Chapter 7

Second-Order Surfaces

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Section 7.1Deriving Equations of Surfaces[42/100%]
Problem 649

Set up the equation of a sphere passing through the circle

x2+y2=11,z=0 x^{2}+y^{2} = 11, \quad z = 0

and tangent to the plane x+y+z−5=0x+y+z-5 = 0.

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Problem 650

Find the center CC and radius rr of the sphere passing through the origin and the line of intersection of the spheres x2+y2+z2−4x−6y−2z+5=0x^{2}+y^{2}+z^{2}-4 x-6 y-2 z + 5 = 0 and x2+y2+z2+2x−4y+2z−10=0x^{2}+y^{2}+z^{2}+2 x-4 y+2 z-10 = 0.

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Problem 651

Find the equation of the sphere of radius 3, tangent to the plane x−2y+2z=4x-2 y+2 z = 4 at the point (0,−1,1)(0,-1,1).

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Problem 652

Write the equation of a circular cone whose vertex is at the origin, whose axis coincides with the axis OzO z, and for which the angle between the axis and a generator of the cone equals γ\gamma.

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Problem 653

Write the equation of a circular cone whose vertex is at the point (x0,y0,z0)(x_{0}, y_{0}, z_{0}), whose axis is parallel to the vector (a,b,c)(a, b, c), and whose generators make an angle φ\varphi with the axis.

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Problem 654

Set up the equation of a circular cone whose vertex is at the point (1,2,3)(1,2,3), whose axis direction vector is (2,2,−1)(2,2,-1), and whose generators make an angle π/6\pi / 6 with the axis.

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Problem 655

Write the equation of a circular cone tangent to the planes OxzO x z and OyzO y z along the lines OxO x and OyO y.

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Problem 656

Write the equation of a circular cone for which the axes OxO x and OyO y are generators, and the axis OzO z makes an angle π/4\pi / 4 with the axis of the cone, which lies in the first and seventh octants. Also write the canonical equation of this cone.

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Problem 657

Set up the equation of a circular cone under the condition that all three coordinate axes are generators of the cone, and the axis of the cone lies in the first and seventh octants.

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Problem 658

Set up the equation of a circular cone tangent to the three coordinate planes, knowing that its axis lies in the first and seventh octants.

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Problem 659

Set up the equation of the cone circumscribed about the sphere with center at the point (0,4,1)(0,4,1) and radius 6, under the condition that the vertex of the cone is at the point (8,0,0)(8,0,0).

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Problem 660

Write the equation of the cone passing through the lines y=±x,z=0y = \pm x, z = 0 and the point (1,2,3)(1,2,3), for which the axis OzO z is an axis of symmetry.

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Problem 661

Write the equation of the cone with vertex at the point (0,0,a)(0,0, a), whose directrix is the hyperbola 2xy=a2,z=02 x y = a^{2}, z = 0. Using a transformation of rectangular coordinates, reduce the resulting equation of the cone to canonical form.

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Problem 662

Write the equation of the cone with vertex at the point (0,0,p)(0,0, p), whose directrix is the parabola y2=2px,z=0y^{2} = 2 p x, z = 0. Using a transformation of rectangular coordinates, reduce the resulting equation to canonical form.

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Problem 663

Write the equation of the cone whose vertex is at the point (0,0,0)(0,0,0), and whose directrix is the hyperbola x2−y24=1,z=3x^{2}-\frac{y^{2}}{4} = 1, z = 3.

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Problem 664

Write the equation of the cone with vertex at the origin, whose directrix is the ellipse

x2a2+y2b2+z2c2=1,xa+yb+zc=1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1, \quad \frac{x}{a}+\frac{y}{b}+\frac{z}{c} = 1
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Problem 665

Write the equation of the cone whose vertex is at the point (1,1,1)(1,1,1), and whose directrix is the parabola y2=2x,z=0y^{2} = 2 x, z = 0.

