d(
x+y = x0 +y0;
y +z = y0 +z0;
x28x29
(x+y)(y +z) = (x0 +y0)(y0 +z0) = x0 +2y0 +z0) = (x+y)+(y +z) = x+2y +z;
x74Px26xcx1fx20x5dx55x1cx33x54x55,x63x14x64x5ex1x1dx1dx18x1cx33xb,x28x29x1cx33x8x7x33.
(3)x54x55y2 +2yz +z2 = 1?x2x2ex47x5fx23x2 +(y +z)2 = 1,x46x47x1x1d
( x = 1;
y +z = 0 x18x29x1cx33xb,x41
xcx54x51x51x56x8(0;?1;1).
x7M(x0;y0;z0)x8x1cx33xbxcx7x8x26,P(x;y;z)x8x3aMxcx1x1d
x?x0
0 =
y?y0
1 =
z?z0
1
xbxcx42x26,xdxex5dx17x13Px18x1cx33xb,x23x29,x25x3d(
x = x0;
y +z = y0 +z0;
x28x29
x2 +(y +z)2 = x02 +(y0 +z0)2 = 1;
x74Px26xcx1fx20x5dx55x1cx33x54x55,x63x14x64x5ex1x1dx1dx18x1cx33xb,x28x29x1cx33x8x7x33.
(4)x78x79x1x1d
( x+y +z = 0;
x?y?z = 0 x18x29x1cx33xb,x41xcx54x51x51x56x8(0;1;?1).
x7M(x0;y0;z0)x8x1cx33xbxcx7x8x26,P(x;y;z)x8x3aMxcx1x1d
x?x0
0 =
y?y0
1 =
z?z0
1
xbxcx42x26,xdxex5dx17x13Px18x1cx33xb,x23x29,x25x3d(
x = x0;
y +z = y0 +z0;
￠ 9 ￠
x28x29
(x+y +z)2?(x?y?z)2 = (x0 +y0 +z0)2?(x0?y0?z0)2 = 0;
x74Px26xcx1fx20x5dx55x1cx33x54x55,x63x14x64x5ex1x1dx1dx18x1cx33xb,x28x29x1cx33x8x7x33.
5,x40x3ax23x7x33xcx4dx54x55x23:
x
1 =
y?1
2 =
z +1
2 ;
x26(1;?2;1)x18x29x23x7x33xb,x58x29x23x7x33xcx54x55.
x4,x28x23x26(1;?2;1)x0x4d x1 = y?1?2 = z +1?2 xcx43x44x74x23x18x23xcx3fx36,x46x47x29x3fx36x23
r = j(1;?3;2)￡(1;?2;?2)jp1+4+4 =
p117
3,
x7x52x29x23x7x33xbxcx42x26P(x;y;z),Px0x4dxcx43x44x1cx23r,x28x29x62
j(x;y?1;z +1)￡(1;?2;?2)j
3 =
p117
3 ;
x64x6ex67x2ex3dx23x7x33x54x55
8x2 +5y2 +5z2 +4xy +4xz?8yz?18y +18z?99 = 0:
(x13,x29x61x1cx2ex4bx3a(1;?2;1)x26xcx1xfx1dx74x4dx73x17x2dx3dx0x29x1cx33x54x55,x1cx2ex47x4bx58x30x42x43x18x23x53x77
x1dx12x58x30x29x7x33x54x55.)
6,x7x7x33xcx77x1dx23
( x = y2 +z2;
x = 2z; xfx1dx54x1x3cx77x1dx46x18xcx3bx33,x58x52x7x33xcx54x55.
x4,x28x23x77x1dx18x3bx33x = 2zxb,x46x47xfx1dxcx54x51x51x56x8(1;0;?2),x3bx29x2ex3dx54x55x228
>>>>
>>><
>>>
>>>>
:
x = x0 +u;
y = y0;
z = z0?2u;
x0 = y02 +z02;
x0 = 2z0;
x79x5ax62x9x67x2ex3dx7x33x54x55:
4x2 +25y2 +z2 +4xz?20x?10z = 0:
7,x58x3fx36x234,x4dx1dx54x55x8x = 2y =?zxcx23x7x33x54x55.x19x17x13x41x1axOyx1fx20x3bx33x46x1bx3dxcx1cx1c
x1dx1dx59xcx33x1ex4dx3c24….
x4,x46x58x23x7x33x34x8x0x4dx1dxcx43x44x4dx3c4xcx26xcxbxc,x28x29x23x7x33xbxcx26P(x;y;z)x5dx55x47x33x54x55fl
flfl(x;y;z)￡?1; 1
2 ;?1
·flfl
fl
q
2+ 14
= 4;
x5fx1fx67x3d
5x2 +8y2 +5z2 +8xz +4yz?4xy = 144:
x29x23x7x33x1axOyx3bx33x46x1bx3dxcx1bx20x8x42x43x21x23,x24x29x21x23xcx22x3fx4dx30x234,x2dxOyx3bx33x19x4dx1dxc
x23x1e x5dx55:
sin = j(0;0;1)￠(2;1;?2)jp22 +12 +22 = 23 ;
x46x47x30x3fx4dxcx30x23
4
sin = 6:
￠ 10 ￠
x29x21x23xcx33x1ex4dx3c
… ￠4￠6 = 24…:
8,x58x19xx4dx2dx3bx33y = kx4dx43x44xcx26xcxbxcx54x55.
x4,x7xax26x23P(x;y;z),x1bx3bx5ex5fx3a
y2 +z2 = (y?k)2;
x74x23
z2 +2ky?k2 = 0:
x0 x1 6–5
1,x58x24x33x54x55:
(1)x77x1d:
( ax2 +by2 = 1;
z = 0; x24x26(x0;y0;z0);
(2)x77x1d:
( f(x;y) = 0;
z = k (6= 0); x24x26(0;0;0);
(3)x77x1d:
( x2 +y2 +(z?5)2 = 9;
y = 2; x24x26(0;0;0).
x4,(1)x4ex3cx77x1dxbxcx26M(x0;y0;z0),x7P(x;y;z)x8x3aMxcx1xfx1dxbxcx26,x1bx62x47x33x54x55x228
>>>
>>>>
<
>>>>
>>>:
x0 = x0 +(x?x0)u;
y0 = y0 +(y?y0)u;
z0 = z0 +(z?z0)u;
ax02 +by02 = 1;
z0 = 0;
x79x5ax62x9x0;y0;z0;ux67x2ex3dx24x33x54x55:
a(z0x?x0z)2 +b(z0y?y0z)2?(z?z0)2 = 0:
(2)x49x5cx25x62x47x33x54x55x22 8
>>>
>>>>
<
>>>>
>>>:
x0 = xu;
y0 = yu;
z0 = zu;
f(x0;y0) = 0;
z0 = k;
x79x5ax62x9x67x2ex3dx24x33x54x55:
f
k xz ;k yz
·
= 0:
(3)x62x47x33x54x55x22 8
>>>>
>>><
>>>>
>>>
:
x0 = xu;
y0 = yu;
z0 = zu;
x02 +y02 +(z0?5)2 = 0;
y0 = 2;
x79x5ax62x9x67x2ex3dx24x33x54x55:
x2 +5y2 +z2?5yz = 0:
￠ 11 ￠
2,x58x47x26P(5;0;0)x23x24x26,x47x1cx1d
( x2 +2y2 = 1;
x+2y?z = 0 x23x77x1dxcx24x33x54x55.
x4,x62x47x33x54x55x22 8
>>>>
>>><
>>>
>>>>
:
x0 = 5+(x?5)u;
y0 = yu;
z0 = zu;
x02 +2y02 = 1;
x0 +2y0?z0 = 0;
x79x5ax62x9x67x2ex3dx24x33x54x55:
x2?146y2?24z2 +4xy?2xz +96yz?10x?20y +10z +25 = 0:
3,x58x47x37x26x23x24x26,x47
( x2?2z +1 = 0;
y?z +1 = 0 x23x77x1dxcx24x33x54x55.
x4,x62x47x33x54x55x22 8
>>>
>>>>
<
>>>>
>>>:
x0 = xu;
y0 = yu;
z0 = zu;
x02?2z0 +1 = 0;
y0?z0 +1 = 0;
x79x5ax62x9x67x2ex3dx24x33x54x55:
x2 +y2?z2 = 0:
4,x40x3ax23x24x33xcx24x26x23(1;2;3),x4dx54x1x3cx3bx332x+2y?z +1 = 0,xfx1dx19x4dxcx23x1ex2330–,x58x6d
x23x24x33xcx54x55.
x4,x7P(x;y;z)x8x24x33xbxcx42x43x26,x26x27x3aPx26xcxfx1dxcx54x51x51x56x8
= (x?1;y?2;z?3):
x2dx23x24xcx4dxcx54x51x51x56x34x8x3bx33xcx19x51x56
” = (2;2;?1):
x57x15x61x8x62
(?;”)
j?jj”j = §cos30
–;
x74
2(x?1)+2(y?2)?(z?3)p
(x?1)2 +(y?2)2 +(z?3)2p4+4+1 = §
p3
2,
x5fx1fx64x6ex67x74x3dx46x58x23x24x33xcx54x55x23
11(x?1)2 +11(y?2)2 +23(z?3)2?32(x?1)(y?2)+16(x?1)(z?3)+16(y?2)(z?3) = 0;
x6f
11x2 +11y2 +23z2?32xy +16xz +16yz?6x?60y?186z +342 = 0:
5,x3axx4dx38yx4dx21x22x53xax3bx33,x1ax41xex28x29x6dx25x1efi,x28x58x41xexcx25x1dx2axexcx1cx33x54x55,x19x2bx30x8
x2cx27x1cx33.
x4,x7x3axx4dxcxax3bx33x23A1y +B1z = 0,x3ayx4dxcxax3bx33x23A2x+B2z = 0,x41xex28x29x6dx1efi,x1b
B1B2p
A21 +B21pA22 +B22 = cosfi:
￠ 12 ￠
x41xexcx25x1dx3ax37x26O,x2dx24x25x1dxcx54x51x51x56
= (0;A1;B1)￡(A2;0;B2) = (A1B2;A2B1;?A1A2);
x28x29x25x1dxbxcx26P(x;y;z)x5dx55 8
><
>:
x = tA1B2;
y = tA2B1;
z =?tA1A2;
x25x3d 8
>>>>
<
>>>>
:
A21 +B21 = 1t2A2
2
y2 +z2;
A22 +B22 = 1t2A2
1
x2 +z2;
(B1B2)2 = x2y2t2z2,
x3bcos2 fi(A21 +B21)(A22 +B22) = (B1B2)2x2ex2fx30x1cx33x54x55cos2 fi(x2y2 +x2z2 +y2z2 +z4) = x2y2,x8x42
x43x24x33.
6,x58x24x26x23(5;0;0)x24x19x35x33x2 +y2 +z2 = 9x18x3cxcx23x24x33x54x55.
x4,x3bx18x3cxcx76x77x2ex3ax29x23x24x33xcx3fx24x1e x5dx55cos = 45,xx4dx8x23x24x33xcx4d,x28x29x4dxcx54x51x51
x56x8(1;0;0),x46x47x29x23x24x33xcx54x55x23
j(x?5;y;z)￠(1;0;0)jp
(x?5)2 +y2 +z2 =
4
5 ;
x5fx1fx23
9(x?5)2?16(y2 +z2) = 0:
7,x13x14,x3ax37x26x24x3cx3cx35x33
x2 +y2 +z2 +2ax+2by +2cz +d = 0; (0 < d < a2 +b2 +c2)
xcx1x1dx46xex59xcx23x24x33x54x55x23
d(x2 +y2 +z2) = (ax+by +cz)2:
x2x3,x40x3ax35x33xcx35x39x23(?a;?b;?c),x46x47x23x24x33xcx4dxcx54x51x51x56x8(a;b;c),x66x35x39x0x37x26xcx43
x44x8pa2 +b2 +c2,x35x3fx36x4dx3cpa2 +b2 +c2?d,x28x29x3ax37x26xcx3cx1dx2dx30x4dx3cpd,x3bx29x2ex3dcos =r
d
a2 +b2 +c2,x1fx20 x8x23x24x33xcx3fx24x1e,x46x47x46x58x23x24x33xcx54x55x23
jax+by +czjp
x2 +y2 +z2pa2 +b2 +c2 =
r d
a2 +b2 +c2 ;
x5fx1fx67x3d
d(x2 +y2 +z2) = (ax+by +cz)2:
8,x13x14ayz +bzx+cxy = 0x50x51x24x33,x63x3bx33x+y +z = 0x19x6dx24x33x25x3cx42x4ex1x1d,x7x1fx25x1ex23
,x24a+b+c = 0,x1b = …2,
x2x3,x28x23x1cx33ayz + bzx + cxy = 0x8x2ex3cx;y;zxcx2exfxexfx54x55,x46x47x8x47x37x26Ox23x24x26xc
x24x33,x2dx3bx33x + y + z = 0x1cx3ax37x26,x46x47x40x25x1dx2fx30x18x1cx33ayz + b(?y? z)z + c(?y? z)y = 0x38
a(?x?z)z +bxz +c(?x?z)x = 0xb,x24x3ax37x26.
x3bx7cx42x43x54x55x5fx1fx3d
cy2?(a?b?c)yz +bz2 = 0 (*)
￠ 13 ￠
x3bx7cx2ex43x54x55x5fx1fx3d
cx2?(b?a?c)xz +az2 = 0 (**)
x21x22x18x40x18x25x1x1dxbx68x52x42x26(x0;y0;z0)x38(x00;y00;z00),x1bx29x40x26xcx1fx20x2ax79x1cx5dx55(*)x38(**),x66x41xe
x5dx55(*)x2ex3d:
cy02?(a?b?c)y0z0 +bz02 = 0 x19 cy002?(a?b?c)y00z00 +bz002 = 0
x5bx4bx57x19x69x9x2ex69,x2ex3dy0y00 = bc z0z00,x45x6e,x3bx3cx41xex1cx5dx55(**),x2ex3dx0x00 = ac z0z00.
x28x29x40x1x1dxcx23x1e x5dx55
cos = x
0x00 +y0y00 +z0z00
px02 +y02 +z02px002 +y002 +z002 = a+b+ccpx02 +y02 +z02px002 +y002 +z002 = 0;
x74 = …2,
9,x13x14,px+py?pz = 0x50x51x42x43x3fx24x1e x23arccos
p6
3 xcx23x24x33.
