Определение вида и расположения поверхности, заданной общим уравнением второго порядка (11.19-11.23)

$(x + y + z)(x - y + 125z) = 1$

$(x + y)(x + y + 1) = 1$

$(x + y)(x + y + 1) = x - y$

$(x + y + z + 1)(x - y + z) = x + z + 1$

$(x + y + z + 1)(x - y + z) = x + 2z + 1$

$(x + y)(x - y) = z$

$(x + y + z)(x - y + z) = 0$

$(x + 2y)(x + 2y + 1) = z$

$(x + y)(x - 75y) = z^{2}$

$(x + y)^{2} = 3x + z$

$(x + 2y + 3z)(2x + 3y + 4z) + (3x + 4y + 75z)^{2} = 1$

$(x + 2y + 3z)(2x + 3y + 4z) - (3x + 4y + 75z)^{2} = 1$

$(x + y) z - x^{2} - x = 0$

$(x + y + z)^{2} + (x + 2y + 3z)^{2} + (2x - y + z)^{2} = 0$

$(x + y + z)^{2} + (x + 2y + 3z)^{2} + (2x + 3y + 4z)^{2} = 0$

$(x + y + z)^{2} + (x - 2y + z)^{2} = 0$

$(x + y + z)^{2} + (x + y)^{2} + (y + z)^{2} = 1987$

$(x + y + z)^{2} + (x + y)^{2} = 1987$

$(x + y + z)^{2} + (x + y)^{2} + (2x + 2y + z)^{2} = 1$

$(x + y)^{2} + z^{2} + 1 = 0$.

$x_{1}^{2} + x_{2}^{2} + 4x_{3}^{2} + 4x_{1} x_{3} + 2x_{2} x_{3} + 2x_{1} + 2x_{2} + 5x_{3} + k = 0$

$2x_{1}^{2} + kx_{2}^{2} + 8x_{1} x_{2} + 4x_{1} x_{3} - 4x_{1} - 8x_{2} - 4x_{3} = 0$

$3x_{1}^{2} + 2x_{2}^{2} + 2x_{3}^{2} + 4x_{1} x_{2} - 2x_{1} x_{3} + 4x_{1} + 4x_{2} + 4x_{3} + k = 0$

$kx_{1}^{2} + 8x_{2}^{2} + x_{3}^{2} + 16x_{1} x_{2} + 4x_{1} x_{3} + 4x_{2} x_{3} - 4x_{1} - 4x_{2} + 2x_{3} = 0$

$3x_{2}^{2} + x_{3}^{2} + 6x_{1} x_{2} + 2x_{1} x_{3} + 4x_{2} x_{3} + kx_{2} + x_{3} + 1 = 0$

(р) $x_{1}^{2} - 2x_{2}^{2} - 3x_{3}^{2} - 4x_{1} x_{2} - 6x_{1} x_{3} - 2x_{1} + 4x_{2} + 6x_{3} + k = 0$.

$2x^{2} + 9y^{2} + 2z^{2} - 4xy + 4yz - 1 = 0$

$4y^{2} - 3z^{2} + 4xy - 4xz + 8yz = 0$

$x^{2} + y^{2} + 4z^{2} - 2xy + 4xz - 4yz - 2x + 2y + 2z = 0$

$x^{2} + y^{2} + z^{2} - xy + xz + yz + 3x + 3y - 3z = 0$

$x^{2} - 3z^{2} - 4yz - 4y + 2z + 5 = 0$

$x^{2} + y^{2} + z^{2} - 2xy - 4x + 4y + 3 = 0$

$y^{2} + 2xz + 2x + 2z + 1 = 0$

$x^{2} + 2y^{2} + 5z^{2} + 4yz + 20y + 20z - 10 = 0$

$- x^{2} + 5y^{2} + 5z^{2} + 8yz + 2x + 12y + 24z + 36 = 0$

$2x^{2} + 5y^{2} + 5z^{2} + 6yz + 4x + 16y + 16z + 10 = 0$

$4x^{2} + 4y^{2} - 4xy - 12x - 12y - 5z + 1 = 0$

$x^{2} + y^{2} + z^{2} + 2xy - 12x + 4y + 6z - 3 = 0$

$4x^{2} + 9y^{2} - 12xy + 2x + 10y + 1 = 0$

$6xy - 8y^{2} - z^{2} + 60y + 2z + 89 = 0$

$5x^{2} + 8y^{2} + 4xy + 2x + 44y - 36z + 65 = 0$

(р) $- x^{2} + y^{2} + z^{2} - 2yz + 2x + 3y - 5z + 1 = 0$

$9y^{2} + 16z^{2} + 24yz + 5x + 10y + 5z + 11 = 0$

$16x^{2} + 9y^{2} - z^{2} - 24xy - 9x - 12y + 4z + 71 = 0$

$2x^{2} + 2y^{2} + z^{2} - 10xy + 20x - 8y + 29 = 0$

$- x^{2} + 7y^{2} - 24yz + 2x + 120y = 0$

$x^{2} - 4y^{2} - 4z^{2} + 10yz + 2x + 2y + 2z + 3 = 0$

$3x^{2} + 4xy + 8x + 8y - 4z = 0$

$- x^{2} - 9y^{2} + 6xy + 50x - 50y - 15z - 100 = 0$

(р) $4x^{2} + y^{2} + 9z^{2} + 4xy - 12xz - 6yz + 2x + 6y - 6z - 5 = 0$.

