Math Problems Vault

Problem 636

Find the plane of symmetry of the surface

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

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Problem 637

Find the planes of symmetry of the surface

x^{2}+y^{2}+z^{2}+4x-2y+9z-1 = 0.

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Problem 638

Find the planes of symmetry of the surface $xy+xz+yz = 1$ .

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Problem 639

In the equation of a second-order surface

Ax^{2}+By^{2}+Cz^{2}+2Dxy+2Exz+2Fyz+2Gx+2Hy+2Iz+K = 0

the values of the determinants

\Delta _{3} = \begin{vmatrix} A & D & E \\ D & B & F \\ E & F & C \end{vmatrix} \qquad \text{and} \qquad \Delta _{4} = \begin{vmatrix} A & D & E & G \\ D & B & F & H \\ E & F & C & I \\ G & H & I & K \end{vmatrix},

which can be called the discriminants of the quadratic forms $f = Ax^{2}+By^{2}+Cz^{2}+2Dxy+2Exz+2Fyz$ and

\Phi = Ax^{2}+By^{2}+Cz^{2}+2Dxy+2Exz+2Fyz+2Gxt+2Hyt+2Izt+Kt^{2},

are invariants, as are the values $p = A+B+C$ , $q = AB+AC+BC-D^{2}-E^{2}-F^{2}$ .

Using this, prove that the equation of a surface with center, when $\Delta_{3} \neq 0$ , can be reduced to the form

\lambda _{1}x_{1}^{2}+\lambda _{2}y_{1}^{2}+\lambda _{3}z_{1}^{2}+\frac{\Delta _{4}}{\Delta _{3}} = 0,

where $\lambda_{1}$ , $\lambda_{2}$ , $\lambda_{3}$ are the roots of the characteristic equation of the form $f$ .

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Problem 640

Prove that when $\lambda_{3} = 0$ , $q \neq 0$ the equation of the surface (the notation $\lambda_{3}$ , $q$ , $\Delta_{4}$ is the same as in problem 639) can be reduced to the form

\lambda _{1}x_{1}^{2}+\lambda _{2}y_{1}^{2} = 2\sqrt{-\frac{\Delta _{4}}{q}} \cdot z.

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Problem 641

Prove that when $\Delta_{4} = 0$ (the notation is the same as in problem 639) the surface is either cylindrical, or the equation represents a point, a line, or one or a pair of planes.

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Problem 642

Simplify the equation of the surface $x^{2}+y^{2}+z^{2}-6x+8y+10z+1 = 0$ and give the formulas for the coordinate transformation by which this simplification is achieved.

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Problem 643

Simplify the equation of the surface $5x^{2}+6y^{2}+7z^{2}-4xy+4yz-10x+8y+14z-6 = 0$ and give the formulas for the coordinate transformation by which this simplification is achieved.

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Problem 644

Simplify the equation of the surface

2x^{2}+5y^{2}+11z^{2}-20xy+4xz+16yz-24x-6y-6z-18 = 0

and give the formulas for the coordinate transformation by which this simplification is achieved.

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Problem 645

Simplify the equation of the surface

x^{2}+2y^{2}-3z^{2}+12xy-8xz-4yz+14x+16y-12z+33 = 0

and give the formulas for the coordinate transformation by which this simplification is achieved.

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Problem 646

Simplify the equation of the surface $3x^{2}-2y^{2}-z^{2}+4xy+8xz-12yz+18x-12y-6z = 0$ and give the formulas for the coordinate transformation by which this simplification is achieved.

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Problem 647

Simplify the equation of the surface $4x^{2}+2y^{2}+3z^{2}+4xz-4yz-10x+4y+6 = 0$ and give the formulas for the coordinate transformation by which this simplification is achieved.

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Problem 648

Simplify the equation of the surface $y^{2}-z^{2}+4xy-4xz-2x+6y+2z+8 = 0$ and give the formulas for the coordinate transformation by which this simplification is achieved.

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Problem 649

Simplify the equation of the surface $4x^{2}+2y^{2}+3z^{2}+4xz-4yz+8x-4y+8z = 0$ and give the formulas for the coordinate transformation by which this simplification is achieved.

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Problem 650

Simplify the equation of the surface $y^{2}-z^{2}+4xy-4xz-3 = 0$ and give the formulas for the coordinate transformation by which this simplification is achieved.

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Problem 651

Simplify the equation of the surface $4x^{2}+y^{2}+4z^{2}-4xy+8xz-4yz-12x-12y+6z = 0$ and give the formulas for the coordinate transformation by which this simplification is achieved.

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Problem 652

Simplify the equation of the surface $7y^{2}-7z^{2}-8xy+8xz = 0$ and give the formulas for the coordinate transformation by which this simplification is achieved.

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Problem 653

Simplify the equation of the surface $5x^{2}+5y^{2}+8z^{2}-8xy-4xz-4yz = 0$ and give the formulas for the coordinate transformation by which this simplification is achieved.

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Problem 654

Simplify the equation of the surface $36x^{2}+9y^{2}+4z^{2}+36xy+24xz+12yz = 49$ and give the formulas for the coordinate transformation by which this simplification is achieved.

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Problem 655

Simplify the equation of the surface $36x^{2}+9y^{2}+4z^{2}+36xy+24xz+12yz = 0$ and give the formulas for the coordinate transformation by which this simplification is achieved.

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Problem 656

Simplify the equation of the surface $x^{2}+y^{2}+2z^{2}+4x-6y-8z+21 = 0$ and give the formulas for the coordinate transformation by which this simplification is achieved.