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Problem 666

Write the equation of the right circular cylinder of radius rr, whose axis is the line

x−x0a=y−y0b=z−z0c \frac{x-x_{0}}{a} = \frac{y-y_{0}}{b} = \frac{z-z_{0}}{c}
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Problem 667

Set up the equation of the right circular cylinder passing through the point (1,−2,1)(1,-2,1), whose axis is the line

x1=y−12=z+3−2 \frac{x}{1} = \frac{y-1}{2} = \frac{z+3}{-2}
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Problem 668

Set up the equation of the cylinder whose generators are tangent to the sphere x2+y2+z2=1x^{2}+y^{2}+z^{2} = 1 and make equal angles with the coordinate axes.

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Problem 669

Write the equation of a parabolic cylinder with parameter p=2/3p = 2/3, knowing the vertex O′(2,1,−1)O^{\prime }(2,1,-1) of the parabola obtained by intersecting the cylinder with a plane perpendicular to its generators, the direction vector e1′=(2/3,2/3,1/3)\mathbf{e}_{1}^{\prime } = (2/3,2/3,1/3) of the axis of this parabola, and the direction vector e2′=(2/3,−1/3,−2/3)\mathbf{e}_{2}^{\prime } = (2/3,-1/3,-2/3) of the tangent at its vertex.

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Problem 670

Write the equation of the cylinder whose directrix is the circle x2+y2=1,z=0x^{2}+y^{2} = 1, z = 0, and whose generators make equal angles with the coordinate axes.

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Problem 671

The directrix of a cylinder is the circle x2+y2=r2x^{2}+y^{2} = r^{2}, z=0z = 0, and its generators are parallel to the vector (0,β,γ)(0, \beta , \gamma ). Write the equation of this cylinder and reduce it to canonical form using a transformation of rectangular coordinates.

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Problem 672

Write the equation of a hyperbolic cylinder with directrix x2−y24=1,z=3x^{2}-\frac{y^{2}}{4} = 1, z = 3 and generators parallel to the vector (1,1,1)(1,1,1).

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Problem 673

Write the equation of a parabolic cylinder whose generators are parallel to the vector (1,1,1)(1,1,1), and whose directrix is the parabola y2=2x,z=0y^{2} = 2 x, z = 0.

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Problem 674

Write the equation of a hyperbolic cylinder with asymptotic planes x+y−z=1,2x−y+z=2x+y-z = 1,2 x-y+z = 2, passing through the point (1,−1,2)(1,-1,2).

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Problem 675

An ellipse with semi-axes aa and b,a>bb, a > b, revolves about its major axis, which coincides with the axis OzO z, so that the center of the ellipse coincides with the origin. Set up the equation of the surface traced by the ellipse as it revolves.

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Problem 676

A hyperbola with semi-axes aa and bb revolves about its real axis, which coincides with the axis OzO z, with the center of the hyperbola coinciding with the origin. Set up the equation of the surface traced by the hyperbola as it revolves.

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Problem 677

A hyperbola with semi-axes aa and bb revolves about its imaginary axis, which coincides with the axis OzO z, with the center of the hyperbola coinciding with the origin. Set up the equation of the surface traced by the hyperbola as it revolves.

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Problem 678

A parabola with parameter pp rotates about its own axis, which coincides with the axis OzO z. The vertex of the parabola coincides with the origin. Find the equation of the surface described by the parabola.

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Problem 679

In the plane OxzO x z a parabola x2=2pz,y=0x^{2} = 2 p z, y = 0 is given. The vertex of another parabola with parameter qq moves along it, its plane remaining at all times parallel to the plane OyzO y z, and its axis parallel to the axis OzO z. Find the equation of the surface described by the moving parabola.

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Problem 680

A line skew to the axis OzO z rotates about it. The angle this line makes with the axis OzO z remains constant and equal to γ\gamma; the common perpendicular to the axis OzO z and this line also retains a constant length rr and lies at all times in the plane OxyO x y. Find the equation of the surface described by the rotating line.