$5x^{2} + 5y^{2} + 3z^{2} + 2xy + 2\sqrt{2} xz + 2\sqrt{2} yz + 26x + 34y + 10\sqrt{2} z + 49 = 0$;

$2x^{2} + 9y^{2} + 2z^{2} - 4xy + 4yz + 4x + 2y - 4z - 1 = 0$

$4y^{2} - 3z^{2} + 4xy - 4xz + 8yz + 4x - 2z + k = 0$;

(1) $k = - 1$;

(2) $k = 5$;

(3) $k = 11$.

$2x^{2} + 2y^{2} + 2z^{2} + 2xy + 2xz + 2yz + 4x - 4y + 4 = 0$

$3x^{2} + 3y^{2} + 3z^{2} - 2xy - 2xz - 2yz - 8x + 8y + k = 0$

(1) $k = 4$;

(2) $k = 8$.

$x^{2} + y^{2} + 4z^{2} - 2xy + 4xz - 4yz - 12x + 12y - 24z + 6 = 0$

$x^{2} + y^{2} + 4z^{2} - 2xy + 4xz - 4yz + 24y - 24z + 12 = 0$

$x^{2} + y^{2} + 4z^{2} - 2xy + 4xz - 4yz + 5x + y - 2z = 0$

$2x^{2} + 2y^{2} + 2z^{2} - 2xy + 2xz + 2yz - 6x - 6y - 12z + k = 0$

(1) $k = 15$;

(2) $k = 18$.

$2x^{2} + 2y^{2} + 2z^{2} - 2xy + 2xz + 2yz + 18z + 18 = 0$;

$x^{2} + 2y^{2} + z^{2} - 2xy - 2yz + 6x - 6y + k = 0$

(1) $k = 8$;

(2) $k = 9$.

$x^{2} + 2y^{2} + z^{2} - 2xy - 2yz + 18x - 6y + 6z - 18 = 0$

$3x^{2} - 7y^{2} + 3z^{2} + 8xy - 8yz - 8xz - 4x + 6y + 8z + k = 0$;

(1) $k = - 12$;

(2) $k = - 3$;

(3) $k = 6$.

$2y^{2} - 3z^{2} - 2\sqrt{3} xy - 4xz + 4\sqrt{3} yz + 50z + k = 0$;

(1) $k = - 75$;

(2) $k = - 70$;

(3) $k = - 80$.

$2x^{2} + 2y^{2} + 2z^{2} + 8xy + 8xz - 8yz - 8x - 4y + 8z + k = 0$;

(1) $k = - 4$;

(2) $k = 2$;

(3) $k = 8$.

$4x^{2} + 4y^{2} - 2z^{2} + 4xy - 8xz + 8yz + 12x - 12y + 24z + k = 0$;

(1) $k = - 42$;

(2) $k = - 36$;

(3) $k = - 30$.

$8y^{2} + 4xy + 2xz + 4yz + 4x + 8y + k = 0$

(1) $k = 0$;

(2) $k = - 9$.

$8y^{2} + 4xy + 2xz + 4yz + 8x + 6y + 4z + 6 = 0$

$4x^{2} + y^{2} + 9z^{2} + 4xy - 12xz - 6yz + 11x + 3y - z + 1 = 0$

$4x^{2} + y^{2} + 9z^{2} + 4xy - 12xz - 6yz + 6x - 2y - 6z + 11 = 0$

$4x^{2} + y^{2} + 9z^{2} + 4xy - 12xz - 6yz + 4x + 2y - 6z + k = 0$;

(1) $k = 1$;

(2) $k = - 13$.

$- x^{2} + 10y^{2} - z^{2} - 8xy + 6xz + 8yz + 24x - 8y - 16z + k = 0$;

(1) $k = - 26$;

(2) $k = - 14$;

(3) $k = - 2$.

$2x^{2} - y^{2} + 2z^{2} + 4xy - 2xz + 4yz + 10x - 2y - 2z + k = 0$;

(1) $k = 2$;

(2) $k = 5$;

(3) $k = 8$.

$x^{2} + y^{2} - 2z^{2} + 10xy + 4xz - 4yz + 13x + 11y + 2z + 6 = 0$.

$x^{2} + y^{2} - 2z^{2} + 10xy + 4xz - 4yz + 24x - 12z + k = 0$;

(1) $k = - 12$;

(2) $k = - 6$.