BDZ_linal_matan
.pdf
|
æ |
2 |
-1 2 ö |
|
æ |
1 4 6 ö |
|||||||
|
ç |
5 -3 3 |
÷ |
|
ç |
|
|
|
|
÷ |
|||
|
ç |
÷ |
|
ç |
-3 -7 -7÷ |
||||||||
23) |
ç |
-1 |
0 |
-2 |
÷ |
24) |
ç |
4 |
8 |
|
7 |
÷ |
|
è |
ø . |
è |
|
ø . |
|||||||||
|
æ |
4 |
5 |
|
6 ö |
|
æ |
0 |
1 |
0ö |
|
||
|
ç |
|
|
|
|
÷ |
|
ç |
-4 4 0 |
÷ |
|
||
|
ç |
-5 -7 -9÷ |
|
ç |
÷ |
|
|||||||
25) |
ç |
2 |
3 |
|
4 |
÷ |
26) |
ç |
-2 |
1 |
2 |
÷ |
|
è |
|
ø . |
è |
ø . |
|||||||||
|
æ |
0 |
1 0ö |
|
|
æ |
1 -3 1 ö |
|
|||||
|
ç |
-4 4 0 |
÷ |
|
|
ç |
|
|
|
÷ |
|
||
|
ç |
÷ |
|
|
ç |
3 -3 -1÷ |
|
||||||
27) |
ç |
-2 |
1 |
2 |
÷ |
|
28) |
ç |
3 |
-5 |
|
÷ |
|
è |
ø . |
|
è |
1 ø . |
|||||||||
|
|
|
|
|
|
|
|
æ |
1 |
0 |
0 |
0ö |
|
|
æ |
1 |
-3 3 ö |
|
ç |
0 0 0 0 |
÷ |
||||||
|
|
ç |
÷ |
||||||||||
|
ç |
-2 -6 13 |
÷ |
|
ç |
1 0 0 0 |
÷ |
||||||
|
ç |
÷ |
|
||||||||||
29) |
ç |
-1 -4 |
|
8 |
÷ |
30) |
ç |
0 |
0 |
0 |
1 |
÷ |
|
è |
|
ø . |
è |
ø . |
9.8. Собственным значениям λ1 и λ2 линейного оператора соответствуют собственные
векторы h1 и h2 . Найти координаты образа вектора x в базисе h1 и h2 и в базисе, в
котором заданы координаты векторов h1 , h2 , x .
|
λ1 = 2, |
λ2 = 3, |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
1) |
h1 = (2;1), |
h2 = (1; 2), x = (4;-3). |
||||||||||||||||||||
|
λ1 = −2, λ2 = 5, |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||
2) |
|
h1 = (-2;1), h2 = (1;-2), x = (2; 3). |
||||||||||||||||||||
|
λ1 |
= −1, λ2 = 2, |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
x = (-1; 2). |
|||||
3) |
|
h1 = (3; 2), |
h2 = (2;1), |
|||||||||||||||||||
|
λ1 |
= 3, λ2 = −2, |
|
|
|
|
|
|
|
|
|
|
|
|
|
x = (5; 4). |
||||||
4) |
|
h1 = (-3; 2), h2 = (3;1), |
||||||||||||||||||||
|
λ1 = −3,λ2 = 1, |
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||
5) |
h1 = (2;1), |
h2 = (1;-2), x = (6; 5). |
||||||||||||||||||||
|
λ1 |
= 4, |
λ2 = −1, |
|
|
|
|
|
|
|
|
|
|
|
|
x = (2; 2). |
||||||
6) |
|
h1 = (1; 2), |
h2 = (-2;1), |
|||||||||||||||||||
|
λ1 |
= 1, |
λ2 = −3, |
|
|
|
|
= (3;1), |
|
|
|
|
||||||||||
7) |
|
h1 |
h2 = (1;1), x = (4; 5). |
|||||||||||||||||||
|
λ1 |
= 3, |
λ2 = 1, |
|
|
|
|
|
|
|||||||||||||
8) |
h1 = (2; 3), h2 = (-2;1), x = (4; 3). |
|||||||||||||||||||||
|
λ1 |
= −4, λ2 = 3, |
|
|
|
|
= (-2; 3), |
|
|
|||||||||||||
9) |
|
h1 |
h2 = (2;3), x = (5;-2). |
10)λ1 = 2, λ2 = 3, h1 = (1; 2), h2 = (-2;1), x = (7; 3).
|
λ1 = 4, λ2 = 5, |
|
|
11) |
h1 = (2;-3), h2 = (-3;1), x = (4; 3). |
12)λ1 = 1, λ2 = 2, h1 = (3;-1), h2 = (2;-1), x = (-5; 2).