x2x3,x37x54x55xcx31x56x52x1bx32x1dx42x5dx55x > 0,y > 0,z > 0,x37x54x55x332xfx3bx54x67x59x23
2(x2 +y2 +z2)?(x+y +z)2 = 0;
x78x79x8x43x24x33,x2dx24x41x19x37x54x55x18x > 0,y > 0,z > 0x52x42x34xax48x8x45x25xc,x28x29x37x1cx33x8x24x33(x2ax79
x47x37x26x23x24x26,x24x35x62x42x43x54x51).
x64x36x3ax37x26xc2x5ex1x1d:
L, x1 = y1 = z1 ; L0, xm = yn = zs ;
x1x1dL0x74x1x1dLx73x17x3dx0x23x24x33xcx54x55x238
>>>>
>>><
>>>
>>>>
:
x0 = mt;
y0 = nt;
z0 = st;
(x?x0)+(y?y0)+(z?z0) = 0;
x2 +y2 +z2 = x02 +y02 +z02;
x79x5ax62x9x0;y0;z0;tx67x3dx0
x2 +y2 +z2 = m
2 +n2 +s2
(m+n+s)2 (x+y +z)
2:
x2a m2 +n2 +s2(m+n+s)2 = 12 x2b,x52x43x23x24x33xcx54x55x19x40x3ax24x33xcx54x55x45x25,x2dx23x24x33xcx3fx24x1e x34x8L0x19L
xcx23x1e,x74
cos = (1;1;1)￠(m;n;s)pm2 +n2 +s2p3 =
p2
p3 =
p6
3,
x0 x1 6–6
1,x40x3ax21x35x33xcx4ex4fx4dx19x1fx20x4dx21x72,x24x69x3ax21x23
( x2
4 +
y2
9 = 1
z = 0
x38x26A(1;2;?p11),x58x21x35x33x54x55.
x4,x78x79x29x21x35x33xcx54x55x23
x2
4 +
y2
9 +
z2
c2 = 1;
￠ 14 ￠
x47x26Axcx1fx20x4ax4b,x25x3dc = 6,x46x47x21x35x33xcx54x55x23
x2
4 +
y2
9 +
z2
36 = 1:
2,x40x3ax24x26x23x37x26,x4ex4fx33x23xOyx33x38zOxx33,x24x3ax26A
1
2 ;?1;2
·
x38B
5
2 ;3;?2
·
,x58x21x23x75
x76x33xcx54x55.
x4,x29x21x23x75x76x33xcx54x55x42x23
y2
b2 +
z2
c2 = 2x;
x3x47A;Bx26xcx1fx20x4ax4b,x25x3db = p2,c = 2p2,x28x29x46x58x21x23x75x76x33xcx54x55x23
y2
2 +
z2
8 = 2x:
3,x58x42x43x2exfx1cx33xcx54x55,x1ax52x43x2exfx1cx33x69x3ax40x5ex75x76x1d(
x2?6y = 0
z = 0 x38
( z2 +4y = 0
x = 0:
x4,x52x43x2exfx1cx33x2ex60x8x21x23x75x76x33x6fx27x1cx75x76x33,x57x15x40x3ax5ex5f,x41xcx54x55x37x62x47x33x2x12:
x2
a +
z2
b = 2y;
x47z = 0x4ax4b,x58x3da = 3,x47x = 0x4ax4b,x58x3db =?2,x28x29x52x8x27x1cx75x1cx33,x1fx54x55x23
x2
3?
z2
2 = 2y:
4,x2akx52x68x7fx70x9x1bx2b,x54x55
(k?3)x2 +y2 = (k +3)z
x50x51x2cx27x1cx33?
x4,x2ak <?3x6f?3 < x < 3x2b,x54x55x50x51x27x1cx75x76x33; x2ak =?3x2b,x54x55x50x51x40x43x18x25x3bx33; x2a
k = 3x2b,x54x55x50x51x75x76x7x33;x2ak > 3x2b,x54x55x50x51x21x23x75x76x33.
5,x40x3ax21x23x75x76x33Ax2 +By2 = 2zx69x3ax23(
x2 +y2 +z2 = 2x+2z;
x = z;
x28x58x1fx54x55.
x4,x23x54x55x2ex47x45x25x31x2x23 (
2x2 +y2 = 4z;
x = z;
x2dx37x1cx33x1ax3bx33x = zx1bx3dxcx1cx1dx8 (
Ax2 +By2 = 2z;
x = z:
x3ex3fx67x74x3dA = 1,B = 12,x28x29x21x23x75x76x33xcx54x55x23
x2 + y
2
2 = 2z:
6,x40x3ax21x23x75x76x33Ax2 +By2 = 2zx69x3ax23(
x2 +y2 +z2 = 2x+2z;
x = z;
x28x58x1fx54x55.
￠ 15 ￠
x4,x52x23xbx40x43x26(1;p2;1),(2;0;2),x41xex1cx18x1cx33xb,x49x41xexcx1fx20x4ax4bx1cx33x54x55,x25x3dA = 1,
B = 12, x46x47x21x23x75x76x33xcx54x55x23x2 + y22 = 2z.
7,x40x3ax21x23x75x76x33x2 + y22 = 2zx38x3bx33x = kzxcx25x1dx8x42x43x23,x28x58x29x23xcx3fx36.
x4,x29x25x1dx1cx2ex50x51x23 (?
x? 1k
·2
+ y22 = 1k2 ;
x = kz:
x52xdx7cx42x43x54x55x8x25x1dx0xOyx3bx33xcx2bx2cx7x33,x41x8x42x43x21x23x7x33,x1fx4ex4fx4dx8x1x1d
( x = 1
k ;
y = 0,x52x5e
x1x1dx19x3bx33x = kzxcx25x26
1
k ;0;
1
k2
·
x34x8x25x1dx23xcx23x39,x12x18xdxex45x2ex49x25x1dxcx54x55x45x25x31x2x23
(?
x? 1k
·2
+y2 +k2
z? 1k2
·2
= 2k2 ;
x = kz:
x66x52x43x54x55x22x2ex47x4fx30,x5dx1ax25x1dx59x23x42x43x47
1
k ;0;
1
k2
·
x23x23x39xcx23,x5cx38x24x35x38k2 = 1,x52x2bx23xcx3f
x36x4dx3cp2.
8,x28x17x13x21x23x75x76x33x19x27x1cx75x76x33xcx62x9x54x55x2ex21x22x39x23
8>
<
>:
x = a(u+v);
y = b(u?v);
z = u2 +v2
x19
8>
<
>:
x = a(u+v);
y = b(u?v);
z = 2uv;
(u;vx23x62x9).
x2x3,x21x23x75x76x33xcx54x55x8 x2a2 + y2b2 = 2z,x53x31x56x3ax6
( x = a(u+v);
y = b(u?v);
x2cx67x4fx30x52x43x31x6x8x2ex3bxc,x4ax4bx54x55x67x2ex3dz = u2 +v2,x3bx3cx1cx33xbxcx26x1ax;yxcx1bx3cx42x50x6d,x28x29
x1cx1au;vx3cx42x50x6d,x63x14x52x50x8x1cx33xcx62x9x54x55.x49x5cx25x2ex47x3dx0x27x1cx75x76x33xcx62x9x54x55.
9,x40x3ax42x75x76x1d
( x2 = 2z;
y = 0 x3bx35x3dxa,x24x24x26x18x75x76x1d
( y2 =?4z;
x = 0 xb,x28x58x1fxbxcx54x55.
x4,x28x23x75x76x1d
( y2 =?4z;
x = 0 x18yOzx3bx33xb,x7x75x76x1d
( x2 = 2z;
y = 0 x3bx35x3dxax2dx67x3ex30xcx1cx33x23
S,x7x52x1fxbx42x26(X;Y;Z),x1bx3bx33y = Y x19x1cx33Sxcx1bx20x42x19x75x76x1d
( x2 = 2z;
y = 0 x18x5c,x35x8x24x26x18
0;Y;?Y 24
x65,x28x29x26(X;Y;Z)x42x6dx5dx55x54x55
X2 = 2
Z?
Y
2
4
;
x74X2? Y 22 = 2Z,x46x47xbxcx54x55x23x2? y22 = 2z,x8x27x1cx75x76x33.
10,x13x14x21x23x75x76x33xbx3fx1x1dx17x18.
x2x3,x7x62x1x1d 8
><
>:
x = x0 +lt;
y = y0 +mt;
z = z0 +nt;
￠ 16 ￠
x41x5dx30x18x21x23x75x76x33 x2a2 + y2b2 = 2zxbx2ax24x40x2a
(x0 +lt)2
a2 +
(y0 +mt)2
b2 = 2(z0 +nt) (*)
x4ex7x8xctx59x11,x71xbx12x8x2ex3ctxcx2exfx54x55,x24t2xcx69x9x8 l2a2 + m2b2,x63x41x2cx4dx3c0,x1b(*)x12x2cx2ex60
x4ex7x8xctx59x11,x53x35x62l = m = 0,x52x2btxcx42xfx11x69x9x23n,x1cx35x60x80,x2cx2ex60.
x0 x1 6–7
1,x58x27x1cx75x76x33x2?y2 = zxbx3ax26(1;?1;0)xcx40x5ex1xfx1dx54x55x47x2dx41xexcx25x1efi.
x4,x7x1fx40x6fx1xfx1dx23 (
x+y =?;
xy = z x38
( x?y =,;
”x+”y = z;
x41xex69x3ax26(1;?1;0),x46x47? = 0,” = 2,x2fx3dx1xfx1dx54x55x23
x
1 =
y
1 =
z
0 x38
x?1
1 =
y +1
1 =
z
4 ;
x25x1ex5dx55cosfi = 0,x74fi = …2,
2,x13x14,x27x1cx75x76x33 x2a2? y2b2 = 2z (a 6= b)xbxcx41x18x29x25xcx1xfx1dxcx25x26xcxbxcx8x42x5ex27x1cx1d.
x2x3,x3bx3cx45x6fxcx7x8x40x5ex1xfx1dx42x8x43x33x1x1d,x44x62x25x26,x28x29x35x45x64x36x43x6fxcx1xfx1d,x2cxcx7(
x
a +
y
b = 2?;
x
a?
y
b
·
= z
x19
( x
a?
y
b = 2”;
”
x
a +
y
b
·
= z
x18x41x54x1,x40x1x1dxcx54x51x51x56x21x22x23
1b ; 1a ;? 2?ab
·
x38
1
b ;
1
a ;
2”
ab
·
,x28x41xex54x1,x46x474?” = a2?b2.
x2a? 6=,x2b,x46x39x25x3dx25x26x5dx558
<
:
x2
a2?
y2
b2 = 4?” = a
2?b2;
z = 2?” = a2?b22 ;
x2a? =,x2b,x25x26x5dx55 8
><
>:
x = 2a?;
z = 2?2;
y = 0:
x2cx10x47x42x7fx1x2,x1dx63x14x25x26x18x27x1cx1d8
<
:
x2
a2?
y2
b2 = a
2?b2;
z = a2?b22 ;
x19
( x2 = 2a2z;
z = 0;
xb.
3,x28x13:
2x2 +y2?z2 +3xy +xz?6z = 0
x8x1x48x33,x19x58x30x1fxbx3ax26M(1;1;1)xcx1xfx1dx54x55.
x4,x37x54x55x2ex5fx23
(x+y +z)(2x+y?z) = 6z;
x1bx1fx1xfx1dx54x55x23 (
(x+y +z) = 6z;
2x+y?z =? x6f
(?(2x+y?z) = 6z;
x+y +z =?;
￠ 17 ￠
x1fx20?;?x23x62x9,x45x28x2cx10?;?x52x1ax1b,x42x62
rank
6
2 1?1
= 2; rank
2 6
1 1 1
= 2;
x63x14x41xex50x70x50x51x40x43x1x1dx6f,x46x47x1cx33x8x1x48x33.
x64x36x3ax26(1;1;1)xcx1xfx1d,x49x1fx20x4ax4bxfx1dx54x55,x2ex3d? = 2,? = 3,x28x29x40x5ex1xfx1dx23(
x+y?2z = 0;
2x+y?z?2 = 0 x38
( 2x+y?3z = 0;
x+y +z?3 = 0:
4,x58x19x56x1x1d
L1,
( y?1 = 0;
x+2z = 0; L2,
( y?z = 0;
x?2 = 0; L3,
x
2 =
y +1
0 =
z
1
x18x25xcxax1x1dx2axexcx1cx33x54x55.
x4,x7xax1x1dx19L1;L3xcx25x26x23(?2u;1;u)x19(2v;?1;v)(x1fx20u;vx23x62x9),x46x47xax1x1dxcx54x55x23
x+2u
2(u+v) =
y?1
2 =
z?u
v?u, (*)
x41x42x19x1x1dL2x18x25,x3bx3cx26(2;0;0)x18L2xb,x24L2xcx54x51x51x56x23(0;1;1),x28x29x57x15x1x1dx18x25xcx5ex5f,
x5cx38 fl
flfl
flfl
fl
0 2(u+v)?2(1+u)
1?2 1
1 v?u u
flfl
flfl
flfl = 0;
x25x3duv =?1,x2dx66(*)x12x2ex47x3dx0 8
<
:
u = 2z?x2(1+y) ;
v =?x?2z2(y?1),
x28x29xax1x1dx46x3ex3axcx1cx33x54x55x23
x2
4 +y
2?z2 = 1:
5,x13x14x49x617.3.
x2x3,x7x8x9x27x1cx33xcx54x55x23 x2a2 + y2b2? z2c2 = 1,x41xcx40x5ex43x6fx1xfx1dx23
(?
0
x
a +
z
c
·
+”0
1+ yb
·
= 0;
0
1? yb
·
+”0
x
a?
z
c
·
= 0
x2d
(?
1
x
a +
z
c
·
+”1
1? yb
·
= 0;
1
1+ yb
·
+”1
x
a?
z
c
·
= 0:
x4ax39x59x42x7ax54x55(
0
a x+
”0
b y +
0
c z +”0 = 0;”
0a x0b y?,0c z +?0 = 0
x2d
(?1
a x?
