Аналитическая геометрия. Часть III. Многомерные пространства. Гиперповерхности второго порядка. Шурыгин В.В. - 57 стр.

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a
0
33
6= 0
x
i
=
p
|a
0
33
|x
i
0
a
0
33
1
◦
. (x
1
)
2
+ (x
2
)
2
= 1
2
◦
. (x
1
)
2
+ (x
2
)
2
= −1
3
◦
. (x
1
)
2
+ (x
2
)
2
= 0
3
◦
x
1
= ±
√
−1x
2
O(0, 0)
II ϕ
I
4
◦
. (x
1
)
2
− (x
2
)
2
= 1
5
◦
. (x
1
)
2
− (x
2
)
2
= 0
5
◦
x
1
= ±x
2
III ϕ
ϕ
(v
1
)
2
a
11
(x
1
)
2
+ 2a
13
x
1
+ 2a
23
x
2
+ a
33
= 0 ⇐⇒
(x
1
+ a
13
)
2
+ 2(a
23
x
2
+
1
2
a
33
−
1
2
(a
13
)
2
) = 0.
a
23
6= 0
x
1
0
= x
1
+ a
13
, x
2
0
= −(a
23
x
2
+
1
2
a
33
−
1
2
(a
13
)
2
),
6
◦
. (x
1
)
2
= 2x
2
a
23
= 0
x
1
0
= x
1
+ a
13
, x
2
0
= x
2
,
/60 a0 6= 0  7- -/2ˆ*/7B0: .3*-+35E-B5,0* 1--3D0,57 xi = p|a0 |xi 0   0


E57*: .-D*60: .3*-+35E-B5,,-* 235B,*,0* 130B-” ,5 a033 < 3*E26>757*  :-
        33                                                        33


927 .-62)07>/8 /6*D2‡ˆ0* 730 235B,*,08 •‰730C0 2 0,D*1/-B -.2/15*:Œ
38D-: / 235B,*,08:0 215E5,A ,5EB5,08 130BAC  E5D5B5*:AC ¨70:0 235B
,*,08:0‹~
                                    ‹
   1◦ . (x1 )2 + (x2 )2 = 1 •¨660./ <
                                             ‹
   2◦ . (x1 )2 + (x2 )2 = −1 •:,0:A” ¨660./ <
                                                                ‹
   3◦ . (x1 )2 + (x2 )2 = 0 •.535 :,0:AC .*3*/*15‡ˆ0C/8 .38:AC <
   @,0:A* .38:A* B /62)5* 3◦ 0:*‡7 235B,*,08 x1 = ±√−1x2  -,0 .*3*
/*15‡7/8 B B*ˆ*/7B*,,-” 7-)1* O(0, 0) <
                                       ) „
   II < 30BA*  D68 1-7-3AC 1B5D3570 ,58 -3:5 ϕ .30B-D07/8 1 B0D2 ?? 
,5EAB5‡7/8 vfakcda xaemf|goasmhvgxg iaej<
   ’*3*,-/8 B ¨7-: /62)5* ,5)56- 1--3D0,57 B ½*,73 130B-” 0 -/2ˆ*/7B688
D56** .3*-+35E-B5,08 1--3D0,57 5,56-90),A* /62)5‡ I  .30B*D*: 235B,*
,0* 130B-” 1 -D,-:2 0E /6*D2‡ˆ0C B0D-B~
                                       ‹
   4◦ . (x1 )2 − (x2 )2 = 1 •90.*3+-65 <
                                                                      ‹
   5◦ . (x1 )2 − (x2 )2 = 0 •.535 B*ˆ*/7B*,,AC .*3*/*15‡ˆ0C/8 .38:AC <
   ’*3*/*15‡ˆ0*/8 .38:A* B /62)5* 5◦ 0:*‡7 235B,*,08 x1 = ±x2 <
                                       ) „
   III < 30BA*  D68 1-7-3AC 1B5D3570 ,58 -3:5 ϕ .30B-D07/8 1 B0D2 ??? 
,5EAB5‡7/8 vfakcda ejfj|goasmhvgxg iaej<
   ’-/6* 7-9-  151 BA+35, +5E0/  B 1-7-3-: 1B5D3570),58 „-3:5 ϕ 0:**7
15,-,0)*/10” B0D (v1)2  235B,*,0* 130B-” .30,0:5*7 B0D
                       a11 (x1 )2 + 2a13 x1 + 2a23 x2 + a33 = 0 ⇐⇒
                                              1      1
                     (x1 + a13 )2 + 2(a23 x2 + a33 − (a13 )2 ) = 0.
      ‹ /60              7- -/2ˆ*/7B0:       2      2
                                           .3*-+35E-B5,0* 1--3D0,57
  
             a23   6= 0 
                   0                   0            1     1
               x1 = x1 + a13 ,       x2 = −(a23 x2 + a33 − (a13 )2 ),
                                                    2     2
B 3*E26>757* )*9- 235B,*,0* .30B-D07/8 1 B0D2
                                  ‹
   6◦ . (x1 )2 = 2x2 •.535+-65 <
   ‹ /60
             a23 = 0  7- -/2ˆ*/7B0: .3*-+35E-B5,0* 1--3D0,57
                                 0                 0
                               x1 = x1 + a13 ,   x2 = x2 ,
                                           ÀÇ

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