13)λ1 = 2, λ2 = 4, h1 = (2; 3), h2 = (1;-2), x = (3; 7).
|
λ1 = 3, λ2 = 2, |
|
|
14) |
h1 = (-2; 3), h2 = (-1;3), x = (3; 6). |
||
|
61 |
PDF created with pdfFactory Pro trial version www.pdffactory.com
15)λ1 = 4, λ2 = 1, h1 = (1; 3), h2 = (2;1), x = (4; 3).
16)λ1 = 5, λ2 = 2, h1 = (4;1), h2 = (3;1), x = (7; 6).
|
λ1 = −1, λ2 = −2, |
|
|
|
|
17) |
h1 = (3;1), h2 = (2; 3), x = (4; 5). |
18)λ1 = 4, λ2 = 1, h1 = (3; 4), h2 = (1;1), x = (7;−6).
19) |
λ1 |
= −3,λ2 = 1, |
h1 = (4;−1), |
|
|
h2 = (2;1), x = (6; 4). |
||
|
λ1 |
= 2, λ2 =1, |
|
|
|
|
|
|
20) |
h1 = (3; 2), |
h2 = (1;1), x = (5; 3). |
21)λ1 = 4, λ2 = 3, h1 = (1;1), h2 = (−3;1), x = (4; 7).
|
λ1 = −5,λ2 = −1, |
|
|
22) |
h1 = (3; 4), h2 = (1; 2), x = (7;1). |
23)λ1 = 3, λ2 = 4, h1 = (5; 3), h2 = (2;1), x = (3; 5).
|
λ1 |
= −1,λ2 = 2, |
|
|
|
|
|
|
|
24) |
h1 = (2; 5), h2 = (1; 2), x = (7; 4). |
||||||||
|
λ1 |
= −4,λ2 = −3, |
|
|
|
|
|||
25) |
h1 = (3; 4), h2 = (1; 2), x = (3;1). |
26)λ1 = 2, λ2 = 5, h1 = (5; 3), h2 = (2;−1), x = (1; 2).
27)λ1 = 3, λ2 = −1, h1 = (−5; 3), h2 = (2; 3), x = (2;1).
28)λ1 = 5, λ2 = 4, h1 = (4; 3), h2 = (2; 3), x = (7;−2).
|
λ1 |
= 4, λ2 = 1, |
|
|
= (7; 4), |
|
|
|
|
x = (2; 5). |
29) |
h1 |
h2 = (2;1), |
||||||||
|
λ1 |
= 1, λ2 = 2, |
|
|
= (6; 5), |
|
|
x = (5; 6). |
||
30) |
h1 |
h2 = (1;1), |
10.Квадратичные формы
10.1.Выписать матрицу квадратичной формы и методом Лагранжа найти канонический вид квадратичной формы.
1)x1x2 + x1x3 + x1x4 + x2 x3 + x2 x4 + x3 x4 .
2)x12 + 3x32 − 2x1x2 + 2x1x3 − 6x2 x3 .
3)x12 + 2x22 + x42 + 4x1x2 + 4x1x3 + 2x1x4 + 2x2 x3 + 2x2 x4 + 2x3x4 .
4)x12 + x22 + 3x32 + 4x1x2 + 2x1x3 + 2x2 x3 .
5)x12 + 2x22 + x32 + 2x1x2 + 4x1x3 + 2x2 x3 .
6)x32 − 3x22 − 2x1x3 + 2x2 x3 − 6x2 x1 .
7) |
2x2 |
+ x2 |
+ x2 |
+ 4x x + 4x x + 2x x + 2x x + 2x x + 2x x |
|||||||||||
1 |
4 |
3 |
4 |
1 |
4 |
2 |
4 |
3 |
1 |
2 |
1 |
3 |
2 |
3 . |
62
PDF created with pdfFactory Pro trial version www.pdffactory.com
8)x22 − 3x12 − 2x2 x3 + 2x2 x1 − 6x1x3 .
9)x12 + 5x22 − 4x32 + 2x1x2 − 4x1x3 .
10)4x12 + x22 + x32 − 4x1x2 + 4x1x3 − 3x2 x3 .
11)x1x2 + x1x3 + x2 x3 .
12)2x12 +18x22 + 8x32 −12x1x2 + 8x1x3 − 27x2 x3 .