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Problem 681

Write the equation of a cone with vertex (2,3,6)(2,3,6), given that the plane Oxy intersects it in an ellipse whose axes are parallel to the axes OxO x and OyO y, the ellipse being tangent to these coordinate axes, and the axis of the cone itself being parallel to the axis OzO z.

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Problem 682

Write the equation of a paraboloid of revolution with parameter p=1/3p = 1 / 3, vertex (1,0,−1)(1,0,-1), and direction vector of the axis of rotation (2/3,1/3,−2/3)(2 / 3,1 / 3,-2 / 3).

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Problem 683

Write the equation of a paraboloid of revolution with parameter pp, vertex at the point O′(x0,y0,z0)O^{\prime }\left(x_{0}, y_{0}, z_{0}\right), and direction vector of the axis u=(a,b,c)\mathbf{u} = (a, b, c).

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Problem 684

An ellipse with semi-axes aa and b,a>bb, a > b, center at the point O′(x0,y0,z0)O^{\prime }\left(x_{0}, y_{0}, z_{0}\right), and direction vector of the major axis u=(α,β,γ)\mathbf{u} = (\alpha , \beta , \gamma ) rotates about its major axis. Write the equation of the resulting ellipsoid of revolution.

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Problem 685

Write the equation of an ellipsoid with vertices (0,0,6)(0,0,6) and (0,0,−2)(0,0,-2), given that the plane OxyO x y intersects it in a circle of radius 3.

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Problem 686

Write the equation of a two-sheeted hyperboloid with vertices (0,0,±6)(0,0, \pm 6), given that the planes OxzO x z and OyzO y z are its planes of symmetry and intersect it in hyperbolas whose asymptotes form angles with the axis OzO z equal to π/6\pi / 6 and π/3\pi / 3, respectively.

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Problem 687

Write the equation of an elliptic paraboloid with vertex (2,3,6(2,3,6) and axis parallel to the axis OzO z, given that the plane OxyO x y intersects it in an ellipse whose axes are parallel to the axes OxO x and OyO y, the ellipse being tangent to these coordinate axes.

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Problem 688

Find the equation of an ellipsoid whose axes coincide with the coordinate axes, given that it contains the circle

x2+y2+z2=9,z=x x^{2}+y^{2}+z^{2} = 9, \quad z = x

and the point (3,1,1)(3,1,1).

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Problem 689

Write the equation of a one-sheeted hyperboloid with equal semi-axes, passing through the lines y=±x,z=0y = \pm x, z = 0 and the point (1,2,3)(1,2,3), for which the axis OzO z is an axis of symmetry.

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Problem 690

Write the equation of a second-order surface passing through three circles:

x2+y2=1,z=0;x2+y2=9,z=1;x2+y2=25,z=2 x^{2}+y^{2} = 1, z = 0 ; \quad x^{2}+y^{2} = 9, z = 1 ; \quad x^{2}+y^{2} = 25, z = 2

and reduce the resulting equation to canonical form.

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Section 7.2Basic Properties of Second-Order Surfaces[23/91%]
Problem 691

Find the center and radius of the circle

{x2+y2+z2−12x+4y−6z+24=02x+2y+z+1=0 \begin{cases} x^{2}+y^{2}+z^{2}-12 x+4 y-6 z+24 = 0 \\ 2 x+2 y+z+1 = 0 \end{cases}
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Problem 692

Write the equations of the planes passing through the line

x−13−1=y+11=z4 \frac{x-13}{-1} = \frac{y+1}{1} = \frac{z}{4}

and tangent to the sphere x2+y2+z2−2x−4y−6z−67=0x^{2}+y^{2}+z^{2}-2 x-4 y-6 z-67 = 0.

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Problem 693

Find the equation of the sphere passing through the circle

{x2+y2+z2+2x−4y+4z−40=02x+2y−z+4=0 \begin{cases} x^{2}+y^{2}+z^{2}+2 x-4 y+4 z-40 = 0 \\ 2 x+2 y-z+4 = 0 \end{cases}

and through the origin.

