Math Problems Vault

Problem 525

Find the locus of points at a distance of $1/\sqrt{3}$ from the line $x = y = z$ .

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Problem 526

Find the locus of points equidistant from the axes $Ox$ and $Oy$ .

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Problem 527

Find the locus of points equidistant from the axis $Ox$ and from the line $y = z = 2$ .

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Problem 528

Find the equation of the locus of lines making an angle of $45^{\circ }$ with the plane $xOy$ and passing through the point $(1,0,0)$ .

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Problem 529

A line slides along the lines

\begin{cases} x = 0, \\ y = 0 \end{cases} \qquad \text{and} \qquad \begin{cases} x = 1, \\ z = 0, \end{cases}

remaining parallel to the plane $x+y+z = 0$ . Find the surface generated by the motion of this line.

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Problem 530

Find the equation of the surface generated by a line that slides along the lines

\begin{cases} x = 3-z, \\ y = 2-2z \end{cases} \qquad \text{and} \qquad \begin{cases} x = -z, \\ y = 3z-3, \end{cases}

remaining parallel to the plane $2x-3y+z+12 = 0$ .

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Problem 531

A line slides along three lines:

\begin{cases} x = 0, \\ z = 0, \end{cases} \qquad \begin{cases} y = 0, \\ z = 1, \end{cases} \qquad \begin{cases} x = 1, \\ y = 1. \end{cases}

Find the equation of the resulting surface.

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Problem 532

Find the surface obtained by a line sliding along the three given lines:

\begin{cases} x = -2z-2, \\ y = -z, \end{cases} \qquad \begin{cases} x = -z+1, \\ y = 2, \end{cases} \qquad x = y = z.

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Problem 533

Find the locus of lines passing through the axis $Ox$ and parallel to the line

\begin{cases} x = 1, \\ y = z. \end{cases}

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Problem 534

Set up the equation of the cylindrical surface whose generators are parallel to the line $2x = -2y = z$ , and whose directrix is given by the equations

\begin{cases} x+y-z = 0, \\ 4y^{2}-2z^{2}+x-8y-8z-2 = 0. \end{cases}

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Problem 535

Find the locus of the perpendiculars dropped from the point $(0,0,1)$ to the generators of the conical surface

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

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Problem 536

Find the equation of the cylindrical surface whose generators are parallel to the line $x = y = z$ , and whose directrix is the circle

\begin{cases} x^{2}+y^{2} = 1, \\ z = 0. \end{cases}

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Problem 537

A line parallel to the plane $xOy$ slides along the axis $Oz$ and along the curve

\begin{cases} x = r\cos \alpha z, \\ y = r\sin \alpha z. \end{cases}

Find the equation of the resulting surface.

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Problem 538

A line parallel to the plane $x+y+z = 0$ slides along the axis $Oz$ and along the circle $x = b$ , $y^{2}+z^{2} = a^{2}$ . Find the resulting surface.

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Problem 539

Find the surface generated by the motion of a line intersecting the circle

\begin{cases} z = 0, \\ x^{2}+y^{2} = 1 \end{cases}

and the lines

x-1 = y = z, \qquad \begin{cases} y = 0, \\ x+1 = 0. \end{cases}

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Problem 540

Set up the equation of the conical surface whose vertex is at the origin, and whose directrix is given by the equations

\begin{cases} z = 1, \\ x^{2}+y^{2} = x. \end{cases}

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Problem 541

Prove that the equation of the conical surface of revolution for which the axes $Ox$ , $Oy$ , $Oz$ are generators has the form $xy+xz+yz = 0$ .

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Problem 542

Find the conical surface whose vertex is at the point $(1,1,1)$ , and whose directrix is given by the equations

\begin{cases} y^{2}+x^{2} = 1, \\ x+y+z = 0. \end{cases}

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Problem 543

A circular disk is parallel to the plane $yOz$ . Its center is at the point $(1,0,2)$ , and its radius equals one. On the axis $Oz$ a luminous point is positioned so that the shadow of the disk on the plane $xOy$ has the shape of a parabola. Find the equation of this parabola.

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Problem 544

A light bulb hangs below the center of a circular lampshade, $10$ cm below it and $50$ cm from the wall. The radius of the lampshade is $15$ cm. Find the outline of the shadow on the wall.

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Problem 545

A disk with center at $(1,0,2)$ and radius one is illuminated by a light source located on the axis $Oz$ so that the shadow of the disk on the plane $xOy$ has the shape of a rectangular hyperbola. Find the point at which the light source is located, if the disk is parallel to $yOz$ .

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Problem 546

The same question, if the shadow has the shape of a hyperbola with the tangent of the angle between the asymptotes equal to $4/3$ .

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Problem 547

The same question for the case of a shadow having the shape of an ellipse whose axis, lying along the axis $Ox$ , equals $8/3$ .

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Problem 548

A cube with center at the origin rotates about a diagonal of length $2a$ , lying along the axis $Oz$ . Find the equation of the surfaces obtained by the rotation of the edges of the cube.

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Problem 549

The axis of rotation of a circular cylinder passes through the point $(a,b,c)$ and is parallel to the vector $\overrightarrow {P}(l,m,n)$ . Prove that the equation of the cylinder has the form

\bigl(m(x-a)-l(y-b)\bigr)^{2}+\bigl(n(x-a)-l(z-c)\bigr)^{2}+\bigl(n(y-b)-m(z-c)\bigr)^{2} = r^{2}(l^{2}+m^{2}+n^{2}).

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Problem 550

Prove that the three curves:

\text{the parabola} \qquad x^{2}+4y^{2}+4xy-6x-2y+3 = 0, \qquad z = 0,

\text{the hyperbola} \qquad x = 0, \qquad 4y^{2}+4z^{2}-10yz-2y+2z+3 = 0

\text{and the ellipse} \qquad y = 0, \qquad x^{2}+4z^{2}-6x+2z+3 = 0

are obtained by intersecting one and the same conical surface with vertex at the point $(1,1,1)$ by the coordinate planes.

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Problem 551

Two spheres of radius $a$ are tangent to each other and to the plane $xOy$ . Find the locus of the centers of spheres tangent to the given spheres and to the plane $xOy$ , if the centers of the given spheres are at the points $(\pm a,0,a)$ .

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Problem 552

Find the locus of the centers of spheres tangent to the two lines:

\begin{cases} x = a+lt, \\ y = b+mt, \\ z = c+nt \end{cases} \qquad \text{and} \qquad \begin{cases} x = a+l_{1}t, \\ y = b_{1}+m_{1}t, \\ z = c_{1}+n_{1}t. \end{cases}

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Problem 553

Find the equation of the cylindrical surface of revolution having as generators the lines

x = y = z, \qquad \begin{cases} y = x+1, \\ z = x+2, \end{cases} \qquad \begin{cases} y = x-2, \\ z = x+1. \end{cases}

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Problem 554

Find the locus of the centers of spheres tangent to the lines

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

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