”1
b y +
1
c z +”1 = 0;”
1a x+?1b y?,1c z +?1 = 0:
x5bx4bx7cx32x4bx3x4cx4dx2ex3cx4bx42x7ax54x55x50x51xcx40x5ex1x1dxcx18x2ex4cx4excx75x10,x3bx3cxbx4fx50x11x1dx76x54x55x22xcx51
x52x53x54xcx35x34x12 fl
flfl
flfl
flfl
flfl
0
a
”0
b
0
c?”0”
0a0b?”0c0
1
a?
”1
b
1
c?”1”
1a?1b?”1c1
flfl
flfl
flfl
flfl
fl
= 0;
x63x14x51x52x53x54xcx55< 4,x28x29x52x40x5ex1xfx1dx42x6dxbx33.
x7x27x1cx75x76x33xcx54x55x23 x2a2? y2b2 = 2z,x41xcx40x5ex43x6fx1xfx1dx23
(?x
a +
y
b
·
+2? = 0;
x
a?
y
b
·
+z = 0
x2d
(,?x
a +
y
b
·
+z = 0;
x
a?
y
b
·
+2” = 0:
￠ 18 ￠
x3bxbx4f4x43x54x55x50x11x3dx0xcx1dx76x54x55x22xcx51x52x53x54x23
A =
0
BB
BB
@
1
a
1
b 0?2
a?
b 1 0”
a
”
b 1 01
a?
1
b 0?2”
1
CC
CC
A;
x31x32x3dj?Aj = 0,x63x14rank?A < 4,x11x42x54x33x2cx67x4fx30rankA = 3 = rank?A,x63x14x52x40x5ex1xfx1dx18x25.
6,x13x14x49x617.4.
x2x3,x7x8x9x27x1cx33xcx54x55x23 x2a2 + y2b2? z2c2 = 1,x41xcx40x5ex45x6fx1xfx1dx23
(?
0
x
a +
z
c
·
+”0
1+ yb
·
= 0;
0
1? yb
·
+”0
x
a?
z
c
·
= 0
x2d
(?
1
x
a +
z
c
·
+”1
1+ yb
·
= 0;
1
1? yb
·
+”1
x
a?
z
c
·
= 0:
x3bxbx4f4x43x54x55x50x11x3dx0xcx1dx76x54x55x22xcx51x52x53x54xcx35x34x12x23
flfl
flfl
flfl
flfl
fl
0
a
”0
b
0
c?”0”
0a0b?”0c0
1
a
”1
b
1
c?”1”
1a1b?”1c1
flfl
flfl
flfl
flfl
fl
= 4(?0”1?”0?1)
2
abc,
x2ax52x40x5ex1xfx1dx2cx21x72x2b,x42x6dx62?0,,0 6=?1,,1,x28x29x51x52x53x54xcx55x4dx3c4,x52x40x5ex1x1dx43x33.
x7x27x1cx75x76x33xcx54x55x23 x2a2? y2b2 = 2z,x41xcx40x5ex45x6fx1xfx1dx23
(?x
a +
y
b
·
+2?0 = 0;
0
x
a?
y
b
·
+z = 0
x2d
(?x
a +
y
b
·
+2?1 = 0;
1
x
a?
y
b
·
+z = 0;
x3bxbx4f4x43x54x55x50x11x3dx0xcx1dx76x54x55x22xcx51x52x53x54xcx35x34x12x23fl
flfl
flfl
flfl
flfl
fl
1
a
1
b 0?2?0?
0
a?
0
b 1 01
a
1
b 0?2?1?
1
a?
1
b 1 0
flfl
flfl
flfl
flfl
flfl
= 4(?01)
2
ab,
x2a?0 6=?1x2b,x51x52x53x54xcx35x34x12x2cx4dx3c,x28x29x41xcx55x4dx3c4,x63x14x52x40x5ex1xfx1dx43x33.
x4ex3cx27x1cx75x76x33xcx42x6fx1xfx1d
(?x
a +
y
b
·
+2? = 0;
x
a?
y
b
·
+z = 0;
x41xcx54x51x51x56x23
1
a ;
1
b ;0
·
￡
a ;?
b ;1
·
=
1
b ;?
1
a ;?
2?
ab
·
,x46x47x52x6fx1x1dx1dx3bx35x3cx3bx33bx+ay = 0.
x45x6ex2ex3a,x11x42x6fx1xfx1d (
”
x
a +
y
b
·
+z = 0;
x
a?
y
b
·
+2” = 0
x1dx3bx35x3cx3bx33bx?ay = 0.
7,x58x8x9x27x1cx33 x2
a2 +
y2
b2?
z2
c2 = 1 (c
2 < a2 +b2)xbxcx41x18x54x1xcx1xfx1dxcx25x26xbxc.
x4,x78x79,x18x41x54x1xcx1xfx1dx5cx2cx45x6f,x7x23
( !?x
a +
z
c
·
=?
1+ yb
·;
x
a?
z
c
·
= !
1? yb
· x38
(?x
a +
z
c
·
=,
1? yb
·;
”
x
a?
z
c
·
=
1+ yb
·;
x1fx20,!;?x2cx45x2bx23x7c,;”x1cx2cx45x2bx23x7c,x52x40x5ex1x1dxcx54x51x21x22x23
2?!2
bc ;
2?!
ac ;
2 +!2
ab
x38
￠ 19 ￠
2?”2
bc ;
2”
ac ;?
”2 + 2
ab
,x28x41xex54x1,x34x62
(?2?!2)( 2?”2)
b2c2 +
4?!”
a2c2?
(?2 +!2)(”2 + 2)
a2b2 = 0:
x5fx1fx67x3d
a2(! +?”)2 +b2(!”)2 +c2(?”?! )2 = (a2 +b2?c2)(? +!”)2:
x3x66x54x55x22x25x3dx25x26x23 8
>>>>
<
>>>>
:
x =?” +!? +!” a;
y = !” +!” b;
z =?”?!? +!” c:
x2ex3ax2 +y2 +z2 = a2 +b2?c2,x53x25x26xbxcx23(
x2
a2 +
y2
b2?
z2
c2 = 1;
x2 +y2 +z2 = a2 +b2?c2:
x0 x1 6–8
1,x39x30x35x33x2 +y2 +z2 = R2x19
(1)x7x33x2 +y2 = a2 (R > a > 0);
(2)x24x33x2 +y2 = z2 tan2 (0 < < …)
x25x1dxcx62x9x54x55.
x4,(1)
8>
<
>:
x = acos ;
y = asin ;
z = pR2?a2;
(0 6 < 2…)x38
8>
<
>:
x = acos ;
y = asin ;
z =?pR2?a2;
(0 6 < 2…);
(2)
8>
<
>:
x = Rsin cos’;
y = Rsin sin’;
z = Rcos ;
(0 6 ’ < 2…)x38
8>
<
>:
x = Rsin cos’;
y = Rsin sin’;
z =?Rcos ;
(0 6 ’ < 2…).
2,x28x50x6dmx23x1ax1bx2bx3bx33x+mz?1 = 0x19x8x9x27x1cx33x2 +y2?z2 = 1x18x25x59,(1)x21x23; (2)x27
x1cx1d.
x4,x2am = 0x2bx25x1dx8x42x4ex18x25x1x1d (
y2?z2 = 0;
x = 1:
x28x29x7m 6= 0,x29x2bx25x1dx23 (
x2 +y2? 1m2 (1?x)2 = 1;
x+mz = 1;
x2ex5fx1fx23 8
><
>:
x2
m4
(m2?1)2
+ y2m2
m2?1
= 1;
x+mz = 1:
x28x29,x2ajmj > 1x2b,x25x1dx8x21x23;x2ajmj < 1x24m 6= 0x2b,x25x1dx8x27x1cx1d.
3,x28x2fx30x3bx35x33x2 +y2 +z2 = 4x19x7x33x2 +y2 = 2xx46x1dx59xcx34xa.
￠ 20 ￠
x50x4cx48x46x44x51x40x3ex3cx3ax4cx4bx4ax32x1bx1cx45x1ax41x17x3cx39
x13x34x31x32x35
x38x3bx3ex19x1ax44
x1cx47x48x34x38x3ax3cx3ex50x42x44x46x54x55x4cx58x59x25x5bx5ex29x63x2bx69
x6cx6ex6fx6cx6a
x66x63x28x5dx5bx25
x6fx2ex6ax68x64x28
x5ex5cx25x59x58x4cx55x54x46
x44x42x40x4fx3cx3ax4cx36x32x47x1bx45x1ax41
x17x3ax39x34x32x1
x10x30x74
x74
x10x30x74
x74
x7dx6dx6bx72
x66x65x63x61x28
x5ex5dx56x5ax59x50
x57x4cx4bx4ax48x46x46x44x43x42x41x40x3ex3ex3dx3bx3bx39x37x36x35x33x31x2ex2ax44x43
x40x3fx3dx3bx39
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xa
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x1ax45x40x1a
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x16x3ex15x39
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x0
x2ax2cx2fx5bx27x24x59
x24x57x23x56x22x56x22x55x22x55
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O
x y
z
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x9x54x48x4ax4bx4dx4ex4f
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xaxbxc
xcxdxd
xfx10x14x2
x46x46x44x44x42x41x41x3fx3fx3dx3dx3bx3bx39x39x36x36x34x33x31x2f
x2dx2ax44x43x42
x40x3fx3ex3cx3b
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x10
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x47x47x47x22x9
x7ax2fxfx1x11x6
xcx13x1x1bx3
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x5cx5ex5fx62
x2ax62x68x6a
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x6bxf
x64x6cx6e
x70x72x72
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x76x76x77
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x78x78x78
x79x78x79
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x7ax79xfx35x34x32x31x47x6
x10x30x10x30
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x10x30x10x30
x10x30x64x2a x63
x2a x64x2a
x64x29 x64x2a
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x5cx26 x5cx26 x5c
x26 x5cx26 x5cx26
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x39x14 x38x15
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x53x22 x54x22 x54x21 x9
O
x
y
z
x04x1x2
4,x28x2fx30x3bx1cx33z = 1?x2?y2x38x3bx33y = x,y = p3x,z = 0x46x1dx34xax18x7cx42x46x47x20xcx26x5ex21.
5,x28x2fx30x2cx4dx12
x2 +y2 6 z 6
p
x2 +y2
x46x50x6dxcx34xaxcx1fx56.
x44x42x42x41x41x3fx3fx3ex3ex3cx3cx3ax3ax38x36x35x49x48x1cx1bx45
x1ax19x17x3bx13x33x10x11
x37x15x17x18x1ax44x1cx1bx1cx48
x34x36x37x39x3ax3bx3dx3cx3ex3fx3fx40x41x42x42x44x44x45x46x48x49x4bx23x56x24x58x25x5axbx5ex29x2ax69
x2ex6fx2ex2cx65x28x27
x26xbx5ax58x24x23x57x56x4ax48x47x46
x45x21
x2fx2dx2cx29xcx27x5bx24x57x4c
x55x48x46x53x52x43x42x41x40x3fx3ex3dx3bx3ax38x4bx34x48x46x1ax5x4x17x15x13x12
x2cx2dx2fxax24xbxaxbx23x56x22x54
x21x53x20x52x20x51x20x50x20x4fx1fx4ex1ex4ex1ex4cx6x7x49x18x44
x1
x52x20 x51x20 x4fx20 x4ex1f x4dx1e x4ax1d x5x18x3x14
xax26xcx29x23x56x22x53x21x52x20x51x20x4fx1fx4ex1ex4dx1cx4ax1x2x17x18x2ax5c
x56x23 x53x22
x6ex6ex6d
x6cx6ax6a
x64x66x63x62
x60x5fx5ex5dx56
x52x4ex55x54x46x51x50x4fx3ax4cx4bx32x2ex2ax43x42x41x20
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x16x16x14
x3ax12x12
x39
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x73x69x2cx66
xdx2ax29x26x25x23x20 x31x47x46x45x43x42x41
x40x3ex17x1ax18
x16x15x13
x13x11x15x2
x19x24x23x24
x44x44x44x44x44x44x44x44x52x9
x74
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x9x23x4c
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x68x2d x68x2c
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x64x2a x62x29
x5fx27 x5cx25
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x62x29 x62x28
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xcx42x4 x42x19
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x42x19 x42x19
x42x18 x42x19
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x30x10x30x10
x30x10x30x10
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x
y
z
O
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x6e
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x10x10x10
xfxexd
xcxbxa
x8x6x31
x11x30
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x42x42x42x42x43x42x42x42x8x1ex39x3ax39 x6cx6b
x70x0
x7ax2fxfx1x11x6
x34x1dx7x40x20x8 xdx65xfx2dx74
x7bx7bx7b
x7bx7bx7b
x7bx7bx7b
x7bx7bx7b
x7bx7bx7b
x6dx42x42x42x43x42x42x42x42
xex4x15x2x0x47x1b
x1dx48x1dx4ax4dx1e x4dx1e x4dx1e x4dx1e x4dx1e x4dx1e x4dx1e x4dx1e x4cx1f
x66x2b x66x2b
x65xexd
x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10x30x10
x
y
z
O
x06x1x2
6,x28x2fx30x3bx2cx4dx12x22
y > 0; z > 0; x > 2y2; x4 + y2 + z4 6 1;
x46x50x6dxcx34xaxcx1fx56.