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Problem 694

Prove that the line of intersection of the two parabolic cylinders y2=x,z2=1−xy^{2} = x, z^{2} = 1-x lies on a circular cylinder. What is the equation of this cylinder?

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Problem 695

Prove that a paraboloid of revolution and a circular cylinder whose axes are parallel intersect in an ellipse whose major axis lies in the plane passing through the axes of the given surfaces.

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Problem 696

Along what curve do the one-sheeted hyperboloid x2a2+y2b2−z2c2=1,a>b\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} = 1, a > b, and the sphere x2+y2+z2=a2x^{2}+y^{2}+z^{2} = a^{2} intersect?

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Problem 697

Prove that the line of intersection of the parabolic cylinder z2=x+yz^{2} = x+y with the paraboloid of revolution x2+y2=zx^{2}+y^{2} = z lies on a sphere. Find the center and radius of this sphere.

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Problem 698

Along what curve do the elliptic paraboloid x2p+y2q=2z,p>q>0\frac{x^{2}}{p}+\frac{y^{2}}{q} = 2 z, p > q > 0, and the sphere x2+y2+z2=2pzx^{2}+y^{2}+z^{2} = 2 p z intersect?

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Problem 699

Prove that the orthogonal projection of any plane section of a paraboloid of revolution onto a plane perpendicular to the axis of the paraboloid is a circle or a line.

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Problem 700

Along what curve do the two ellipsoids

x2a2+y2b2+z2c2=1,x2b2+y2a2+z2c2=1, where a>b? \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1, \quad \frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{c^{2}} = 1, \quad \text{ where } a > b?

intersect?

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Problem 701

Find the line of intersection of the surfaces x2+y2−z2=a2x^{2}+y^{2}-z^{2} = a^{2}, x2−y2=2azx^{2}-y^{2} = 2 a z.

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Problem 702

Find the acute angle between the generators of the cone x2+y2−−z2=0x^{2}+y^{2}- -z^{2} = 0 along which it is intersected by the plane 5x+10y−11z=05 x+10 y-11 z = 0.

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Problem 703

Determine the type and position of the line of intersection of the plane x=9x = 9 and the hyperboloid

x29+y28−z22=1 \frac{x^{2}}{9}+\frac{y^{2}}{8}-\frac{z^{2}}{2} = 1
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Problem 704

Write the equation of the plane parallel to the plane OyzO y z and intersecting the one-sheeted hyperboloid

x29+y24−z2=1 \frac{x^{2}}{9}+\frac{y^{2}}{4}-z^{2} = 1

in a hyperbola whose real semi-axis equals 1.

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Problem 705

Find the necessary and sufficient condition for the plane z=ax+by+cz = a x+b y+c to intersect the paraboloid of revolution x2+y2=2pz(p>0)x^{2}+y^{2} = 2 p z(p > 0) in a real ellipse.

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Problem 706

Along what curve do the hyperbolic paraboloid

x29−y24=2z \frac{x^{2}}{9}-\frac{y^{2}}{4} = 2 z

and the plane 2x+3y−6=02 x+3 y-6 = 0 intersect?

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Problem 707

Find the equation of a second-order surface, given that it intersects the plane OxyO x y in the circle x2+y2−12x−−18y+32=0,z=0x^{2}+y^{2}-12 x- -18 y+32 = 0, z = 0, and the planes OxzO x z and OyzO y z in parabolas whose axes are parallel to the positive direction of the axis OzO z, with the parameter of the parabola lying in the plane OxzO x z equal to 1.

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Problem 708

Write the equation of a second-order surface intersecting the plane OxyO x y in a parabola, and the planes OxzO x z and OyzO y z in circles of radius rr tangent to the positive coordinate semi-axes. Using a transformation of rectangular coordinates, reduce the equation to canonical form.

