Задача 10. Привести квадратичную форму к каноническому виду методом Лагранжа.

10.1.x12 + 4x1x2 + 4x1x3 + 4x2 x3 + 4x32.

10.2.4x12 + 4x1x2 + 8x1x3 - 3x22 + 4x32 .

10.3.4x12 + 8x1x2 + 4x1x3 + x32.

10.4.4x12 + 8x1x2 + 4x1x3 + 3x22 - 2x32.

10.5.x12 + 4x1x2 + 4x1x3 + 3x22 + 4x2 x3 + x32.

10.6.x12 + 4x1x2 + 4x2 x3 + x32.

10.7.x12 + 2x1x2 + 2x1x3 - 3x22 - 6x2 x3 - 2x32.

10.8.x12 + 4x1x2 + 2x1x3 + 3x22 + 2x2 x3 + x32.

263

10.9.x12 + 4x1x3 − x22 − 2x2 x3 + 4x32.

10.10.x12 + 2x1x2 + 2x1x3 + x32.

10.11.x12 + 4x1x2 + 4x1x3 + 8x22 +12x2 x3 + 4x32.

10.12.4x12 + 4x1x2 + 8x1x3 + 5x22 + 8x2 x3 + 4x32.

10.13.4x12 + 8x1x2 + 4x1x3 + 8x22 + 8x2 x3 + x32.

10.14.4x12 + 8x1x2 + 4x1x3 + 5x22 + 8x2 x3 + 4x32.

10.15.x12 + 4x1x2 + 4x1x3 + 5x22 +12x2 x3 + 7x32.

10.16.x12 + 4x1x2 + 4x1x3 + 8x22 +16x2 x3 + 7x32.

10.17.x12 + 2x1x2 + 2x1x3 + 5x22 +10x2 x3 + 4x32.

10.18.x12 + 4x1x2 + 2x1x3 + 5x22 + 6x2 x3 + x32.

10.19.x12 + 4x1x3 + x22 + 2x2 x3 + 4x32.

10.20.x12 + 2x1x2 + 2x1x3 + 2x22 + 4x2 x3 + x32.

10.21.x12 + 4x1x2 + 4x1x3 + 4x2 x3 + 2x32.

10.22.4x12 + 4x1x2 + 4x1x3 − 3x22 + 2x32.

10.23.4x12 + 8x1x2 + 4x1x3 + x32.

10.24.4x12 + 8x1x2 + 4x1x3 + 3x22 − 4x32.

10.25.x12 + 4x1x2 + 4x1x3 + 3x22 + 4x2 x3 − x32.

10.26.x12 + 4x1x2 + 4x1x3 − x32.

10.27.x12 + 2x1x2 + 2x1x3 − 3x22 − 6x2 x3 − 4x32.

10.28.x12 + 4x1x2 + 2x1x3 + 3x22 + 2x2 x3 − x32.

10.29.x12 + 4x1x2 − x22 − 2x2 x3 + 2x32.

10.30.x12 + 2x1x2 + 2x1x3 − x32.

10.31.x12 + 2x1x2 + 2x1x3 + 2x22 + 4x2 x3 + 3x32.

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Задача 11. Привести квадратичную форму к каноническому виду ортогональным преобразованием.

11.1.4x22 − 3x32 + 4x1x2 − 4x1x3 + 8x2 x3.

11.2.4x12 + 4x22 + x32 − 2x1x2 + 23x2 x3.

11.3.2x12 + 2x22 + 2x32 + 8x1x2 + 8x1x3 − 8x2 x3.

11.4.2x12 + 9x22 + 2x32 − 4x1x2 + 4x2 x3.

11.5.−4x12 − 4x22 + 2x32 − 4x1x2 + 8x1x3 −8x2 x3.

11.6.x12 + x22 + 4x32 + 2x1x2 − 23x2 x3.

11.7.4x12 + 4x22 + x32 + 2x1x2 − 4x1x3 + 4x2 x3.

11.8.3x12 + x22 − 32 x32 + 23x1x2 − x1x3 + 3x2 x3.

11.9.−x12 − x22 − 3x32 − 2x1x2 − 6x1x3 + 6x2 x3.

11.10.x12 − 7x22 + x32 − 4x1x2 − 2x1x3 − 4x2 x3.

11.11.5 42 x12 + 5 42 x22 + 3 22 x32 + 22 x1x2 + x1x3 + x2 x3.

11.12.3x12 − 7x22 + 3x32 + 8x1x2 − 8x1x3 −8x2 x3.