￠ 21 ￠
x0x8x3 x9xaxbxcxdxexfx10x11
x0 x1 6–1
1,x28x21x22x4bx29x4dx2ax2bx2cx2dx29x2ex4dx2ax2bx2cx2fx30x41x30x4dx3c2,3,4xcx30x54x31x47x2dx29x32x33x31.
x4:
x70x74
x74
x74
x52x53x52x53
x52x53x52x53
x52x74
x74
x74
x70
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x52x53x52x53
x52
x2ex2dx2ex2d
x2ex2dx52x53x52x53x52x53
x52x53x52
x2dx2ex2dx2e
x2dx2e
x2ex2dx2ex2d
x2ex2dx74
x74
x74
x70
x47x1b x47x1b x46x1c
x47x1b x59x25 x59x25 x5ax24 x59x25
x59x25 x59x25
x30x10x30x10
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x30x10x30x10
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x0x74
x74
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x74
x74
x74
x0x44x44x44x44x44x44x44x44x44x44x44x44x24
x23x24x44x44x44x44x44x44x44x44x44x44x44x44
x24x23x24
x24x23x24
x74
x74
x74
x74
x0
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x52x21 x52x21 x52x21 x52x21 x52x21 x52x21 x52x21 x52x21x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10
x3bx3bx3bx3bx3bx3bx3bx3bx3bx3bx3cx3bx3bx3bx3b
x3bx3bx3bx3bx3bx4dx8
x69x68x68
x69x68x68
x68x69x68
x68x69x68
x68x68x69
x68
x14xexfxe
xfxexf
xexfxe
xfxexf
xexfxe
xfxexf
xe
xfxfxf
xexfxf
xfxfxf
xexfxf
xexexe
xexexd
xex14x1
x25x59x25x59x24x5a
x25x59x25x59x25x59
x24x5ax25x59x25x59
x25x59xb
x37x38x37x37x37x38x37x37x37
x38x37x37x37x38x37x37x4c
x2ax63x68x67
x67x68x67
x67x68x67
x67x67x68
x67x67x68
x67
x11x7x7x7
x7x7x7
x7x6x7
x7x7x7
x7x7x7
x6x7x7
x7x7
x70x70x71
x70x70x71
x70
x13x13x13
x12x13x13
x13x13x13
x13x12x13
x13x15
x21x52x21x52x21x52x21x52x21x52x21x52x21x52x21x52x21x52x21x53x21x52x21x52
x21x52x52
x0 x1 6–2
1,x21x22x34x33x34x5ex5fx58x35x33x54x55:
(1)x42x1x36xcx40x37x26x23A(2;?3;5)x38B(4;1;?3);
(2)x35x39x18x1x1d x?42 = y +8?4 = z?21 xb,x24x3ax26(2;?3;6)x38(6;3;?2);
(3)x3ax26(?1;2;5),x24x193x43x1fx20x3bx33x18x3c;
(4)x3ax26(p2;p2;2),x24x3dx3ex23:
( x2 +y2 = 4;
z = 0:
x4,(1)x35x39x1fx20C
2+4
2 ;
3+1
2 ;
5?3
2
·
= (3;?1;1),x3fx36
R =
p
(3?2)2 +(?1+3)2 +(1?5)2 = p21;
x46x47x35x33x54x55x23(x?3)2 +(y +1)2 +(z?1)2 = 21.
(2)x28x35x39x18x40x3ax1x1dxb,x53x41xcx1fx20x42x23(4+2t;?8?4t;2+t),x45x28x26(2;?3;6)x38(6;3;?2)x18
x35x33xb,x46x47x41xex0x35x39xcx43x44x18x4d,x74
(4+2t?2)2 +(?8?4t+3)2 +(2+t?6)2 = (4+2t?6)2 +(?8?4t?3)2 +(2+t+2)2;
x25x3dt =?2,x66x2dx35x39x1fx20x8(0;0;0),x24x3fx36x4dx3c7,x35x33x54x55x23x2 +y2 +z2 = 49.
(3)x35x39x19x26(?1;2;5)x18x45x42x46x47x48,x28x29x2ex7x41xcx1fx20x23(?a;a;a),x1bx35x33x54x55x23
(x+a)2 +(y?a)2 +(z?a)2 = a2:
￠ 1 ￠
x49(?1;2;5)xcx1fx20x4ax4b,x3da2?8a+15 = 0,x25x3da = 5x6f3,x74x35x33x54x55x23
(x+5)2 +(y?5)2 +(z?5)2 = 25 x47x2d (x+3)2 +(y?3)2 +(z?3)2 = 9:
(4)x40x3ax23x4cx3cx1fx20x3bx33xOyxb,x23x39x8x37x26,x28x29x35x39x42x6dx18zx4dxb,x7x35x39x1fx20x23(0;0;t),x1b
4+t2 = 2+2+(2?t)2,x25x3dt = 1,x46x47x35x33x54x55x23x2 +y2 +(z?1)2 = 5.
2,x58x33x34x23xcx23x39x2dx3fx36:
(1)
( x2 +y2 +z2 = 4;
x+y +z?3 = 0;
(2)
( x2 +y2 +z2 = 5;
x2 +y2 +z2 +x+2y +3z?7 = 0:
x4,(1) x46x4excx23x2ex47x4fx59x8x7cx42x43x54x55x46x50x6dxcx35x33x19x7cx2ex43x54x55x50x6dxcx3bx33xcx25x1d,x2dx35x39
x8x37x26,x46x47x23x39x42x18x37x26x51x52x43x3bx33x46x53xcx54x1dxcx54x55xb,x29x54x1dxcx54x51x51x56x8(1;1;1),x53x54x1dx54
x55x23
8>
<
>:
x = t;
y = t;
z = t:
x54x1dx19x3bx33xcx25x26x8(1;1;1),x29x74x35x39,x57x15x57x58x6dx6e,x35xcx3fx36x23p4?3 = 1.
(2)x7cx2ex43x54x55x59x5ax7cx42x43x54x55x67x2ex3dx+2y+3z?2 = 0,x5bx4bx19(1)x49x5cxcx54x19,x2ex3ax23x39x34x8
x29x54x55x46x50x6dxcx3bx33x19x1x1d
8>
<
>:
x = t;
y = 2t;
z = 3t
xcx25x26,x25x3dx23x39x1fx20x23
1
7 ;
2
7 ;
3
7
·
,x3fx36
q
5? 27 =
p231
7,
3,x58x13,8
><
>:
x = acos2 t;
y = asin2 t;
z = ap2sintcost;
0 6 t < … (a > 0)
x50x51x42x23,x58x29x23xcx23x39x38x3fx36.
x4,x29x1cx1dxbxcx7x8x42x26(x;y;z)x5dx55x54x55x2 +y2 +z2 = a2x47x2dx+y?a = 0,x53x1cx1dx8x35x33x19
x3bx33xcx25x1d(x23),(
x2 +y2 +z2 = a2;
x+y?a = 0;
x6fx1fx42x5ex21,x23x13x29x1cx1dx50x8x23,x7(x;y;z)x8x23xbx7x8x42x26,x3cx8y = a?x,x2 + (a?x)2 + z2 = a2.
x67x12x2ex5fx232
x? a2
·2
+z2 = a22, x28x29x17x180 6 < 2…x1ax3d
8<
:
x = a2 + a2 cos = acos2 2 ;
z =
p2a
2 sin =
p2asin
2 cos
2 ;
x66x2dy = a?x = asin2 2, x1et = 2,x34x60x3dx0x61x7xcx62x9x54x55,x63x14x5dx55x23x54x55xcx26x1dx8x61x7x1cx1dxb
xcx26,x28x29x40x3ax1cx1dx50x8x23.
x1fx23x39x42x8x1x1d
8>
<
>:
x = t;
y = t;
z = 0
x19x3bx33x+y?a = 0xcx25x26,x74
a
2 ;
a
2 ;0
·
,x3fx36x1bx23
r
a2? a24? a24 =
p2
2 a.
4,x58x13,x40x43x35x33
Si, x2 +y2 +z2 +Aix+Biy +Ciz +Di = 0; (i = 1;2);
￠ 2 ￠
x25x1dx23x46x18x3bx33x23
(A1?A2)x+(B1?B2)y +(C1?C2)z +(D1?D2) = 0:
x2x3,x7x52x40x43x35x33xcx25x1dx23xbxc3x43x2cx45x26Mj(aj;bj;cj) (j = 1;2;3),x1b(
a2j +b2j +c2j +A1aj +B1bj +C1cj +D1 = 0;
a2j +b2j +c2j +A2aj +B2bj +C2cj +D2 = 0; (j = 1;2;3):
x49x40x12x18x59x3d
(A1?A2)aj +(B1?B2)bj +(C1?C2)cj +(D1?D2) = 0; (j = 1;2;3):
x52x63x14x23xbxc3x43x26M1;M2;M3x1dx18x3bx33
(A1?A2)x+(B1?B2)y +(C1?C2)z +(D1?D2) = 0
xb,x28x29x64x43x23x1cx18x29x3bx33xb.
5,x40x3ax40x35x33:
Si, x2 +y2 +z2 +2Uix+2Viy +2Wiz +di = 0; (i = 1;2)
x58x13x52x40x35x33x29x25(x18x25x26x65xcx3cx3bx33x54x1)xcx5ex5fx8:
2(U1U2 +V1V2 +W1W2) = d1 +d2:
x2x3,x7x26M(x0;y0;z0)x23x40x43x35x33xcx7x8x42x43x25x26,x1bx3aMx26xcx3cx3bx33xcx19x51x56x34x8x3ax52x43x26
xcx35x3fx36,x4bx66x54x19x2cx67x4fx30x52x40x43x35x33xcx35x39x8(?Ui;?Vi;?Wi) (i = 1;2),x28x29x35x3fx36xcx54x51x51x56
x8(x0 +Ui;y0 +Vi;z0 +Wi) (i = 1;2),x52x40x43x35x33x29x25x4dx68x3cx52x40x43x35x3fx36x29x25,x74
(x0 +U1)(x0 +U2)+(y0 +V1)(y0 +V2)+(z0 +W1)(z0 +W2) = 0;
x5x6x3d
x20 +y20 +z20 +(U1 +U2)x0 +(V1 +V2)y0 +(W1 +W2)z0 +U1U2 +V1V2 +W1W2 = 0:
x3bx3cMx18x35x33xcx25x1dxb,x28x29x41xcx1fx20x45x2bx5dx55x40x43x35x33xcx54x55,x49x29x40x43x54x55x18x7fx67x14x472x2ex3d
x20 +y20 +z20 +(U1 +U2)x0 +(V1 +V2)y0 +(W1 +W2)z0 + d1 +d22 = 0:
x4ax4bx2x33x4dx12x67x74x3d
2(U1U2 +V1V2 +W1W2) = d1 +d2:
x52x43x5ex5fx8x5bx21x24x5cx5dxc.
6,x58x13x40x23
S1,
( x2 +y2 = 4;
z = 0 x19S2,
( x2 +y2 +z2?2x?2y?2 = 0;
x+y +z = 1
x18x45x42x35x33xb.
x2x3,x69x3aS1xcx35x33xcx35x39x42x6dx18zx4dxb(x62x6ax6bx611(4)),x28x29x1fx1fx20x23(0;0;a),x3x58S2xcx23
x39,x41x8x1x1d
8>
<
>:
x = 1+t;
y = 1+t;
z = t
x19x3bx33x + y + z = 1xcx25x26
2
3 ;
2
3 ;?
1
3
·
,x2dx35x39x19x23x39xcx6cx1dx42x6dx19x3b
x33x + y + z = 1xcx19x51x56x3bx35,x74 23, 23,
13?a
·
= 1, 1, 1,x46x47a =?1,x74x35x39x23(0;0;?1),x45
x28(2;0;0)x18S1xb,x66x2dx18x35x33xb,x58x3dx35x3fx36x4dx3cp22 +12 = p5,x28x29S1x18x35x33
x2 +y2 +(z +1)2 = 5
￠ 3 ￠
xb,x29x54x55x59x5ax54x55x+y +z = 1xc2x6ex67x3dx0
x2 +y2 +z2?2x?2y?2 = 0;
x63x14x29x35x33x19x3bx33x+y +z = 1xcx25x1dx34x8S2,x28x29S1x19S2x18x45x42x43x35x33xb.
7,x13x14,x3ax23:
( S = x2 +y2 +z2 +2ux+2vy +2wz +d = 0; (u2 +v2 +w2?d > 0);
E = Ax+By +Cz +D = 0; (A2 +B2 +C2 6= 0)
xcx35x33x6fx54x55x2ex50x51x23,S +2?E = 0 (?x23x62x9).
x2x3,x4ex3cx62x9?,x63x54x55S +2?E = 0x50x70x50x51x42x43x35x33,x1bx23
( S = 0;
E = 0 x42x6dx18x29x35x33xb.
x4ex7x8x42x43x3ax23
( S = 0;
E = 0 xcx35x33
x2 +y2 +z2 +2px+2qy +2tz +r = 0;
x40x3ax23x42x18x40x43x35xcx25x1d (
x2 +y2 +z2 +2ux+2vy +2wz +d = 0;
x2 +y2 +z2 +2px+2qy +2tz +r = 0
xb,x3bx7c4x61x3a,x29x23x42x18x3bx33
(p?u)x+(q?v)y +(t?w)z + r?d2 = 0
xb,x71E = 0x1cx3ax29x23,x28x29x40x43x3bx33x21x72,x74x17x18x70x9?x1ax3d
p?u =?A; q?v =?B; t?w =?C; r?d2 =?D:
x25x30p;q;t;r,x4ax4bx35x33x54x55x74x3d
x2 +y2 +z2 +2ux+2vy +2wz +d+2?(Ax+By +Cz +D) = 0:
x0 x1 6–3
1,x58x33x34x73x17x1cx33xcx54x55:
(1)x1x1d x2 = y1 = z?10 x74x1x1dx = y = zx73x17;
(2)x1x1d x?11 = y?3 = z3 x74zx4dx73x17;
(3)x75x76x1d
( y2 = 2px;
z = 0 x74x41xcx77x1dx73x17;
(4)x1cx1d
( x2 = y;
x+z = 0 x74x1x1d
x
1 =
y
2 =
z
1 x73x17.
x4,(1)x78x79x37x26Ox18x73x17x4dxb,x24x4dxcx54x51x51x56x8? = (1;1;1),x62x78(3.1),x2ex47x3dx0x54x55x22
8>
<
>:
(x?x0)+(y?y0)+(z?z0) = 0;
x2 +y2 +z2 = x02 +y02 +z02;
x0
2 =
y0
1 =
z0?1
0 ;
x18x54x55x22x20x79x5ax62x9x0;y0;z0x67x2ex3d
x2 +y2 +z2?1 = 59 (x+y +z?1)2;
x28x29x46x58x73x17x1cx33xcx54x55x23
2(x2 +y2 +z2)?5(xy +xz +yz)+5(x+y +z)?7 = 0:
￠ 4 ￠
(2)x78x79x37x26Ox18x73x17x4dxb,x24x4dxcx54x51x51x56x8? = (0;0;1),x45xbx61,x2ex47x3dx0x54x55x22
8>
<
>:
z?z0 = 0;
x2 +y2 +z2 = x02 +y02 +z02;
x0?1
1 =
y0
3 =
z0
3 ;
x28x29z0 = z,y0 =?z,x0 = 1+ z3,x4ax4bx54x55x22x79x5ax62x9x0;y0;z0x67x2ex3d
x2 +y2 +z2 =
1+ z3
·2
+z2 +z2;
x28x29x46x58x73x17x1cx33xcx54x55x23
9x2 +9y2?10z2?6z?9 = 0:
(3)x75x76x1dxcx77x1dxcx42x7ax54x55x23
( x =?p
2 ;
z = 0; x1bx1fx20x77x54x55x23
x+ p2
0 =
y
1 =
z
0,
x52x4dxbxcx42x26(x0;y0;z0) =
p2 ;0;0
·
,x34x2ex7bx30x47x33x54x55x22
8>
>>>>
<
>>>>
>:
y?y0 = 0;?
x+ p2
·2
+y2 +z2 =
x0 + p2
·2
+y02 +z02;
y02 = 2px0;
z0 = 0:
x66x54x55x22x79x5ax62x9x0;y0;z0,x34x60x3dx0x73x17x1cx33xcx54x55
y4?4p2x2 +2p2y2?4p2z2?4p3x = 0:
(4)x78x79x37x26Ox18x73x17x4dxb,x28x29x2ex3dx54x55x22:8
>>>
><
>>>
>:
(x?x0)+2(y?y0)+(z?z0) = 0;
x2 +y2 +z2 = x02 +y02 +z02;
x02 = y0;
x0 +z0 = 0:
x66x54x55x22x79x5ax62x9x0;y0;z0,x34x60x3dx0x73x17x1cx33xcx54x55
3x2 +3z2?4xy?2xz?4yz?4x?8y?4z = 0:
2,x57x15k;lxcx2cx45x52x1b(x7cx6fx7dx7c)x7ex10x1x1d
L, x1 = yk = z?l0
x74xx4dx73x17x46x59x1cx33Sx8x1ax7fx1cx33.
x4,x21x47x33x0x7fx1x2x7ex10.