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Problem 709

Find the locus of foci of the hyperbolas obtained by intersecting the hyperbolic paraboloid

x2p−y2q=2z \frac{x^{2}}{p}-\frac{y^{2}}{q} = 2 z

with planes parallel to the plane Oxy.

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Problem 710

Find the equation of a second-order surface passing through the points (0,0,0),(1,1,−1),(0,0,1)(0,0,0),(1,1,-1),(0,0,1), for which the planes

x+y+z=0,2x−y−z=0,y−z+1=0 x+y+z = 0, \quad 2 x-y-z = 0, \quad y-z+1 = 0

are planes of symmetry. Write the canonical equation of this surface and indicate its type.

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Problem 711

Find the equation of a second-order surface for which the planes

x+y+z=0,2x−y−z−2=0,y−z+1=0 x+y+z = 0, \quad 2 x-y-z-2 = 0, \quad y-z+1 = 0

are planes of symmetry and which passes through the points (1,0,0),(0,−1,0),(1,1,−1)(1,0,0),(0,-1,0),(1,1,-1).

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Problem 712

Find the type of surface consisting of the lines along which mutually perpendicular planes passing through two given lines intersect, depending on the relative position of the latter.

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Problem 713

Find the locus of points such that the ratio of the distances from them to two given skew lines is the same number kk.

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Section 7.3Reducing a Surface to Canonical Form[6/100%]
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:

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(1)

x2+2xy+y2−z2+2z−1=0x^{2}+2 x y+y^{2}-z^{2}+2 z-1 = 0

(2)

3x2+3y2+3z2−6x+4y−1=03 x^{2}+3 y^{2}+3 z^{2}-6 x+4 y-1 = 0;

(3)

3x2+3y2−6x+4y−1=03 x^{2}+3 y^{2}-6 x+4 y-1 = 0;

(4)

3x2+3y2−3z2−6x+4y+4z+3=03 x^{2}+3 y^{2}-3 z^{2}-6 x+4 y+4 z+3 = 0;

(5)

4x2+y2−4xy−36=04 x^{2}+y^{2}-4 x y-36 = 0.

Problem 715

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

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(1)

x2+4y2+9z2−6x+8y−36z=0x^{2}+4 y^{2}+9 z^{2}-6 x+8 y-36 z = 0;

(2)

4x2−y2−z2+32x−12z+44=04 x^{2}-y^{2}-z^{2}+32 x-12 z+44 = 0;

(3)

3x2−y2+3z2−18x+10y+12z+14=03 x^{2}-y^{2}+3 z^{2}-18 x+10 y+12 z+14 = 0;

(4)

6y2+6z2+5x+6y+30z−11=06 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:

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(1)

z2=2xyz^{2} = 2 x y

(2)

z=xyz = x y

(3)

z2=3x+4yz^{2} = 3 x+4 y;

(4)

z2=3x2+4xyz^{2} = 3 x^{2}+4 x y

(5)

z2=x2+2xy+y2+1z^{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:

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(1)

x2−2y2+z2+4xy−4yz−8zx−14x−4y+14z+18=0x^{2}-2 y^{2}+z^{2}+4 x y-4 y z-8 z x-14 x-4 y+14 z+18 = 0;

(2)

5x2+8y2+5z2−4xy+4yz+8zx−6x+6y+6z+10=05 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)

2yz+2zx+2xy+2x+2y+2z+1=02 y z+2 z x+2 x y+2 x+2 y+2 z+1 = 0;

(4)

3x2+3y2+3z2−2xy−2xz−2yz−2x−2y−2z−1=03 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)

2x2+6y2+2z2+8xz−4x−8y+3=02 x^{2}+6 y^{2}+2 z^{2}+8 x z-4 x-8 y+3 = 0;

(6)

5x2+2y2+5z2−4xy−4yz−2zx+10x−4y−2z+4=05 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)

x2+y2+z2−xy−yz−zx−1=0x^{2}+y^{2}+z^{2}-x y-y z-z x-1 = 0;

(8)