13)12x1x2 −12x12 − 3x22 −12x32 − 24x1x3 + 8x2 x3 .
14)x1x2 + x2 x3 + x3x4 + x4 x1 .
15)3x12 + 2x22 − x32 − 2x42 − 2x1x2 − 4x2 x3 + 2x2 x4 .
16)2x22 +18x32 + 8x12 −12x2 x3 + 8x1x2 − 27x1x3 .
17)x22 + 5x32 − 4x12 + 2x2 x3 − 4x1x2 .
18)2x32 +18x12 + 8x22 −12x3x1 − 8x2 x3 − 27x1x2 .
19)2x1x2 + 4x1x3 − x22 − 8x32 .
20)x22 + 5x32 − 4x12 + 2x2 x3 − 4x1x2 .
21)x1x2 .
22)2x12 + 3x22 + 4x32 − 2x1x2 + 4x1x3 − 3x2 x3 .
23)3x12 − 2x22 + 2x32 + 4x1x2 − 3x1x3 − x2 x3 .
24)x12 + 4x22 + 6x42 − 2x1x2 + 2x2 x3 − 2x3x4 .
25)3x12 − 8x22 + 2x32 + 8x1x2 − 3x1x3 − 2x2 x3 .
26)x12 − 5x32 − 4x12 + 2x2 x3 − 8x1x2 .
27)4x12 + 4x22 − 8x1x2 + x13 − x2 x3 .
28)x12 + 2x22 + x32 − x1x2 + 4x3x2 .
29)x12 + 4x22 − x1x2 + 4x2 x3 .
30)x22 + 5x12 − x32 + 2x2 x3 − 4x1x2 .
10.2. Определить тип квадратичной формы (положительно определенная, отрицательно определенная, неотрицательная, неположительная, знакопеременная).
1) |
x2 |
+ 2x2 |
+ 3x2 |
− 2x x |
2 |
− 4x |
2 |
x |
1 |
2 |
3 |
1 |
|
3 . |
63
PDF created with pdfFactory Pro trial version www.pdffactory.com
2)−2x12 − 5x22 − 6x32 − 4x1x2 + 4x1x3 + 4x2 x3 .
3)x12 + 3x22 + 2x1x2 + 5x32 − 2x1x3 − 4x2 x3 .
4)2x1x2 + 4x1x3 − 3x12 − 2x22 − 9x32 − 8x2 x3 .
5)2x12 + 5x22 + x32 + 6x1x2 + 2x1x3 + 4x2 x3 .
6)2x1x2 − 2x12 − 4x1x3 − x22 + 4x2 x3 − 4x32 .
7)3x12 + 2x1x2 + 2x22 − 2x1x3 − 2x2 x3 + x32 .
8)4x1x3 − 2x12 − x22 − 3x32 − 2x1x2 + 2x2 x3 .
9)x12 + 2x1x2 − 2x22 + 4x1x3 − 4x32 + 2x2 x3 .
10)2x1x3 − x12 − 4x1x2 + 2x2 x3 − 5x22 − 3x32 .
11)2x12 + 2x1x2 + x22 + 6x32 + 6x1x3 + 4x2 x3 .
12)x12 + x22 + 4x1x3 − 2x2 x3 + 6x32 .
13)x12 + 2x1x2 + 2x22 + 4x1x3 − 2x2 x3 −13x32 .
14)2x1x2 − x12 − x22 − 2x1x3 + 4x2 x3 − 2x32 .
15)4x12 − 2x1x2 + x22 + 4x1x3 + 2x2 x3 + 3x32 .
16)6x12 + 8x1x2 + 4x1x3 + x2 x3 + 4x22 + x32 .
17)2x1x2 − 5x12 − 4x1x3 + 2x2 x3 − 3x32 .
18)3x12 + 4x1x2 + x22 + 2x1x3 + 4x2 x3 + x32 .
19)4x1x2 − x12 − x22 + 4x1x3 − 4x2 x3 + 2x32 .
20)2x1x3 − 5x12 − 4x1x2 − x22 + 2x2 x3 − 3x32 .
21)2x1x2 − 3x12 − x22 + 4x1x3 + 2x2 x3 − 5,5x32 .
22)x12 + 4x1x2 + 2x22 − 8x1x2 − 4x2 x3 + x32 .
23)2x12 − 2x1x2 + 4x22 + 2x1x3 − 4x2 x3 + 3x32 .