(i) k = l = 0x2b,Lxcx54x55x59x23 x1 = y0 = z0,Lx34x8xx4d,x28x29x74xx4dx73x17x3x79x8xx4dx6cx4;
(ii) k = 0,l 6= 0x2b,Lxcx54x55x23
( z = l;
y = 0; Lx8x1fx20x3bx33xOzxbxcx1cx1d,x57x15x53.1xcx7ex10,xOzx1f
x20x3bx33xbxcx1cx1dx74xx4dx73x17x3dx0xcx73x17x1cx33xcx54x55x2ex47x4bpy2 +z2x4ax6x54x55x20xczx2dx3dx0,x28x29x73x17
x1cx1dxcx54x55x23y2 +z2 = l2,x8x42x43x23x7x33;
(iii) k 6= 0,l = 0x2b,Lxcx54x55x8
( y = kx;
z = 0; Lx8x1fx20x3bx33xOyxbxcx1cx1d,x45x6e,x73x17x1cx33xcx54x55x2e
x47x4bpy2 +z2x4ax6x54x55x20xcyx2dx3dx0,x74x23y2 +z2 = k2x2,x52x8x23x24x33;
￠ 5 ￠
(iv) k 6= 0,l 6= 0x2b,x28x37x26x18x73x17x4dxb,x2ex3dx47x33x54x55x22
8>
<
>:
x?x0 = 0;
x2 +y2 +z2 = x02 +y02 +z02;
x0
1 =
y0
k =
z0?l
0 ;
x79x5ax62x9x67x3dx0x1cx33x54x55
y2 +z2
l2?
k2x2
l2 = 1;
x52x8x8x9x27x1cx33.
3,x13x14,x0x6dx1x1dx2dx6dx1x1dxbx42x6dx26xcx43x44x3bx54x38x8x44x9xcxax26xbxcx8x42x73x17x1cx33.
x2x3,x7x6dx1x1dx23zx4d,x6dx26x23x37x26O,x7P(x;y;z)x8x5dx55x5ex5fxcx26,x1bPxcx1fx20x5dx55x47x33x54x55:
(x2 +y2)+(x2 +y2 +z2) = k2;
x78x79x52x8x1cx1d (
2x2 +z2?k2 = 0;
y = 0
x74zx4dx73x17x2dx3dxcx73x17x1cx33x54x55.
4,x58x13,yz +zx+xy = a2x8x73x17x1cx33,x24x58x73x17x4d.
x2x3,x28x23
(x2 +y2 +z2)+2(yz +zx+xy)?(x2 +y2 +z2) = 2a2;
x2ex3d
(x+y +z)2?(x2 +y2 +z2) = 2a2:
x4ex7x8x70x9p (jpj > p2jaj),x1cx1d (
x2 +y2 +z2 = p2?2a2;
x+y +z?p = 0
x8x42x43x23,x23x39x18x1x1dx = y = zxb,x28x29x52x8x42x43x73x17x1cx33,x73x17x4dx8x = y = z.
x1cx2ex47x1ax1cx1d (
yz +zx+xy = a2;
2x?y?z = 0
x74x1x1dx = y = zx73x17x2dx3dx0x1cx33yz +zx+xy = a2.
5,x58x13,8
><
>:
x = a(cosu+cosv);
y = a(sinu+sinv);
z = b(u?v)
x8x73x17x1cx33,x52xda;b 6= 0x24a;bx8x44x9.
x2x3,x28x23
x2 +y2 = a2 +2a2(cosucosv +sinusinv) = a2 +2a2 cos(u?v)
= a2 +2a2 cos zb ;
x78x79x41x8x1cx1d (
x2 = a2 +2a2 cos zb ;
y = 0
x74zx4dx73x17x2dx3d.
￠ 6 ￠
6,x58x1cx1d
( x = z2;
x2 +y2 = 1 x74x1x1d
8>
<
>:
x = 2t;
y = 0;
z = 3t
x73x17xex59xcx73x17x1cx33xcx54x55.
x4,x73x17x4dx69x3ax37x26O,x28x29x2ex3dx54x55x22
8>
>>>
<
>>>
>:
2(x?x0)+3(z?z0) = 0;
x2 +y2 +z2 = x02 +y02 +z02;
x0 = z02;
x02 +y02 = 1:
x79x5ax62x9x67x2ex3dx73x17x1cx33x54x55
2x+3z +2?2x2?2y2?2z2 = §3
p
x2 +y2 +z2?1:
7,x13x14x1cx33F(x;y;z) = (x2 +y2 +z2)2?16(x2 +z2) = 0x8x42x43x73x17x1cx33.
x2x3,x52x8x1cx1d (
(x2 +y2)2?16x2 = 0;
z = 0
x74yx4dx73x17x2dx3dx0xcx1cx33.
x0 x1 6–4
1,x40x3ax7x33x77x1dx54x55x23 (
x2 +y2 +z2 = 4;
x2 +(y?3)2 +z2 = 4;
xfx1dx54x51x231, 1, (?1),x28x58x1fx54x55.
x4,x7x52x26M(x0;y0;z0)x18x77x1dxb,P(x;y;z)x23x7x33x3aMxcxfx1dxbxcx26,x1bx62
8
>>>>
>>><
>>>
>>>>
:
x = x0 +u;
y = y0 +u;
z = z0?u;
x02 +y02 +z02 = 4;
x02 +(y0?3)2 +z02 = 4;
x25x3d 8
><
>:
x0 = x?y +y0;
z0 = z +y?y0;
y0 = 32 ;
x2fx30x7x33x54x55
(2x?2y +3)2 +(2z +2y?3)2 = 7:
2,x40x3ax7x33x77x1dx54x55x23 (
y = x2 +z2;
y = 2z;
xfx1dx54x1x3cx77x1dx46x18x3bx33,x28x58x29x7x33x54x55.
x4,x28x23xfx1dx54x1x3cx77x1dx46x18x3bx33y?2z = 0,x46x47xfx1dx54x51x23(0;1;?2),x19xbx61x49x5c,x2ex3dx54x55
￠ 7 ￠
x22 8
>>>>
>>><
>>>
>>>>
:
x = x0;
y = y0 +u;
z = z0?2u;
y0 = x02 +z02;
y0 = 2z0;
x25x3d 8
>><
>>:
x0 = x;
y0 = 25 (2y +z);
z0 = 15 (2y +z);
x2fx30x7x33x54x55
2
5 (2y +z) = x
2 + 1
25 (2y +z)
2;
x5x6x67x3d
25x2 +4y2 +z2 +4yz?20y?10z = 0:
3,x58x1cx1d (
x2 +2y2 +z2 = 1;
x2 +z2 = y
x4exOyx3bx33xcx10x2cx7x33x54x55.
x4,xfx1dxcx54x51x51x56x8(0;0;1),x2ex3dx54x55x228
>>>>
>>><
>>>
>>>>
:
x = x0;
y = y0;
z = z0 +u;
x02 +2y02 +z02 = 1;
x02 +z02 = y0;
x25x3d 8
><
>:
x0 = x;
y0 = y;
2y02 +y0?1 = 0;
x3bx3cy =?1x2cx72x61x8,x28x29x7x33x54x55x23
y = 12 ;
p2
2 6 x 6
p2
2
!
:
x11x25,x49x1cx1dx54x55x22xcx7cx42x43x54x55x59x5ax7cx2ex43x54x55,x2ex3d2y2?y?1 = 0,x52x8xfx1dx19zx4dx3bx35xc
x7x33,x2dx24x69x3ax40x3ax1cx1d,x74x23x46x58xcx10x2cx7x33.
4,x28x63x14x33x34x54x55x46x50x51xcx1cx33x8x7x33:
(1) (x+y)(y +z) = a2; (2) (x+y)(y +z) = x+2y +z;
(3) y2 +2yz +z2 = 1?x2; (4) (x+y +z)2 = (x?y?z)2.
x4,(1) x28x23x1x1d
( x+y = a;
y +z = a x18x29x1cx33xb,x41xcx54x51x231, (?1), 1,x24x26
a
2 ;
a
2 ;
a
2
·
x18x29x1x1d
xb,x2dx3bx33x?y +z? a2 = 0x19x1cx33(x+y)(y +z) = a2xcx25x1dx23
( (x+y)(y +z) = a2;
x?y +z = a2,
￠ 8 ￠
xdxex47x52x5ex1cx1dx23x77x1d,x471, (?1), 1x23xfx1dx54x51,x2ex58x3dx54x55x22
8
>>>>
>>><
>>>>
>>>
:
x = x0 +u;
y = y0?u;
z = z0 +u;
(x0 +y0)(y0 +z0) = a2;
x0?y0 +z0 = a2 ;
x79x5ax62x9x67,x3dx0x7x33x54x55(x+y)(y +z) = a2,x46x47x37x12xcx1cx33x8x7x33.
(x13,x47x33x0x43x14x61xdxex49x15x50x6dxfx1dx54x51,x79x67x13x14x69x3ax1cx33xbx7x8x42x26xcx19xfx1dx54x51x3bx35xcx1
x1dx1dx18x1cx33xb,x4bx52x16xcx54x19x12x13x14x1cx33x8x7x33.)
(2) x28x23(x + y)(y + z) = (x + y) + (y + z),x46x47x1x1d
( x+y = 0;
y +z = 0 x18x29x1cx33xb,x1fx54x51x51x56x8
(1;?1;1).
x7M(x0;y0;z0)x8x1cx33xbxcx7x8x26,P(x;y;z)x8x3aMxcx1x1d
x?x0
1 =
y?y0
1 =
z?z0
1
xbxcx42x26,xdxex5dx17x13Px18x1cx33xb,x23x29,x25x3d(
x+y = x0 +y0;
y +z = y0 +z0;
x28x29
(x+y)(y +z) = (x0 +y0)(y0 +z0) = x0 +2y0 +z0) = (x+y)+(y +z) = x+2y +z;
x74Px26xcx1fx20x5dx55x1cx33x54x55,x63x14x64x5ex1x1dx1dx18x1cx33xb,x28x29x1cx33x8x7x33.
(3)x54x55y2 +2yz +z2 = 1?x2x2ex47x5fx23x2 +(y +z)2 = 1,x46x47x1x1d
( x = 1;
y +z = 0 x18x29x1cx33xb,x41
xcx54x51x51x56x8(0;?1;1).
x7M(x0;y0;z0)x8x1cx33xbxcx7x8x26,P(x;y;z)x8x3aMxcx1x1d
x?x0
0 =
y?y0
1 =
z?z0
1
xbxcx42x26,xdxex5dx17x13Px18x1cx33xb,x23x29,x25x3d(
x = x0;
y +z = y0 +z0;
x28x29
x2 +(y +z)2 = x02 +(y0 +z0)2 = 1;
x74Px26xcx1fx20x5dx55x1cx33x54x55,x63x14x64x5ex1x1dx1dx18x1cx33xb,x28x29x1cx33x8x7x33.
(4)x78x79x1x1d
( x+y +z = 0;
x?y?z = 0 x18x29x1cx33xb,x41xcx54x51x51x56x8(0;1;?1).
x7M(x0;y0;z0)x8x1cx33xbxcx7x8x26,P(x;y;z)x8x3aMxcx1x1d
x?x0
0 =
y?y0
1 =
z?z0
1
xbxcx42x26,xdxex5dx17x13Px18x1cx33xb,x23x29,x25x3d(
x = x0;
y +z = y0 +z0;
￠ 9 ￠
x28x29
(x+y +z)2?(x?y?z)2 = (x0 +y0 +z0)2?(x0?y0?z0)2 = 0;
x74Px26xcx1fx20x5dx55x1cx33x54x55,x63x14x64x5ex1x1dx1dx18x1cx33xb,x28x29x1cx33x8x7x33.
5,x40x3ax23x7x33xcx4dx54x55x23:
x
1 =
y?1
2 =
z +1
2 ;
x26(1;?2;1)x18x29x23x7x33xb,x58x29x23x7x33xcx54x55.
x4,x28x23x26(1;?2;1)x0x4d x1 = y?1?2 = z +1?2 xcx43x44x74x23x18x23xcx3fx36,x46x47x29x3fx36x23
r = j(1;?3;2)￡(1;?2;?2)jp1+4+4 =
p117
3,
x7x52x29x23x7x33xbxcx42x26P(x;y;z),Px0x4dxcx43x44x1cx23r,x28x29x62
j(x;y?1;z +1)￡(1;?2;?2)j
3 =
p117
3 ;
x64x6ex67x2ex3dx23x7x33x54x55
8x2 +5y2 +5z2 +4xy +4xz?8yz?18y +18z?99 = 0:
(x13,x29x61x1cx2ex4bx3a(1;?2;1)x26xcx1xfx1dx74x4dx73x17x2dx3dx0x29x1cx33x54x55,x1cx2ex47x4bx58x30x42x43x18x23x53x77
x1dx12x58x30x29x7x33x54x55.)