4xy+4yz+4zx+4x+4y+4z+3=04 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:

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(1)

4x2+y2+4z2−4xy+4yz−8zx−28x+2y+16z+45=04 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)

2x2+5y2+2z2−2xy−4xz+2yz+2x−10y−2z−1=02 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)

7x2+7y2+16z2−10xy−8yz−8zx−16x−16y−8z+72=07 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)

4x2+4y2−8z2−10xy+4yz+4zx−16x−16y+10z−2=04 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)

2x2−7y2−4z2+4xy+20yz−16zx+60x−12y+12z−90=02 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)

7x2+6y2+5z2−4xy−4yz−6x−24y+18z+30=07 x^{2}+6 y^{2}+5 z^{2}-4 x y-4 y z-6 x-24 y+18 z+30 = 0;

(7)

2x2+2y2−5z2+2xy−2x−4y−4z+2=02 x^{2}+2 y^{2}-5 z^{2}+2 x y-2 x-4 y-4 z+2 = 0;

(8)

2x2+2y2+3z2+4xy+2yz+2zx−4x+6y−2z+3=02 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)

x2+5y2+z2+2xy+2yz+6zx−2x+6y+2z=0x^{2}+5 y^{2}+z^{2}+2 x y+2 y z+6 z x-2 x+6 y+2 z = 0;

(10)

x2−2y2+z2+4xy+4yz−10zx+2x+4y−10z−1=0x^{2}-2 y^{2}+z^{2}+4 x y+4 y z-10 z x+2 x+4 y-10 z-1 = 0;

(11)

2x2+y2+2z2−2xy+2yz+4x+4z=02 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):

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(1)

4x2+6y2+4z2+4xz−8y−4z+3=04 x^{2}+6 y^{2}+4 z^{2}+4 x z-8 y-4 z+3 = 0

(2)

x2+5y2+z2+2xy+6xz+2yz−2x+6y−10z=0x^{2}+5 y^{2}+z^{2}+2 x y+6 x z+2 y z-2 x+6 y-10 z = 0;

(3)

x2+y2−3z2−2xy−6xz−6yz+2x+2y+4z=0x^{2}+y^{2}-3 z^{2}-2 x y-6 x z-6 y z+2 x+2 y+4 z = 0;

(4)

x2−2y2+z2+4xy−8xz−4yz−14x−4y+14z+16=0x^{2}-2 y^{2}+z^{2}+4 x y-8 x z-4 y z-14 x-4 y+14 z+16 = 0;

(5)

2x2+y2+2z2−2xy−2yz+x−4y−3z+2=02 x^{2}+y^{2}+2 z^{2}-2 x y-2 y z+x-4 y-3 z+2 = 0;

(6)

x2−2y2+z2+4xy−10xz+4yz+x+y−z=0x^{2}-2 y^{2}+z^{2}+4 x y-10 x z+4 y z+x+y-z = 0;

(7)

2x2+y2+2z2−2xy−2yz+4x−2y=02 x^{2}+y^{2}+2 z^{2}-2 x y-2 y z+4 x-2 y = 0;

(8)

x2−2y2+z2+4xy−10xz+4yz+2x+4y−10z−1=0x^{2}-2 y^{2}+z^{2}+4 x y-10 x z+4 y z+2 x+4 y-10 z-1 = 0;

(9)

x2+y2+4z2+2xy+4xz+4yz−6z+1=0x^{2}+y^{2}+4 z^{2}+2 x y+4 x z+4 y z-6 z+1 = 0;

(10)

4xy+2x+4y−6z−3=04 x y+2 x+4 y-6 z-3 = 0

(11)

xy+xz+yz+2x+2y−2z=0x y+x z+y z+2 x+2 y-2 z = 0.