24)3x12 + 2x1x2 + 2x22 − 4x2 x3 − x32 .
25)2x12 + 4x1x2 + 5x22 + 5x32 − 4x1x3 − 8x2 x3 .
26)x12 + 4x1x2 + 5x22 − 2x1x3 − 2x2 x3 + 2x32 .
64
PDF created with pdfFactory Pro trial version www.pdffactory.com
27)x12 + x22 + 6x1x2 + 2x2 x3 + 2x1x3 + 5x32 .
28)4x1x2 − x12 − 5x22 + 2x1x3 + 2x2 x3 −10x32 .
29)2x12 − 2x1x2 + 4x22 + 2x1x3 − 4x2 x3 + 3x32 .
30)x12 + 2x1x2 + x22 + 2x1x3 + 2x2 x3 + 3x32 .
11.Евклидовы пространства. Кривые и поверхности второго порядка
11.1. Используя процесс ортогонализации, перейти от базиса a, b, c к ортонормированному базису.
1) |
a = (3; 1; 2) , |
b |
= (1; 3; 1) , |
c (−1; 2; 4) . |
|
a = (1; 3; 0) , |
|
= (2; −1; 1) , |
c (1; −1; 2) . |
2) |
b |
|||
|
a = (2; 1; −1) , |
|
= (4; 3; 2) , |
c (1; −1; 1) . |
3) |
b |
|||
|
a = (4; 1; 1) , |
|
= (2; −1; − 3) , c ( −1; 2; −1) . |
|
4) |
b |
|||
|
a = ( − 2; 3; 1) , |
|
= (1; 3; −1) , |
c (2; 4; 1) . |
5) |
b |
|||
|
a = (1; 2; −1) , |
|
= (5; 1; 1) , |
c (2; −1; 3) . |
6) |
b |
|||
|
a = (3; 2;1) , |
|
= (− 2; 2; 1) , |
c (3; 1; −1) . |
7) |
b |
|||
|
a = (3; 1; 2) , |
|
= (2; 1; 1) , |
c (2; −1; 4) . |
8) |
b |
|||
|
a = (4; 2; 1) , |
|
= (−1; 2; 1) , |
c (−1; 1; 2) . |
9) |
b |
|||
|
a = ( −1; 2; 1) , |
|
= (2;1; 3) , |
c (1;1; −1) . |
10) |
b |
|||
|
a = (1;1; 4) , |
|
= (0; − 3; 2) , |
c (2;1; −1) . |
11) |
b |
|||
|
a = (1; − 2; 0) , |
|
= (1;1; 3) , |
c (1;1; 4) . |
12) |
b |
|||
|
a = (1; 0; 5) , |
|
= (−1; 3; 2) , |
c (1; −1;1) . |
13) |
b |
|||
|
a = (1; 3; − 2) , |
|
= (0; −1; 2) , |
c (3; 3; 4) . |
14) |
b |
|||
|
a = (2; 3;1) , |
|
= (−1; 0;1) , |
c (2; 5; − 3) . |
15) |
b |
|||
|
a = (2; 3;1) , |
|
= (2; 2; 3) , |
c (4;1; 2) . |
16) |
b |
|||
|
a = (4; 2; 3) , |
|
= (3; 2; −1) , |
c (4;1; 2) . |
17) |
b |
|||
|
a = (1; 2; −1) , |
|
= (3; 0; 2) , |
c (−1; 1; 1) . |
18) |
b |
|||
|
a = (1; 4;1) , |
|
= (− 3; 2; 0) , |
c (1; −1; 2) . |
19) |
b |
|||
|
|
|
|
65 |
PDF created with pdfFactory Pro trial version www.pdffactory.com
20)a = (2;1; − 2) ,
21)a = (0; 5;1) ,
22)a = (2; 2; −1) ,
23)a = (2; 2;1) ,
24)a = (2;1; 3) ,
25)a = (2; 3;1) ,
26)a = (1; −1; 2) ,
27)a = (1; 3; 6) ,
28)a = (−1; 2; 6) ,
29)a = (4; 7; 2) ,
30)a = (2; 2;1) ,
|
b |
= (3; −1; 1) , |
c (4;1; 0) . |
|
|
= (3; 2;1) , |
c (−1;1; 0) . |
b |
|||
|
|
= (0; − 2;1) , |
c(1; 3;1) . |
b |
|||
|
|
= (1; − 2; 0) , |
c (− 3; 2; 5) . |
b |
|||
|
|
= (3; 5; 3) , |
c (4; 2;1) . |
b |
|||
|
|
= (1; −1; 2) , |
c (2; −1; 0) . |
b |
|||
|
|
= (3; 2; 0) , |
c ( −1;1;1) . |
b |
|||
|
|
= (2; 2;1) , |
c (−1; 0;1) . |
b |
|||
|
|
= (2; − 3; 0) , |
c ( −1; 5; 8) . |
b |
b = (7; −1; − 2) , c (3; 3;1) . b = (1; 2; 3) , c (4;1; 0) .