6,x7x7x33xcx77x1dx23
( x = y2 +z2;
x = 2z; xfx1dx54x1x3cx77x1dx46x18xcx3bx33,x58x52x7x33xcx54x55.
x4,x28x23x77x1dx18x3bx33x = 2zxb,x46x47xfx1dxcx54x51x51x56x8(1;0;?2),x3bx29x2ex3dx54x55x228
>>>>
>>><
>>>
>>>>
:
x = x0 +u;
y = y0;
z = z0?2u;
x0 = y02 +z02;
x0 = 2z0;
x79x5ax62x9x67x2ex3dx7x33x54x55:
4x2 +25y2 +z2 +4xz?20x?10z = 0:
7,x58x3fx36x234,x4dx1dx54x55x8x = 2y =?zxcx23x7x33x54x55.x19x17x13x41x1axOyx1fx20x3bx33x46x1bx3dxcx1cx1c
x1dx1dx59xcx33x1ex4dx3c24….
x4,x46x58x23x7x33x34x8x0x4dx1dxcx43x44x4dx3c4xcx26xcxbxc,x28x29x23x7x33xbxcx26P(x;y;z)x5dx55x47x33x54x55fl
flfl(x;y;z)￡?1; 1
2 ;?1
·flfl
fl
q
2+ 14
= 4;
x5fx1fx67x3d
5x2 +8y2 +5z2 +8xz +4yz?4xy = 144:
x29x23x7x33x1axOyx3bx33x46x1bx3dxcx1bx20x8x42x43x21x23,x24x29x21x23xcx22x3fx4dx30x234,x2dxOyx3bx33x19x4dx1dxc
x23x1e x5dx55:
sin = j(0;0;1)￠(2;1;?2)jp22 +12 +22 = 23 ;
x46x47x30x3fx4dxcx30x23
4
sin = 6:
￠ 10 ￠
x29x21x23xcx33x1ex4dx3c
… ￠4￠6 = 24…:
8,x58x19xx4dx2dx3bx33y = kx4dx43x44xcx26xcxbxcx54x55.
x4,x7xax26x23P(x;y;z),x1bx3bx5ex5fx3a
y2 +z2 = (y?k)2;
x74x23
z2 +2ky?k2 = 0:
x0 x1 6–5
1,x58x24x33x54x55:
(1)x77x1d:
( ax2 +by2 = 1;
z = 0; x24x26(x0;y0;z0);
(2)x77x1d:
( f(x;y) = 0;
z = k (6= 0); x24x26(0;0;0);
(3)x77x1d:
( x2 +y2 +(z?5)2 = 9;
y = 2; x24x26(0;0;0).
x4,(1)x4ex3cx77x1dxbxcx26M(x0;y0;z0),x7P(x;y;z)x8x3aMxcx1xfx1dxbxcx26,x1bx62x47x33x54x55x228
>>>
>>>>
<
>>>>
>>>:
x0 = x0 +(x?x0)u;
y0 = y0 +(y?y0)u;
z0 = z0 +(z?z0)u;
ax02 +by02 = 1;
z0 = 0;
x79x5ax62x9x0;y0;z0;ux67x2ex3dx24x33x54x55:
a(z0x?x0z)2 +b(z0y?y0z)2?(z?z0)2 = 0:
(2)x49x5cx25x62x47x33x54x55x22 8
>>>
>>>>
<
>>>>
>>>:
x0 = xu;
y0 = yu;
z0 = zu;
f(x0;y0) = 0;
z0 = k;
x79x5ax62x9x67x2ex3dx24x33x54x55:
f
k xz ;k yz
·
= 0:
(3)x62x47x33x54x55x22 8
>>>>
>>><
>>>>
>>>
:
x0 = xu;
y0 = yu;
z0 = zu;
x02 +y02 +(z0?5)2 = 0;
y0 = 2;
x79x5ax62x9x67x2ex3dx24x33x54x55:
x2 +5y2 +z2?5yz = 0:
￠ 11 ￠
2,x58x47x26P(5;0;0)x23x24x26,x47x1cx1d
( x2 +2y2 = 1;
x+2y?z = 0 x23x77x1dxcx24x33x54x55.
x4,x62x47x33x54x55x22 8
>>>>
>>><
>>>
>>>>
:
x0 = 5+(x?5)u;
y0 = yu;
z0 = zu;
x02 +2y02 = 1;
x0 +2y0?z0 = 0;
x79x5ax62x9x67x2ex3dx24x33x54x55:
x2?146y2?24z2 +4xy?2xz +96yz?10x?20y +10z +25 = 0:
3,x58x47x37x26x23x24x26,x47
( x2?2z +1 = 0;
y?z +1 = 0 x23x77x1dxcx24x33x54x55.
x4,x62x47x33x54x55x22 8
>>>
>>>>
<
>>>>
>>>:
x0 = xu;
y0 = yu;
z0 = zu;
x02?2z0 +1 = 0;
y0?z0 +1 = 0;
x79x5ax62x9x67x2ex3dx24x33x54x55:
x2 +y2?z2 = 0:
4,x40x3ax23x24x33xcx24x26x23(1;2;3),x4dx54x1x3cx3bx332x+2y?z +1 = 0,xfx1dx19x4dxcx23x1ex2330–,x58x6d
x23x24x33xcx54x55.
x4,x7P(x;y;z)x8x24x33xbxcx42x43x26,x26x27x3aPx26xcxfx1dxcx54x51x51x56x8
= (x?1;y?2;z?3):
x2dx23x24xcx4dxcx54x51x51x56x34x8x3bx33xcx19x51x56
” = (2;2;?1):
x57x15x61x8x62
(?;”)
j?jj”j = §cos30
–;
x74
2(x?1)+2(y?2)?(z?3)p
(x?1)2 +(y?2)2 +(z?3)2p4+4+1 = §
p3
2,
x5fx1fx64x6ex67x74x3dx46x58x23x24x33xcx54x55x23
11(x?1)2 +11(y?2)2 +23(z?3)2?32(x?1)(y?2)+16(x?1)(z?3)+16(y?2)(z?3) = 0;
x6f
11x2 +11y2 +23z2?32xy +16xz +16yz?6x?60y?186z +342 = 0:
5,x3axx4dx38yx4dx21x22x53xax3bx33,x1ax41xex28x29x6dx25x1efi,x28x58x41xexcx25x1dx2axexcx1cx33x54x55,x19x2bx30x8
x2cx27x1cx33.
x4,x7x3axx4dxcxax3bx33x23A1y +B1z = 0,x3ayx4dxcxax3bx33x23A2x+B2z = 0,x41xex28x29x6dx1efi,x1b
B1B2p
A21 +B21pA22 +B22 = cosfi:
￠ 12 ￠
x41xexcx25x1dx3ax37x26O,x2dx24x25x1dxcx54x51x51x56
= (0;A1;B1)￡(A2;0;B2) = (A1B2;A2B1;?A1A2);
x28x29x25x1dxbxcx26P(x;y;z)x5dx55 8
><
>:
x = tA1B2;
y = tA2B1;
z =?tA1A2;
x25x3d 8
>>>>
<
>>>>
:
A21 +B21 = 1t2A2
2
y2 +z2;
A22 +B22 = 1t2A2
1
x2 +z2;
(B1B2)2 = x2y2t2z2,
x3bcos2 fi(A21 +B21)(A22 +B22) = (B1B2)2x2ex2fx30x1cx33x54x55cos2 fi(x2y2 +x2z2 +y2z2 +z4) = x2y2,x8x42
x43x24x33.
6,x58x24x26x23(5;0;0)x24x19x35x33x2 +y2 +z2 = 9x18x3cxcx23x24x33x54x55.
x4,x3bx18x3cxcx76x77x2ex3ax29x23x24x33xcx3fx24x1e x5dx55cos = 45,xx4dx8x23x24x33xcx4d,x28x29x4dxcx54x51x51
x56x8(1;0;0),x46x47x29x23x24x33xcx54x55x23
j(x?5;y;z)￠(1;0;0)jp
(x?5)2 +y2 +z2 =
4
5 ;
x5fx1fx23
9(x?5)2?16(y2 +z2) = 0:
7,x13x14,x3ax37x26x24x3cx3cx35x33
x2 +y2 +z2 +2ax+2by +2cz +d = 0; (0 < d < a2 +b2 +c2)
xcx1x1dx46xex59xcx23x24x33x54x55x23
d(x2 +y2 +z2) = (ax+by +cz)2:
x2x3,x40x3ax35x33xcx35x39x23(?a;?b;?c),x46x47x23x24x33xcx4dxcx54x51x51x56x8(a;b;c),x66x35x39x0x37x26xcx43
x44x8pa2 +b2 +c2,x35x3fx36x4dx3cpa2 +b2 +c2?d,x28x29x3ax37x26xcx3cx1dx2dx30x4dx3cpd,x3bx29x2ex3dcos =r
d
a2 +b2 +c2,x1fx20 x8x23x24x33xcx3fx24x1e,x46x47x46x58x23x24x33xcx54x55x23
jax+by +czjp
x2 +y2 +z2pa2 +b2 +c2 =
r d
a2 +b2 +c2 ;
x5fx1fx67x3d
d(x2 +y2 +z2) = (ax+by +cz)2:
8,x13x14ayz +bzx+cxy = 0x50x51x24x33,x63x3bx33x+y +z = 0x19x6dx24x33x25x3cx42x4ex1x1d,x7x1fx25x1ex23
,x24a+b+c = 0,x1b = …2,
x2x3,x28x23x1cx33ayz + bzx + cxy = 0x8x2ex3cx;y;zxcx2exfxexfx54x55,x46x47x8x47x37x26Ox23x24x26xc
x24x33,x2dx3bx33x + y + z = 0x1cx3ax37x26,x46x47x40x25x1dx2fx30x18x1cx33ayz + b(?y? z)z + c(?y? z)y = 0x38
a(?x?z)z +bxz +c(?x?z)x = 0xb,x24x3ax37x26.
x3bx7cx42x43x54x55x5fx1fx3d
cy2?(a?b?c)yz +bz2 = 0 (*)
￠ 13 ￠
x3bx7cx2ex43x54x55x5fx1fx3d
cx2?(b?a?c)xz +az2 = 0 (**)
x21x22x18x40x18x25x1x1dxbx68x52x42x26(x0;y0;z0)x38(x00;y00;z00),x1bx29x40x26xcx1fx20x2ax79x1cx5dx55(*)x38(**),x66x41xe
x5dx55(*)x2ex3d:
cy02?(a?b?c)y0z0 +bz02 = 0 x19 cy002?(a?b?c)y00z00 +bz002 = 0
x5bx4bx57x19x69x9x2ex69,x2ex3dy0y00 = bc z0z00,x45x6e,x3bx3cx41xex1cx5dx55(**),x2ex3dx0x00 = ac z0z00.
x28x29x40x1x1dxcx23x1e x5dx55
cos = x
0x00 +y0y00 +z0z00
px02 +y02 +z02px002 +y002 +z002 = a+b+ccpx02 +y02 +z02px002 +y002 +z002 = 0;
x74 = …2,
9,x13x14,px+py?pz = 0x50x51x42x43x3fx24x1e x23arccos
p6
3 xcx23x24x33.
x2x3,x37x54x55xcx31x56x52x1bx32x1dx42x5dx55x > 0,y > 0,z > 0,x37x54x55x332xfx3bx54x67x59x23
2(x2 +y2 +z2)?(x+y +z)2 = 0;
x78x79x8x43x24x33,x2dx24x41x19x37x54x55x18x > 0,y > 0,z > 0x52x42x34xax48x8x45x25xc,x28x29x37x1cx33x8x24x33(x2ax79
x47x37x26x23x24x26,x24x35x62x42x43x54x51).
x64x36x3ax37x26xc2x5ex1x1d:
L, x1 = y1 = z1 ; L0, xm = yn = zs ;
x1x1dL0x74x1x1dLx73x17x3dx0x23x24x33xcx54x55x238
>>>>
>>><
>>>
>>>>
:
x0 = mt;
y0 = nt;
z0 = st;
(x?x0)+(y?y0)+(z?z0) = 0;
x2 +y2 +z2 = x02 +y02 +z02;
x79x5ax62x9x0;y0;z0;tx67x3dx0
x2 +y2 +z2 = m
2 +n2 +s2
(m+n+s)2 (x+y +z)
2:
x2a m2 +n2 +s2(m+n+s)2 = 12 x2b,x52x43x23x24x33xcx54x55x19x40x3ax24x33xcx54x55x45x25,x2dx23x24x33xcx3fx24x1e x34x8L0x19L
xcx23x1e,x74
cos = (1;1;1)￠(m;n;s)pm2 +n2 +s2p3 =
p2
p3 =
p6
3,
x0 x1 6–6
1,x40x3ax21x35x33xcx4ex4fx4dx19x1fx20x4dx21x72,x24x69x3ax21x23
( x2
4 +
y2
9 = 1
z = 0
x38x26A(1;2;?p11),x58x21x35x33x54x55.
x4,x78x79x29x21x35x33xcx54x55x23
x2
4 +
y2
9 +
z2
c2 = 1;
￠ 14 ￠
x47x26Axcx1fx20x4ax4b,x25x3dc = 6,x46x47x21x35x33xcx54x55x23
x2
4 +
y2
9 +
z2
36 = 1:
2,x40x3ax24x26x23x37x26,x4ex4fx33x23xOyx33x38zOxx33,x24x3ax26A
1
2 ;?1;2
·
x38B
5
2 ;3;?2
·
,x58x21x23x75
x76x33xcx54x55.
x4,x29x21x23x75x76x33xcx54x55x42x23
y2
b2 +
z2
c2 = 2x;
x3x47A;Bx26xcx1fx20x4ax4b,x25x3db = p2,c = 2p2,x28x29x46x58x21x23x75x76x33xcx54x55x23
y2
2 +
z2
8 = 2x:
3,x58x42x43x2exfx1cx33xcx54x55,x1ax52x43x2exfx1cx33x69x3ax40x5ex75x76x1d(
x2?6y = 0
z = 0 x38
( z2 +4y = 0
x = 0:
x4,x52x43x2exfx1cx33x2ex60x8x21x23x75x76x33x6fx27x1cx75x76x33,x57x15x40x3ax5ex5f,x41xcx54x55x37x62x47x33x2x12:
x2
a +
z2
b = 2y;
x47z = 0x4ax4b,x58x3da = 3,x47x = 0x4ax4b,x58x3db =?2,x28x29x52x8x27x1cx75x1cx33,x1fx54x55x23
x2
3?
z2
2 = 2y:
4,x2akx52x68x7fx70x9x1bx2b,x54x55
(k?3)x2 +y2 = (k +3)z
x50x51x2cx27x1cx33?
x4,x2ak <?3x6f?3 < x < 3x2b,x54x55x50x51x27x1cx75x76x33; x2ak =?3x2b,x54x55x50x51x40x43x18x25x3bx33; x2a
k = 3x2b,x54x55x50x51x75x76x7x33;x2ak > 3x2b,x54x55x50x51x21x23x75x76x33.