Section 7.4Orthogonal Invariants of Second-Order Surfaces[18/83%]
Problem 720

Using the invariants

I1=a11+a22+a33,I2=∣a11a12a12a22∣+∣a22a23a23a33∣+∣a11a13a13a33∣I3=∣a11a12a13a12a22a23a13a23a33∣I4=∣a11a12a13a1a12a22a23a2a13a23a33a3a1a2a3a∣ \begin{aligned} & I_{1} = a_{11}+a_{22}+a_{33}, \quad I_{2} =\left|\begin{smallmatrix} a_{11} & a_{12} \\ a_{12} & a_{22} \end{smallmatrix}\right|+\left|\begin{smallmatrix} a_{22} & a_{23} \\ a_{23} & a_{33} \end{smallmatrix}\right|+\left|\begin{smallmatrix} a_{11} & a_{13} \\ a_{13} & a_{33} \end{smallmatrix}\right| \\ & I_{3} =\left|\begin{smallmatrix} a_{11} & a_{12} & a_{13} \\ a_{12} & a_{22} & a_{23} \\ a_{13} & a_{23} & a_{33} \end{smallmatrix}\right| \\ & I_{4} =\left|\begin{smallmatrix} a_{11} & a_{12} & a_{13} & a_{1} \\ a_{12} & a_{22} & a_{23} & a_{2} \\ a_{13} & a_{23} & a_{33} & a_{3} \\ a_{1} & a_{2} & a_{3} & a \end{smallmatrix}\right| \end{aligned}

find the necessary and sufficient conditions for the second-order surface given by the general equation

a11x2+a22y2+a33z2+2a12xy+2a23yz+2a13zx++2a1x+2a2y+2a3z+a=0 \begin{aligned} a_{11} x^{2}+a_{22} y^{2}+a_{33} z^{2}+2 a_{12} x y+2 a_{23} y z & +2 a_{13} z x+ \\ + & 2 a_{1} x+2 a_{2} y+2 a_{3} z+a = 0 \end{aligned}

to be:

?
(1)

an ellipsoid;

(2)

an imaginary ellipsoid;

(3)

an imaginary cone;

(4)

a one-sheeted hyperboloid;

(5)

a two-sheeted hyperboloid;

(6)

a cone;

(7)

an elliptic paraboloid;

(8)

a hyperbolic paraboloid.

Problem 721

(Orthogonal semi-invariants). Prove that:

?
(1)

the functions

I3∗=∣a11a12a1a12a22a2a1a2a∣+∣a22a23a2a23a33a3a2a3a∣+∣a11a13a1a13a33a3a1a3a∣I2∗=∣a11a1a1a∣+∣a22a2a2a∣+∣a33a3a3a∣ \begin{aligned} I_{3}^{*} & =\left|\begin{smallmatrix} a_{11} & a_{12} & a_{1} \\ a_{12} & a_{22} & a_{2} \\ a_{1} & a_{2} & a \end{smallmatrix}\right|+\left|\begin{smallmatrix} a_{22} & a_{23} & a_{2} \\ a_{23} & a_{33} & a_{3} \\ a_{2} & a_{3} & a \end{smallmatrix}\right|+\left|\begin{smallmatrix} a_{11} & a_{13} & a_{1} \\ a_{13} & a_{33} & a_{3} \\ a_{1} & a_{3} & a \end{smallmatrix}\right| \\ I_{2}^{*} & =\left|\begin{smallmatrix} a_{11} & a_{1} \\ a_{1} & a \end{smallmatrix}\right|+\left|\begin{smallmatrix} a_{22} & a_{2} \\ a_{2} & a \end{smallmatrix}\right|+\left|\begin{smallmatrix} a_{33} & a_{3} \\ a_{3} & a \end{smallmatrix}\right| \end{aligned}

of the coefficients of a second-degree polynomial in three variables

a11x2+a22y2+a33z2+2a23yz+2a13zx++2a1x+2a2y+2a3z+a=0 \begin{aligned} a_{11} x^{2}+a_{22} y^{2}+a_{33} z^{2}+2 a_{23} y z & +2 a_{13} z x+ \\ & +2 a_{1} x+2 a_{2} y+2 a_{3} z+a = 0 \end{aligned}

are invariants of a homogeneous orthogonal transformation of variables;

