All textbooks/Collection of problems in analytical geometry and linear algebra/4 · Quadric surfaces/11 · General theory of surfaces of the second order

Subsection 11.2

Determining the type and location of a surface given by a general equation of the second order (11.19-11.23)

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Problem 11.19

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)

Using the result of problem 11. 18, determine the type of the surface given, in a general Cartesian coordinate system, by the equation:

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(1)

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

(2)

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

(3)

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

(4)

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

(5)

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

(6)

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

(7)

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

(8)

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

(9)

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

(10)

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

(11)

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

(12)

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

(13)

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

(14)

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

(15)

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

(16)

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

(17)

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

(18)

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

(19)

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

(20)

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

Problem 11.20

In a general Cartesian coordinate system a hyperboloid is given by the equation $(x + y + z)(x - y + z) - (2x - y + 2z)^{2} = 1$ . Find the equation of its asymptotic cone.

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Problem 11.21

(1)(2)(3)(4)(5)(6)

A surface is given in a general Cartesian coordinate system by an equation containing a parameter $k$ . Determine the type of the surface for all values of $k$ .

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(1)

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

(2)

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

(3)

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

(4)

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

(5)

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

(6)

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

Problem 11.22

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)

A surface is given by an equation in a rectangular Cartesian coordinate system. Find the canonical coordinate system and the canonical equation of this surface. Determine the type of the surface.

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(1)

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

(2)

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

(3)

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

(4)

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

(5)

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

(6)

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

(7)

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

(8)

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

(9)

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

(10)

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

(11)

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

(12)

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

(13)

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

(14)

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

(15)

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

(16)

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

(17)

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

(18)

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

(19)

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

(20)

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

(21)

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

(22)

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

(23)

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

(24)

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

Problem 11.23

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)

A surface is given in a rectangular Cartesian coordinate system by an equation containing a parameter $k$ . For the given value of $k$ , find the canonical coordinate system and the canonical equation of the surface. Determine the type of the surface for all possible values of $k$ . If the surface is a line, a plane, or a pair of planes, find the linear equations of these sets in the original coordinate system.

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(1)

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

(2)

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

(3)

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

(1) $k = - 1$ ;

(2) $k = 5$ ;

(3) $k = 11$ .

(4)

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

(5)

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

(1) $k = 4$ ;

(2) $k = 8$ .

(6)

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

(7)

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

(8)

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

(9)

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

(1) $k = 15$ ;

(2) $k = 18$ .

(10)

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

(11)

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

(1) $k = 8$ ;

(2) $k = 9$ .

(12)

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

(13)

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

(1) $k = - 12$ ;

(2) $k = - 3$ ;

(3) $k = 6$ .

(14)

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

(1) $k = - 75$ ;

(2) $k = - 70$ ;

(3) $k = - 80$ .

(15)

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

(1) $k = - 4$ ;

(2) $k = 2$ ;

(3) $k = 8$ .

(16)

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

(1) $k = - 42$ ;

(2) $k = - 36$ ;

(3) $k = - 30$ .

(17)

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

(1) $k = 0$ ;

(2) $k = - 9$ .

(18)

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

(19)

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

(20)

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

(21)

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

(1) $k = 1$ ;

(2) $k = - 13$ .

(22)

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

(1) $k = - 26$ ;

(2) $k = - 14$ ;

(3) $k = - 2$ .

(23)

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

(1) $k = 2$ ;

(2) $k = 5$ ;

(3) $k = 8$ .

(24)

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

(25)

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

(1) $k = - 12$ ;

(2) $k = - 6$ .