5,x40x3ax21x23x75x76x33Ax2 +By2 = 2zx69x3ax23(
x2 +y2 +z2 = 2x+2z;
x = z;
x28x58x1fx54x55.
x4,x23x54x55x2ex47x45x25x31x2x23 (
2x2 +y2 = 4z;
x = z;
x2dx37x1cx33x1ax3bx33x = zx1bx3dxcx1cx1dx8 (
Ax2 +By2 = 2z;
x = z:
x3ex3fx67x74x3dA = 1,B = 12,x28x29x21x23x75x76x33xcx54x55x23
x2 + y
2
2 = 2z:
6,x40x3ax21x23x75x76x33Ax2 +By2 = 2zx69x3ax23(
x2 +y2 +z2 = 2x+2z;
x = z;
x28x58x1fx54x55.
￠ 15 ￠
x4,x52x23xbx40x43x26(1;p2;1),(2;0;2),x41xex1cx18x1cx33xb,x49x41xexcx1fx20x4ax4bx1cx33x54x55,x25x3dA = 1,
B = 12, x46x47x21x23x75x76x33xcx54x55x23x2 + y22 = 2z.
7,x40x3ax21x23x75x76x33x2 + y22 = 2zx38x3bx33x = kzxcx25x1dx8x42x43x23,x28x58x29x23xcx3fx36.
x4,x29x25x1dx1cx2ex50x51x23 (?
x? 1k
·2
+ y22 = 1k2 ;
x = kz:
x52xdx7cx42x43x54x55x8x25x1dx0xOyx3bx33xcx2bx2cx7x33,x41x8x42x43x21x23x7x33,x1fx4ex4fx4dx8x1x1d
( x = 1
k ;
y = 0,x52x5e
x1x1dx19x3bx33x = kzxcx25x26
1
k ;0;
1
k2
·
x34x8x25x1dx23xcx23x39,x12x18xdxex45x2ex49x25x1dxcx54x55x45x25x31x2x23
(?
x? 1k
·2
+y2 +k2
z? 1k2
·2
= 2k2 ;
x = kz:
x66x52x43x54x55x22x2ex47x4fx30,x5dx1ax25x1dx59x23x42x43x47
1
k ;0;
1
k2
·
x23x23x39xcx23,x5cx38x24x35x38k2 = 1,x52x2bx23xcx3f
x36x4dx3cp2.
8,x28x17x13x21x23x75x76x33x19x27x1cx75x76x33xcx62x9x54x55x2ex21x22x39x23
8>
<
>:
x = a(u+v);
y = b(u?v);
z = u2 +v2
x19
8>
<
>:
x = a(u+v);
y = b(u?v);
z = 2uv;
(u;vx23x62x9).
x2x3,x21x23x75x76x33xcx54x55x8 x2a2 + y2b2 = 2z,x53x31x56x3ax6
( x = a(u+v);
y = b(u?v);
x2cx67x4fx30x52x43x31x6x8x2ex3bxc,x4ax4bx54x55x67x2ex3dz = u2 +v2,x3bx3cx1cx33xbxcx26x1ax;yxcx1bx3cx42x50x6d,x28x29
x1cx1au;vx3cx42x50x6d,x63x14x52x50x8x1cx33xcx62x9x54x55.x49x5cx25x2ex47x3dx0x27x1cx75x76x33xcx62x9x54x55.
9,x40x3ax42x75x76x1d
( x2 = 2z;
y = 0 x3bx35x3dxa,x24x24x26x18x75x76x1d
( y2 =?4z;
x = 0 xb,x28x58x1fxbxcx54x55.
x4,x28x23x75x76x1d
( y2 =?4z;
x = 0 x18yOzx3bx33xb,x7x75x76x1d
( x2 = 2z;
y = 0 x3bx35x3dxax2dx67x3ex30xcx1cx33x23
S,x7x52x1fxbx42x26(X;Y;Z),x1bx3bx33y = Y x19x1cx33Sxcx1bx20x42x19x75x76x1d
( x2 = 2z;
y = 0 x18x5c,x35x8x24x26x18
0;Y;?Y 24
x65,x28x29x26(X;Y;Z)x42x6dx5dx55x54x55
X2 = 2
Z?
Y
2
4
;
x74X2? Y 22 = 2Z,x46x47xbxcx54x55x23x2? y22 = 2z,x8x27x1cx75x76x33.
10,x13x14x21x23x75x76x33xbx3fx1x1dx17x18.
x2x3,x7x62x1x1d 8
><
>:
x = x0 +lt;
y = y0 +mt;
z = z0 +nt;
￠ 16 ￠
x41x5dx30x18x21x23x75x76x33 x2a2 + y2b2 = 2zxbx2ax24x40x2a
(x0 +lt)2
a2 +
(y0 +mt)2
b2 = 2(z0 +nt) (*)
x4ex7x8xctx59x11,x71xbx12x8x2ex3ctxcx2exfx54x55,x24t2xcx69x9x8 l2a2 + m2b2,x63x41x2cx4dx3c0,x1b(*)x12x2cx2ex60
x4ex7x8xctx59x11,x53x35x62l = m = 0,x52x2btxcx42xfx11x69x9x23n,x1cx35x60x80,x2cx2ex60.
x0 x1 6–7
1,x58x27x1cx75x76x33x2?y2 = zxbx3ax26(1;?1;0)xcx40x5ex1xfx1dx54x55x47x2dx41xexcx25x1efi.
x4,x7x1fx40x6fx1xfx1dx23 (
x+y =?;
xy = z x38
( x?y =,;
”x+”y = z;
x41xex69x3ax26(1;?1;0),x46x47? = 0,” = 2,x2fx3dx1xfx1dx54x55x23
x
1 =
y
1 =
z
0 x38
x?1
1 =
y +1
1 =
z
4 ;
x25x1ex5dx55cosfi = 0,x74fi = …2,
2,x13x14,x27x1cx75x76x33 x2a2? y2b2 = 2z (a 6= b)xbxcx41x18x29x25xcx1xfx1dxcx25x26xcxbxcx8x42x5ex27x1cx1d.
x2x3,x3bx3cx45x6fxcx7x8x40x5ex1xfx1dx42x8x43x33x1x1d,x44x62x25x26,x28x29x35x45x64x36x43x6fxcx1xfx1d,x2cxcx7(
x
a +
y
b = 2?;
x
a?
y
b
·
= z
x19
( x
a?
y
b = 2”;
”
x
a +
y
b
·
= z
x18x41x54x1,x40x1x1dxcx54x51x51x56x21x22x23
1b ; 1a ;? 2?ab
·
x38
1
b ;
1
a ;
2”
ab
·
,x28x41xex54x1,x46x474?” = a2?b2.
x2a? 6=,x2b,x46x39x25x3dx25x26x5dx558
<
:
x2
a2?
y2
b2 = 4?” = a
2?b2;
z = 2?” = a2?b22 ;
x2a? =,x2b,x25x26x5dx55 8
><
>:
x = 2a?;
z = 2?2;
y = 0:
x2cx10x47x42x7fx1x2,x1dx63x14x25x26x18x27x1cx1d8
<
:
x2
a2?
y2
b2 = a
2?b2;
z = a2?b22 ;
x19
( x2 = 2a2z;
z = 0;
xb.
3,x28x13:
2x2 +y2?z2 +3xy +xz?6z = 0
x8x1x48x33,x19x58x30x1fxbx3ax26M(1;1;1)xcx1xfx1dx54x55.
x4,x37x54x55x2ex5fx23
(x+y +z)(2x+y?z) = 6z;
x1bx1fx1xfx1dx54x55x23 (
(x+y +z) = 6z;
2x+y?z =? x6f
(?(2x+y?z) = 6z;
x+y +z =?;
￠ 17 ￠
x1fx20?;?x23x62x9,x45x28x2cx10?;?x52x1ax1b,x42x62
rank
6
2 1?1
= 2; rank
2 6
1 1 1
= 2;
x63x14x41xex50x70x50x51x40x43x1x1dx6f,x46x47x1cx33x8x1x48x33.
x64x36x3ax26(1;1;1)xcx1xfx1d,x49x1fx20x4ax4bxfx1dx54x55,x2ex3d? = 2,? = 3,x28x29x40x5ex1xfx1dx23(
x+y?2z = 0;
2x+y?z?2 = 0 x38
( 2x+y?3z = 0;
x+y +z?3 = 0:
4,x58x19x56x1x1d
L1,
( y?1 = 0;
x+2z = 0; L2,
( y?z = 0;
x?2 = 0; L3,
x
2 =
y +1
0 =
z
1
x18x25xcxax1x1dx2axexcx1cx33x54x55.
x4,x7xax1x1dx19L1;L3xcx25x26x23(?2u;1;u)x19(2v;?1;v)(x1fx20u;vx23x62x9),x46x47xax1x1dxcx54x55x23
x+2u
2(u+v) =
y?1
2 =
z?u
v?u, (*)
x41x42x19x1x1dL2x18x25,x3bx3cx26(2;0;0)x18L2xb,x24L2xcx54x51x51x56x23(0;1;1),x28x29x57x15x1x1dx18x25xcx5ex5f,
x5cx38 fl
flfl
flfl
fl
0 2(u+v)?2(1+u)
1?2 1
1 v?u u
flfl
flfl
flfl = 0;
x25x3duv =?1,x2dx66(*)x12x2ex47x3dx0 8
<
:
u = 2z?x2(1+y) ;
v =?x?2z2(y?1),
x28x29xax1x1dx46x3ex3axcx1cx33x54x55x23
x2
4 +y
2?z2 = 1:
5,x13x14x49x617.3.
x2x3,x7x8x9x27x1cx33xcx54x55x23 x2a2 + y2b2? z2c2 = 1,x41xcx40x5ex43x6fx1xfx1dx23
(?
0
x
a +
z
c
·
+”0
1+ yb
·
= 0;
0
1? yb
·
+”0
x
a?
z
c
·
= 0
x2d
(?
1
x
a +
z
c
·
+”1
1? yb
·
= 0;
1
1+ yb
·
+”1
x
a?
z
c
·
= 0:
x4ax39x59x42x7ax54x55(
0
a x+
”0
b y +
0
c z +”0 = 0;”
0a x0b y?,0c z +?0 = 0
x2d
(?1
a x?
”1
b y +
1
c z +”1 = 0;”
1a x+?1b y?,1c z +?1 = 0:
x5bx4bx7cx32x4bx3x4cx4dx2ex3cx4bx42x7ax54x55x50x51xcx40x5ex1x1dxcx18x2ex4cx4excx75x10,x3bx3cxbx4fx50x11x1dx76x54x55x22xcx51
x52x53x54xcx35x34x12 fl
flfl
flfl
flfl
flfl
0
a
”0
b
0
c?”0”
0a0b?”0c0
1
a?
”1
b
1
c?”1”
1a?1b?”1c1
flfl
flfl
flfl
flfl
fl
= 0;
x63x14x51x52x53x54xcx55< 4,x28x29x52x40x5ex1xfx1dx42x6dxbx33.
x7x27x1cx75x76x33xcx54x55x23 x2a2? y2b2 = 2z,x41xcx40x5ex43x6fx1xfx1dx23
(?x
a +
y
b
·
+2? = 0;
x
a?
y
b
·
+z = 0
x2d
(,?x
a +
y
b
·
+z = 0;
x
a?
y
b
·
+2” = 0:
￠ 18 ￠
x3bxbx4f4x43x54x55x50x11x3dx0xcx1dx76x54x55x22xcx51x52x53x54x23
A =
0
BB
BB
@
1
a
1
b 0?2
a?
b 1 0”
a
”
b 1 01
a?
1
b 0?2”
1
CC
CC
A;
x31x32x3dj?Aj = 0,x63x14rank?A < 4,x11x42x54x33x2cx67x4fx30rankA = 3 = rank?A,x63x14x52x40x5ex1xfx1dx18x25.
6,x13x14x49x617.4.
x2x3,x7x8x9x27x1cx33xcx54x55x23 x2a2 + y2b2? z2c2 = 1,x41xcx40x5ex45x6fx1xfx1dx23
(?
0
x
a +
z
c
·
+”0
1+ yb
·
= 0;
0
1? yb
·
+”0
x
a?
z
c
·
= 0
x2d
(?
1
x
a +
z
c
·
+”1
1+ yb
·
= 0;
1
1? yb
·
+”1
x
a?
z
c
·
= 0:
x3bxbx4f4x43x54x55x50x11x3dx0xcx1dx76x54x55x22xcx51x52x53x54xcx35x34x12x23
flfl
flfl
flfl
flfl
fl
0
a
”0
b
0
c?”0”
0a0b?”0c0
1
a
”1
b
1
c?”1”
1a1b?”1c1
flfl
flfl
flfl
flfl
fl
= 4(?0”1?”0?1)
2
abc,
x2ax52x40x5ex1xfx1dx2cx21x72x2b,x42x6dx62?0,,0 6=?1,,1,x28x29x51x52x53x54xcx55x4dx3c4,x52x40x5ex1x1dx43x33.
x7x27x1cx75x76x33xcx54x55x23 x2a2? y2b2 = 2z,x41xcx40x5ex45x6fx1xfx1dx23
(?x
a +
y
b
·
+2?0 = 0;
0
x
a?
y
b
·
+z = 0
x2d
(?x
a +
y
b
·
+2?1 = 0;
1
x
a?
y
b
·
+z = 0;
x3bxbx4f4x43x54x55x50x11x3dx0xcx1dx76x54x55x22xcx51x52x53x54xcx35x34x12x23fl
flfl
flfl
flfl
flfl
fl
1
a
1
b 0?2?0?
