Math Problems Vault

Problem 714

Determine the type of the surface and its position relative to the original coordinate system, using a transformation of the left-hand side of its equation:

?

(1)

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

(2)

$3 x^{2}+3 y^{2}+3 z^{2}-6 x+4 y-1 = 0$ ;

(3)

$3 x^{2}+3 y^{2}-6 x+4 y-1 = 0$ ;

(4)

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

(5)

$4 x^{2}+y^{2}-4 x y-36 = 0$ .

Problem 715

(1)(2)(3)(4)

Determine the type and position of the surface, using a translation of the coordinate system:

?

(1)

$x^{2}+4 y^{2}+9 z^{2}-6 x+8 y-36 z = 0$ ;

(2)

$4 x^{2}-y^{2}-z^{2}+32 x-12 z+44 = 0$ ;

(3)

$3 x^{2}-y^{2}+3 z^{2}-18 x+10 y+12 z+14 = 0$ ;

(4)

$6 y^{2}+6 z^{2}+5 x+6 y+30 z-11 = 0$ .

Problem 716

Determine the type of the surface and its position relative to the coordinate system, using a rotation of the coordinate system about one of its axes:

?

(1)

$z^{2} = 2 x y$

(2)

$z = x y$

(3)

$z^{2} = 3 x+4 y$ ;

(4)

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

(5)

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

Problem 717

Prove that each of the following surfaces is a surface of revolution, determine its type, write the canonical equation, and find the position of the surface relative to the original coordinate system:

?

(1)

$x^{2}-2 y^{2}+z^{2}+4 x y-4 y z-8 z x-14 x-4 y+14 z+18 = 0$ ;

(2)

$5 x^{2}+8 y^{2}+5 z^{2}-4 x y+4 y z+8 z x-6 x+6 y+6 z+10 = 0$ ;

(3)

$2 y z+2 z x+2 x y+2 x+2 y+2 z+1 = 0$ ;

(4)

$3 x^{2}+3 y^{2}+3 z^{2}-2 x y-2 x z-2 y z-2 x-2 y-2 z-1 = 0$ ;

(5)

$2 x^{2}+6 y^{2}+2 z^{2}+8 x z-4 x-8 y+3 = 0$ ;

(6)

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

(7)

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

(8)

$4 x y+4 y z+4 z x+4 x+4 y+4 z+3 = 0$ .

Problem 718

Determine the type of each of the following surfaces, write its canonical equation, and find the canonical coordinate system:

?

(1)

$4 x^{2}+y^{2}+4 z^{2}-4 x y+4 y z-8 z x-28 x+2 y+16 z+45 = 0$ ;

(2)

$2 x^{2}+5 y^{2}+2 z^{2}-2 x y-4 x z+2 y z+2 x-10 y-2 z-1 = 0$ ;

(3)

$7 x^{2}+7 y^{2}+16 z^{2}-10 x y-8 y z-8 z x-16 x-16 y-8 z+72 = 0$ ;

(4)

$4 x^{2}+4 y^{2}-8 z^{2}-10 x y+4 y z+4 z x-16 x-16 y+10 z-2 = 0$ ;

(5)

$2 x^{2}-7 y^{2}-4 z^{2}+4 x y+20 y z-16 z x+60 x-12 y+12 z-90 = 0$ ;

(6)

$7 x^{2}+6 y^{2}+5 z^{2}-4 x y-4 y z-6 x-24 y+18 z+30 = 0$ ;

(7)

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

(8)

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

(9)

$x^{2}+5 y^{2}+z^{2}+2 x y+2 y z+6 z x-2 x+6 y+2 z = 0$ ;

(10)

$x^{2}-2 y^{2}+z^{2}+4 x y+4 y z-10 z x+2 x+4 y-10 z-1 = 0$ ;

(11)

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

Problem 719

Determine the affine type of the surface using Lagrange's method (the coordinate system is affine):

?

(1)

$4 x^{2}+6 y^{2}+4 z^{2}+4 x z-8 y-4 z+3 = 0$

(2)

$x^{2}+5 y^{2}+z^{2}+2 x y+6 x z+2 y z-2 x+6 y-10 z = 0$ ;

(3)

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

(4)

$x^{2}-2 y^{2}+z^{2}+4 x y-8 x z-4 y z-14 x-4 y+14 z+16 = 0$ ;

(5)

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

(6)

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

(7)

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

(8)

$x^{2}-2 y^{2}+z^{2}+4 x y-10 x z+4 y z+2 x+4 y-10 z-1 = 0$ ;

(9)

$x^{2}+y^{2}+4 z^{2}+2 x y+4 x z+4 y z-6 z+1 = 0$ ;

(10)

$4 x y+2 x+4 y-6 z-3 = 0$

(11)

$x y+x z+y z+2 x+2 y-2 z = 0$ .