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Problem 657

Set up the equation of the locus of the vertices of the surfaces $4y^{2}-2z^{2}+x+ay+bz = 0$ .

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Problem 658

Determine $\lambda$ and $\mu$ so that the equation

x^{2}-y^{2}+3z^{2}+(\lambda x+\mu y)^{2} = 1

represents a cylinder of revolution.

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Problem 659

Find the condition on $a$ , $b$ , and $c$ under which the equation

a(x^{2}+2yz)+b(y^{2}+2xz)+c(z^{2}+2xy) = 1

represents a surface of revolution.

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Problem 660

Find the axis of rotation of the surface

y^{2}+(z^{2}-2z)(1-\lambda ^{2})+2\lambda xz-2x = 0.

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Problem 661

Find $c$ for which the cone $x^{2}-2xy+cz^{2} = 0$ is a cone of revolution, and give the equations of its axis.

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Problem 662

Prove that the cone with vertex at the point $\left(1,\dfrac {1}{2},0\right)$ and with directrix $x^{2}+y^{2}-2x-y = 0$ , $z = 1$ is a cone of revolution.

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Problem 663

Investigate the character of the surface

x^{2}+(2m^{2}+1)(y^{2}+z^{2})-2xy-2xz-2yz = 2m^{2}-3m+1

as $m$ varies from $-\infty$ to $+\infty$ .

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Problem 664

For what relation between $\alpha$ and $\beta$ does the equation

x^{2}+y^{2}-z^{2}+2\alpha xz+2\beta yz-2x-4y+2z = 0

represent a conical surface?

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Problem 665

Find the equation of the locus of points outside a cube with side $a$ , for which the product of the distances to three faces of the cube meeting at a vertex equals the product of the distances to the three other faces of the cube.

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Problem 666

Determine the kind of surface obtained by the motion of a line parallel to the plane $Ax+By+Cz+D = 0$ and sliding along the lines

\begin{cases} x = 0, \\ z = a; \end{cases} \qquad \text{and} \qquad \begin{cases} y = 0, \\ z = -a. \end{cases}

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Problem 667

The vectors $\overrightarrow {P}(a,b,c)$ , $\overrightarrow {P}_{1}(a_{1},b_{1},c_{1})$ , and $\overrightarrow {P}_{2}(a_{2},b_{2},c_{2})$ are mutually perpendicular. Prove that the surface

A(ax+by+cz+d)^{2}+B(a_{1}x+b_{1}y+c_{1}z+d_{1})^{2}+C(a_{2}x+b_{2}y+c_{2}z+d_{2})^{2}+D = 0

has as its principal directions the vectors $\overrightarrow {P}$ , $\overrightarrow {P}_{1}$ , $\overrightarrow {P}_{2}$ .

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Problem 668

A second-order surface passes through the points $(0,0,0)$ , $(1,1,-1)$ , $(0,0,1)$ and has planes of symmetry $x+y+z = 0$ , $2x-y-z = 0$ , $y-z+1 = 0$ . Find its equation.

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Problem 669

A second-order cylinder passes through the points $(1,0,-1)$ , $(2,0,2)$ . Its axis is given by the equations $x = -y = z$ , and its planes of symmetry by the equations $x+2y+z = 0$ and $x = z$ . Find its equation.

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Problem 670

$x = 0$ and $y = 0$ are planes of symmetry of a second-order surface. The points $(0,1,1)$ , $(2,0,1)$ , $(1,1,2)$ , and $(2,0,3)$ lie on the surface. Find its equation.

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Problem 671

A second-order surface of revolution passes through the points $(1,0,0)$ , $(1,1,1)$ , $(0,1,2)$ . The axis of rotation is the axis $Oz$ . Find the equation of the surface.

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Problem 672

The generators of a parabolic cylinder are parallel to the line $2x = 2y = -z$ , the equation of the plane of symmetry is $x+y+z = 0$ , and the points $(1,1,1)$ , $(1,-1,1)$ lie on the surface. Find the equation of the cylinder.

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Problem 673

$s_{1}(x,y,z) = 0$ , $s_{2}(x,y,z) = 0$ , $s_{3}(x,y,z) = 0$ are the equations of three diametral planes of a surface. Moreover, each of them is conjugate to the line of intersection of the other two planes. Prove that the equation of the surface can be written in the form:

As_{1}^{2}+Bs_{2}^{2}+Cs_{3}^{2}+D = 0.

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Problem 674

Three chords of an ellipsoid, passing through the center, such that each of them is conjugate to the plane passing through the other two, can be called three mutually conjugate axes of the ellipsoid. Prove that the volume of the parallelepiped built on three mutually conjugate axes of the ellipsoid is a constant quantity.

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Problem 675

Prove that the equation

(ax+by+cz+d)^{2} \pm (\alpha x+\beta y+\gamma z+\delta )^{2} = px+qy+rz+s

represents a paraboloid for which $px+qy+rz+s = 0$ is a tangent plane, and the combined equations

ax+by+cz+d = 0, \qquad \alpha x+\beta y+\gamma z+\delta = 0

give the diameter passing through the point of tangency.

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Problem 676

Find the equation of the paraboloid tangent to the plane $x+y+z = 0$ at the origin, having the diametral plane $x = y$ , and passing through the points $(1,1,2)$ , $(0,2,2)$ , $(3,1,1)$ , $\left(\dfrac {9}{2},2,2\right)$ .

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