11.2. Привести уравнение кривой второго порядка к каноническому виду с помощью поворота системы координат и параллельного переноса. Отметить в старой системе координат центр кривой и направления осей новой системы координат. Построить кривую.
1)а) 17x2 −12xy + 8y2 − 20 = 0 ;
б) 9x2 + y2 + 6xy −12x − 4y + 3 = 0 .
2)а) 35x2 − 30xy − 5y2 + 4 = 0 ;
б) 4x2 − 4xy + y2 + 8x − 4y + 3 = 0 .
3)а) 4x2 + y2 − 4xy + 4x + 8y = 0 ;
б) 9x2 + 4y2 −12xy + 6x − 4y +1 = 0 .
4)а) 7x2 + 60xy + 32y2 − 52 = 0 ;
б) 5x2 − 2xy + y2 + 6x − 2y + 2 = 0 .
5)а) 16x2 + 8xy + y2 − 6x + 24y = 0 ; б) x2 − 6xy + 9y2 + 4x −12y − 51 = 0 .
6)а) 37x2 + 32xy +13y2 − 45 = 0 ;
66
PDF created with pdfFactory Pro trial version www.pdffactory.com
б)
7)а)
б)
8)а)
б)
9)а)
б)
10)а)
б)
11)а)
б)
12)а)
б)
13)а)
б)
14)а)
б)
15)а)
б)
16)а)
б)
17)а)
б)
18)а)
б)
9x2 +12xy + 4y2 − 3x − 2y − 2 = 0 .
9x2 + 6xy + y2 − 8x + 24y = 0 ;
9x2 +12xy + 4y2 −12x − 8y + 4 = 0 .
37x2 −18xy +13y2 − 40 = 0 ;
5x2 − 24xy +10y2 + 2x + 2y +1 = 0 .
3y2 + 4xy + 4 = 0 ;
x2 + 2xy + y2 − 4x − 4y + 3 = 0 .
13x2 − 32xy + 37 y2 − 45 = 0 ;
x2 −10xy + 25y2 + x − 5y − 6 = 0 .
5x2 + 4xy + 8y2 − 32x − 56y + 80 = 0 ;
x2 − 4xy + 4y2 −1 = 0 .
9x2 + 24xy +16y2 − 230x +110y − 475 = 0 ;
x2 + 4xy + y2 − 9 = 0 .
5x2 +12xy − 22x −12y −19 = 0 ;
x2 − 5xy + 4y2 + x + 2y − 2 = 0 .
x2 + 2xy + y2 −10x − 6y + 25 = 0 ; x2 + 2xy + y2 + x + y = 0 .
5x2 + 8xy + 5y2 −18x −18y + 9 = 0 ;
4x2 −12xy + 9y2 − 2x + 3y − 2 = 0 .
5x2 + 6xy + 5y2 −16x −16y −16 = 0 ;
9x2 −12xy + 4y2 −1 = 0 .
6xy + 8y2 −12x − 26y +11 = 0 ;
x2 + 6xy + 9y2 − 4 = 0 .
7x2 +16xy − 23y2 −14x −16y − 218 = 0 ;
9x2 + 6xy + y2 −1 = 0 .
67
PDF created with pdfFactory Pro trial version www.pdffactory.com
19)а)
б)
20)а)
б)
21)а)
б)
22)а)
б)
23)а)
б)
24)а)
б)
25)а)
б)
26)а)
б)
27)а)
б)
28)а)
б)
29)а)
б)
30)а)
б)
7x2 − 24xy − 38x + 24y +175 = 0 ;
4x2 +12xy + 9y2 − 4 = 0 .
9x2 + 24xy +16y2 − 40x + 30y = 0 ;
16x2 − 8xy + y2 −1 = 0 .
x2 + 2xy + y2 − 8x + 4 = 0 ;
4x2 +12xy + 9y2 − 9 = 0 .
4x2 − 4xy + y2 − 2x −14y + 7 = 0 ;
x2 + 4xy + 4y2 − 9 = 0 .