11.13.x12 + 5x22 + x32 − 4x1x2 + 52x1x3 + 2x2 x3.

11.14.x12 + x22 + x32 − 43 x1x2 − 8 32 x2 x3.

11.15.−2x12 + 2x22 − 2x32 − 4x1x2 + 52x1x3 + 2x2 x3.

11.16.−(12)x12 + 5x22 − (12)x32 − 4x1x2 + 3x1x3 + 4x2 x3.

11.17.x12 + x22 + x32 − 4x1x3 + 4x2 x3.

11.18.−2x12 + 2x22 − 2x32 + 4x1x2 − 6x1x3 + 4x2 x3.

11.19.2x12 + 3x22 + 2x32 − 8x1x2 − 42x1x3 + 22x2 x3.

11.20.−4x12 + x22 − 4x32 + 4x1x2 − 4x1x3 + 4x2 x3.

265

11.21.10x12 +14x22 + 7x32 −10x1x2 − 2x1x3 − 52x2 x3.

11.22.(32)x12 − 5x22 + (32)x32 + 4x1x2 − x1x3 − 4x2 x3.

11.23.x12 + x22 + 2x32 + 4x1x2 + 22x1x3 − 22x2 x3.

11.24.2x22 − 3x32 − 23x1x2 − 4x1x3 + 43x2 x3.

11.25.x12 + x22 + x32 + 43 x1x2 + 8 32 x2 x3.

11.26.x12 + x32 + 8x1x2 + 42x1x3 − 22x2 x3.

11.27.5x12 +13x22 + 5x32 + 4x1x2 + 8x2 x3.

11.28.2x12 + 2x22 + 2x32 + 23 x1x2 + 4 32 x2 x3.

11.29.5x12 + 4x22 + 2x32 − 4x1x2 − 22x1x3 + 42x2 x3.

11.30.−2x12 + 5x22 − 2x32 + 4x1x2 + 4x2 x3.

11.31.−3x12 + 9x22 + 3x32 + 2x1x2 + 8x1x3 + 4x2 x3.

Задача 12. Исследовать кривую второго порядка и построить ее.

12.1.−x2 − y2 + 4xy + 2x − 4y +1 = 0.

12.2.2x2 + 2y2 − 2xy − 2x − 2y +1 = 0.

12.3.4xy + 4x − 4y = 0.

12.4.−2x2 − 2y2 + 2xy − 6x + 6y + 3 = 0.

12.5.−3x2 − 3y2 + 4xy − 6x + 4y + 2 = 0.

12.6.−2xy − 2x − 2y +1 = 0.

12.7.−x2 − y2 − 4xy − 4x − 2y + 2 = 0.

12.8.−4x2 − 4y2 + 2xy +10x −10y +1 = 0.

12.9.4xy + 4x − 4y − 2 = 0.

12.10.x2 + y2 + 2xy −8x −8y +1 = 0.

266

12.11.x2 + y2 + 4xy − 8x − 4y +1 = 0.

12.12.x2 + y2 − 2xy − 2x + 2y − 7 = 0.

12.13.2xy + 2x + 2y − 3 = 0.

12.14.4x2 + 4y2 + 2xy +12x +12y +1 = 0.

12.15.3x2 + 3y2 + 4xy + 8x +12y +1 = 0.

12.16.x2 + y2 − 8xy − 20x + 20y +1 = 0.

12.17.3x2 + 3y2 − 2xy − 6x + 2y +1 = 0.

12.18.4xy + 4x + 4y +1 = 0.

12.19.3x2 + 3y2 − 4xy + 6x − 4y − 7 = 0.

12.20.−4xy − 4x + 4y + 6 = 0.

12.21.5x2 + 5y2 − 2xy +10x − 2y +1 = 0.

12.22.2x2 + 2y2 + 4xy + 8x + 8y +1 = 0.

12.23.−x2 − y2 + 2xy + 2x − 2y +1 = 0.

12.24.2x2 + 2y2 − 4xy − 8x + 8y +1 = 0.

12.25.3x2 + 3y2 + 2xy −12x − 4y +1 = 0.

12.26.−4xy + 8x + 8y +1 = 0.

12.27.2x2 + 2y2 − 2xy + 6x − 6y − 6 = 0.

12.28.x2 + y2 + 4xy + 4x + 2y − 5 = 0.

12.29.4xy + 4x − 4y + 4 = 0.

12.30.3x2 + 3y2 − 4xy + 4x + 4y +1 = 0.

12.31.x2 + y2 − 4xy + 4x − 2y +1 = 0.

267