0
a?
0
b 1 01
a
1
b 0?2?1?
1
a?
1
b 1 0
flfl
flfl
flfl
flfl
flfl
= 4(?01)
2
ab,
x2a?0 6=?1x2b,x51x52x53x54xcx35x34x12x2cx4dx3c,x28x29x41xcx55x4dx3c4,x63x14x52x40x5ex1xfx1dx43x33.
x4ex3cx27x1cx75x76x33xcx42x6fx1xfx1d
(?x
a +
y
b
·
+2? = 0;
x
a?
y
b
·
+z = 0;
x41xcx54x51x51x56x23
1
a ;
1
b ;0
·
￡
a ;?
b ;1
·
=
1
b ;?
1
a ;?
2?
ab
·
,x46x47x52x6fx1x1dx1dx3bx35x3cx3bx33bx+ay = 0.
x45x6ex2ex3a,x11x42x6fx1xfx1d (
”
x
a +
y
b
·
+z = 0;
x
a?
y
b
·
+2” = 0
x1dx3bx35x3cx3bx33bx?ay = 0.
7,x58x8x9x27x1cx33 x2
a2 +
y2
b2?
z2
c2 = 1 (c
2 < a2 +b2)xbxcx41x18x54x1xcx1xfx1dxcx25x26xbxc.
x4,x78x79,x18x41x54x1xcx1xfx1dx5cx2cx45x6f,x7x23
( !?x
a +
z
c
·
=?
1+ yb
·;
x
a?
z
c
·
= !
1? yb
· x38
(?x
a +
z
c
·
=,
1? yb
·;
”
x
a?
z
c
·
=
1+ yb
·;
x1fx20,!;?x2cx45x2bx23x7c,;”x1cx2cx45x2bx23x7c,x52x40x5ex1x1dxcx54x51x21x22x23
2?!2
bc ;
2?!
ac ;
2 +!2
ab
x38
￠ 19 ￠
2?”2
bc ;
2”
ac ;?
”2 + 2
ab
,x28x41xex54x1,x34x62
(?2?!2)( 2?”2)
b2c2 +
4?!”
a2c2?
(?2 +!2)(”2 + 2)
a2b2 = 0:
x5fx1fx67x3d
a2(! +?”)2 +b2(!”)2 +c2(?”?! )2 = (a2 +b2?c2)(? +!”)2:
x3x66x54x55x22x25x3dx25x26x23 8
>>>>
<
>>>>
:
x =?” +!? +!” a;
y = !” +!” b;
z =?”?!? +!” c:
x2ex3ax2 +y2 +z2 = a2 +b2?c2,x53x25x26xbxcx23(
x2
a2 +
y2
b2?
z2
c2 = 1;
x2 +y2 +z2 = a2 +b2?c2:
x0 x1 6–8
1,x39x30x35x33x2 +y2 +z2 = R2x19
(1)x7x33x2 +y2 = a2 (R > a > 0);
(2)x24x33x2 +y2 = z2 tan2 (0 < < …)
x25x1dxcx62x9x54x55.
x4,(1)
8>
<
>:
x = acos ;
y = asin ;
z = pR2?a2;
(0 6 < 2…)x38
8>
<
>:
x = acos ;
y = asin ;
z =?pR2?a2;
(0 6 < 2…);
(2)
8>
<
>:
x = Rsin cos’;
y = Rsin sin’;
z = Rcos ;
(0 6 ’ < 2…)x38
8>
<
>:
x = Rsin cos’;
y = Rsin sin’;
z =?Rcos ;
(0 6 ’ < 2…).
2,x28x50x6dmx23x1ax1bx2bx3bx33x+mz?1 = 0x19x8x9x27x1cx33x2 +y2?z2 = 1x18x25x59,(1)x21x23; (2)x27
x1cx1d.
x4,x2am = 0x2bx25x1dx8x42x4ex18x25x1x1d (
y2?z2 = 0;
x = 1:
x28x29x7m 6= 0,x29x2bx25x1dx23 (
x2 +y2? 1m2 (1?x)2 = 1;
x+mz = 1;
x2ex5fx1fx23 8
><
>:
x2
m4
(m2?1)2
+ y2m2
m2?1
= 1;
x+mz = 1:
x28x29,x2ajmj > 1x2b,x25x1dx8x21x23;x2ajmj < 1x24m 6= 0x2b,x25x1dx8x27x1cx1d.
3,x28x2fx30x3bx35x33x2 +y2 +z2 = 4x19x7x33x2 +y2 = 2xx46x1dx59xcx34xa.
￠ 20 ￠
x50x4cx48x46x44x51x40x3ex3cx3ax4cx4bx4ax32x1bx1cx45x1ax41x17x3cx39
x13x34x31x32x35
x38x3bx3ex19x1ax44
x1cx47x48x34x38x3ax3cx3ex50x42x44x46x54x55x4cx58x59x25x5bx5ex29x63x2bx69
x6cx6ex6fx6cx6a
x66x63x28x5dx5bx25
x6fx2ex6ax68x64x28
x5ex5cx25x59x58x4cx55x54x46
x44x42x40x4fx3cx3ax4cx36x32x47x1bx45x1ax41
x17x3ax39x34x32x1
x10x30x74
x74
x10x30x74
x74
x7dx6dx6bx72
x66x65x63x61x28
x5ex5dx56x5ax59x50
x57x4cx4bx4ax48x46x46x44x43x42x41x40x3ex3ex3dx3bx3bx39x37x36x35x33x31x2ex2ax44x43
x40x3fx3dx3bx39
x38x36x35x34x2
x7fx7ex7b
x6dx78x76
x74x6ax70
x6e
x0x31x4
x5x8x9
xcxex10
x39x3
x31x34x36x37x3ax3cx3dx40x41x43x45x47x49x4bx4dx50x52x56x5cx5dx5e
x5fx60x64x68
x65x66x67x70
x72x74x6b
x78x79x7c
x7dx7f
x1fx3ax38x36x34x4ax30x2e
x46x45x44x26x41
x40x3fx1ax16
x3bx11x39xe
xcx35x8
x7x5x3
x1
x13x18x1ax5x46x5x1bx2ex1bx47
x2dx2cx2cx2a
x29x28x26
x24x41x20
x3fx3ex3dx18
x3bx14x39x38
x37xcx8
x33x32x31xf
x7fx7dx7b
x79x77x75
x73x71x6e
x6dx6ax63
x66x62x61x60
x5fx5cx58x57
x5ax52x50x4ex4ax48
x21x51x20x1fx7x6x7x5x4x2x1
x74
x0x30x10
x6x2dx2ex2dx2e
x2d x52x53x52x53x52x53
xa
x7ax2fxfx1x11x6
x25x5x34x1dx7 x9x23x4cxbx5b
x23 x56x22 x53x22 x52x20 x50x20 x4fx1e x4dx1e x4ax1d x47x1a x18x3e
x35x12x2dx6d
x29x63x25x5ax23x56x22x54
x21x52x20x51x1fx4fx1ex4ex1dx4cx1cx48
x19x44x14x39
x10x30 x68x2b x5d
x26
x2ex6fx2ax66
x25x5cx23x57x22x55x21x52x20x51x1fx4fx1fx4dx1dx4cx1cx49x44x1bx14x3b
x1x32
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x0x10
x6fx10x2dx6b
x2bx66x28x61
x26x5cx25x59x24x57x23x56
x22x55x22x53x22x52x21x52x21x50x20x50x1fx50x1fx4fx1ex4ex1ex4dx1ex4bx1dx4bx1dx49x1cx47
x1ax45x40x1a
x15x3cx13x37
x10x33
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x10x30
x1bx47x1cx47x1cx46
x1bx47x1bx47x1cx46
x1bx47x1bx47
x59x25 x59x25 x5ax24
x59x25 x59x25 x59x24
x5ax24 x59x25 x24xb
x6fx10x2dx6b
xcxdx64x24x5ax21x55x20x51x1fx4ex1dx4cx1cx49x1bx45x19x42
x16x3ex15x39
x14x37x12x35
x11x33x11x31
x0
x2ax2cx2fx5bx27x24x59
x24x57x23x56x22x56x22x55x22x55
x22x55x22x54x22x54x55x22
O
x y
z
x03x1x2
x9x54x48x4ax4bx4dx4ex4f
x51x53x54x56
x58x5dx5cx5f
x60x62x66
x68x6ax66
x6ex71x72
x6ax76x78
x7bx7cx7f
x0x2x1
x3x3x3
x4x4x5
x5x6x6
x6x7x7
x8x8x9
x9x9xa
xaxbxc
xcxdxd
xfx10x14x2
x46x46x44x44x42x41x41x3fx3fx3dx3dx3bx3bx39x39x36x36x34x33x31x2f
x2dx2ax44x43x42
x40x3fx3ex3cx3b
x3ax14x3
x70x30
x10
x2x11x11
x47x47x47x22x9
x7ax2fxfx1x11x6
xcx13x1x1bx3
x8x20x40x9x23x4c
x5cx5ex5fx62
x2ax62x68x6a
x6bx6dx6d
x6fx6fx70
x71x71x72
x72x73x73
x73x74x74
x74x75x75
x74x76x75
x6bxf
x64x6cx6e
x70x72x72
x73x74x74
x75x76x76
x76x76x77
x77x78x77
x78x78x78
x79x78x79
x78x79x79
x7ax79xfx35x34x32x31x47x6
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x10x30
x10x30x64x2a x63
x2a x64x2a
x64x29 x64x2a
x64x2a x64x2a
x64x2a xcxe
x5cx26 x5cx26 x5c
x26 x5cx26 x5cx26
x5cx26 x5cx26 x5c
x26 x5cx26 x5c
x39x14 x38x15
x39x14 x39x14
x38x15 x38x14
x39x14 x39x14
x38x15 x53x22 x54x21 x53x22 x54x22 x54x21 x53x22 x54x22 x53x22 x53x22 x54x21
x53x22 x54x22 x54x21 x9
O
x
y
z
x04x1x2
4,x28x2fx30x3bx1cx33z = 1?x2?y2x38x3bx33y = x,y = p3x,z = 0x46x1dx34xax18x7cx42x46x47x20xcx26x5ex21.
5,x28x2fx30x2cx4dx12
x2 +y2 6 z 6
p
x2 +y2
x46x50x6dxcx34xaxcx1fx56.
x44x42x42x41x41x3fx3fx3ex3ex3cx3cx3ax3ax38x36x35x49x48x1cx1bx45
x1ax19x17x3bx13x33x10x11
x37x15x17x18x1ax44x1cx1bx1cx48
x34x36x37x39x3ax3bx3dx3cx3ex3fx3fx40x41x42x42x44x44x45x46x48x49x4bx23x56x24x58x25x5axbx5ex29x2ax69
x2ex6fx2ex2cx65x28x27
x26xbx5ax58x24x23x57x56x4ax48x47x46
x45x21
x2fx2dx2cx29xcx27x5bx24x57x4c
x55x48x46x53x52x43x42x41x40x3fx3ex3dx3bx3ax38x4bx34x48x46x1ax5x4x17x15x13x12
x2cx2dx2fxax24xbxaxbx23x56x22x54
x21x53x20x52x20x51x20x50x20x4fx1fx4ex1ex4ex1ex4cx6x7x49x18x44
x1
x52x20 x51x20 x4fx20 x4ex1f x4dx1e x4ax1d x5x18x3x14
xax26xcx29x23x56x22x53x21x52x20x51x20x4fx1fx4ex1ex4dx1cx4ax1x2x17x18x2ax5c
x56x23 x53x22
x6ex6ex6d
x6cx6ax6a
x64x66x63x62
x60x5fx5ex5dx56
x52x4ex55x54x46x51x50x4fx3ax4cx4bx32x2ex2ax43x42x41x20
x3ex1ax3dx3c
x16x16x14
x3ax12x12
x39
x78x78x78
x77x76x74
x73x69x2cx66
xdx2ax29x26x25x23x20 x31x47x46x45x43x42x41
x40x3ex17x1ax18
x16x15x13
x13x11x15x2
x19x24x23x24
x44x44x44x44x44x44x44x44x52x9
x74
x30x10
x16x3x2ax5 x8x20x40
x9x23x4c
x7ax2fxfx1x11x6
x11x33x11x33
x11x33x11x33
x12x33x12x33
x12x34x35x12
x13x34x1x2x35
x14x36x3ax14
x7x1ex1b
x68x2d x68x2c
x67x2c x67x2b
x66x2b x65xexd
x64x2a x62x29
x5fx27 x5cx25
x57x24 x54x22 x51x1f x4dx1e
x67x2c x67x2c
x67x2c x67x2c
x67x2c x67x2c
x35x12 x34x12
x35x12 x34x12
x35x12 x34x12
x35x12 x34x12
x35x12 x34x12
x35x12 x34x12x62x29 x62x28
x62x29 x62x29
x62x28 x62x29
x62x29 x62x28
x62x29 x62x29
xcx42x4 x42x19
x42x19 x41x19
x42x19 x42x19
x42x18 x42x19
x42x19
x30x10x30x10
x30x10x30x10
x30x10x30x10
x
y
z
O
x05x1x2
x6e
x3x2x11x11
x10x10x10
xfxexd
xcxbxa
x8x6x31
x11x30
x3x4x17x1ex1e
x1dx1ex1d
x1ex1ex1d
x1ex1ex1d
x1ex1dx1e
x42x42x42x42x43x42x42x42x8x1ex39x3ax39 x6cx6b
x70x0
x7ax2fxfx1x11x6
x34x1dx7x40x20x8 xdx65xfx2dx74
x7bx7bx7b
x7bx7bx7b
x7bx7bx7b
x7bx7bx7b
x7bx7bx7b
x6dx42x42x42x43x42x42x42x42
xex4x15x2x0x47x1b
x1dx48x1dx4ax4dx1e x4dx1e x4dx1e x4dx1e x4dx1e x4dx1e x4dx1e x4dx1e x4cx1f
x66x2b x66x2b
x65xexd
x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10x30x10
x
y
z
O
x06x1x2
6,x28x2fx30x3bx2cx4dx12x22
y > 0; z > 0; x > 2y2; x4 + y2 + z4 6 1;
x46x50x6dxcx34xaxcx1fx56.
￠ 21 ￠