4x2 + 3xy + 4y2 − 5x + 2y − 7 = 0 ;
6xy +10y2 + 3x + y − 2 = 0 .
4x2 − 4xy + y2 − 8x + 6y − 2 = 0 ;
2x2 − 5xy −12y2 − x + 26y −10 = 0 .
4xy − 3y2 + 6x + 6y +1 = 0 ;
3x2 + xy − 2y2 − 5x + 5y − 2 = 0 .
x2 + 2xy + y2 + y = 0 ;
4x2 +16xy +15y2 − 8x − 22y − 5 = 0 .
x2 − 4xy + y2 − 4x + 2y − 2 = 0 ;
4x2 − 4xy + y2 − 6x + 3y − 4 = 0 .
x2 + 4xy + 4y2 − 6x − 8y = 0 ;
x2 + 4xy + 4y2 + 3x + 6y − 4 = 0 .
10x2 − 8xy +10y2 − 28x + 20y + 65 = 0 ;
x2 − y2 + 2x + 4y − 3 = 0 .
x2 − 2xy + y2 − 2x − 2y + 4 = 0 ;
3x2 − 8xy − 3y2 + 8x + 6y − 3 = 0 .
68
PDF created with pdfFactory Pro trial version www.pdffactory.com
11.3. Привести уравнение поверхности второго порядка к каноническому виду. Сделать схематический рисунок поверхности в новой системе координат.
1)x2 + y2 + 5z2 − 6xy + 2xz − 2yz = 0 .
2)2xy + 2xz + 2yz − 2x − 6y = 0 .
3)x2 + y2 − 6xy + 2y + 2z = 0 .
4)3y2 + 3z2 + 4xy + 4xz − 2yz + 2x + 6z = 0 .
5)2x2 + y2 + 2z2 − 2xy + 2yz + 4x − 2y = 0 .
6)5x2 + 8z2 + 4xz − 32x − 56z = 0 .
7)2x2 + 2y2 + 3z2 + 4xy + 2xz + 2yz − 4x − 6y − 2z = 0 .
8)y2 + 2xy + 4xz + 2yz − 4x − 2y = 0 .
9)2xy − 2xz + 2yz + 4x + 6z = 0 .
10)6x2 + 5y2 + 7z2 − 4xy + 4xz + 6x + 2z = 0 .
11)x2 + z2 + 2xz + 4x = 0 .
12)2x2 + 6y2 + 2z2 − 2xy + 6xz − 2yz + 2x + 6y + 2z = 0 .
13)x2 + y2 + z2 + 2xy + 2xz + 2yz + 4x − 4z = 0 .
14)x2 + y2 + 4z2 + 2xy + 4xz + 4yz − 6z = 0 .
15)x2 − 2y2 + z2 + 4xy + 4yz −10zx + 2x + 4y −10z −1 = 0 .
16)2x2 + 5y2 + 2z2 − 4xy − 2xz + 4yz − 8x = 0 .
17)2x2 + 2y2 + 2z2 + 2xy + 2xz + 2yz − 3y = 0 .
18)x2 + y2 + z2 + 4xy + 2x + 2z = 0 .
19)−x2 + y2 − 5z2 + 6xz + 4yz −14y = 0 .
20)2x2 + y2 + z2 + 6yz + 8x = 0 .
21)13x2 + 27 y2 − 48xy + 2z = 0 .
22)5y2 + 8z2 + 4yz − 32y − 56z = 0 .
23)5x2 + 9z2 +12xy − 22x −12y + 6z = 0 .
24)3x2 + 2y2 + 3z2 − 2xy + 2yz = 0 .
69
PDF created with pdfFactory Pro trial version www.pdffactory.com
25)x2 + 3y2 − 3z2 + 6xz + 4yz −10x = 0 .
26)2x2 + 2y2 − 5z2 + 2xy − 2x − 4y − 4z = 0 .
27)2x2 + 2y2 + 3z2 + 4xy + 2yz + 2zx − 4x + 6y − 2z + 3 = 0 .
28)x2 + 5y2 + z2 + 2xy + 2yz + 6zx − 2x + 6y + 2z = 0 .
29)2x2 + 5y2 + 2z2 − 2xy − 4xz + 2yz + 2x −10y − 2z −1 = 0 .
30)x2 − 2y2 + z2 + 4xy + 4yz −10zx + 2x + 4y −10z = 0 .
70
PDF created with pdfFactory Pro trial version www.pdffactory.com