(2)

I3∗I_{3}^{*} is an invariant of an inhomogeneous orthogonal transformation of variables, if

I4=∣a11a12a13a1a12a22a23a2a13a23a33a3a1a2a3a∣=0,I3=∣a11a12a13a12a22a23a13a23a33∣=0 I_{4} =\left|\begin{smallmatrix} a_{11} & a_{12} & a_{13} & a_{1} \\ a_{12} & a_{22} & a_{23} & a_{2} \\ a_{13} & a_{23} & a_{33} & a_{3} \\ a_{1} & a_{2} & a_{3} & a \end{smallmatrix}\right| = 0, \quad I_{3} =\left|\begin{smallmatrix} a_{11} & a_{12} & a_{13} \\ a_{12} & a_{22} & a_{23} \\ a_{13} & a_{23} & a_{33} \end{smallmatrix}\right| = 0
(3)

I2∗I_{2}^{*} is an invariant of an inhomogeneous orthogonal transformation of variables, if I3=I4=0I_{3} = I_{4} = 0 and

I2=∣a11a12a12a22∣+∣a22a23a23a33∣+∣a11a13a13a33∣=0,I3∗=0 I_{2} =\left|\begin{smallmatrix} a_{11} & a_{12} \\ a_{12} & a_{22} \end{smallmatrix}\right|+\left|\begin{smallmatrix} a_{22} & a_{23} \\ a_{23} & a_{33} \end{smallmatrix}\right|+\left|\begin{smallmatrix} a_{11} & a_{13} \\ a_{13} & a_{33} \end{smallmatrix}\right| = 0, \quad I_{3}^{*} = 0
Problem 722

Using the invariants I1,I2,I3,I4I_{1}, I_{2}, I_{3}, I_{4} and semi-invariants I2∗,I3∗I_{2}^{*}, I_{3}^{*}, find the necessary and sufficient conditions for the second-order surface given by the general equation to be:

?
(1)

an elliptic cylinder;

(2)

an imaginary elliptic cylinder;

(3)

a pair of imaginary intersecting planes;

(4)

a hyperbolic cylinder;

(5)

a pair of real intersecting planes;

(6)

a parabolic cylinder;

(7)

a pair of real parallel planes;

(8)

a pair of imaginary parallel planes;

(9)

a pair of coincident planes.

Problem 723

Without performing coordinate changes, find the canonical equations of the surfaces:

?
(1)

3x2+3y2+4xy+2xz−2yz−6x−14y+9=03 x^{2}+3 y^{2}+4 x y+2 x z-2 y z-6 x-14 y+9 = 0

(2)

3x2+3y2+4xy+2xz−2yz−8x−2y−6z+8=03 x^{2}+3 y^{2}+4 x y+2 x z-2 y z-8 x-2 y-6 z+8 = 0;

(3)

3x2+3y2+4xy+2xz−2yz−2x−8y+2z+5=03 x^{2}+3 y^{2}+4 x y+2 x z-2 y z-2 x-8 y+2 z+5 = 0;

(4)

−z2+4xy−2xz+2yz+2x−2y+4z=0-z^{2}+4 x y-2 x z+2 y z+2 x-2 y+4 z = 0;

(5)

4x2+4y2+2z2+2xy+2xz−2yz−10x−10y−10=04 x^{2}+4 y^{2}+2 z^{2}+2 x y+2 x z-2 y z-10 x-10 y-10 = 0;

(6)

2x2+2y2+z2+xy+xz−yz−3x−2y−3z+5=02 x^{2}+2 y^{2}+z^{2}+x y+x z-y z-3 x-2 y-3 z+5 = 0;

(7)

3x2+3y2+2z2−2xy+4xz−4yz−1=03 x^{2}+3 y^{2}+2 z^{2}-2 x y+4 x z-4 y z-1 = 0.

Problem 724

Find the largest angle φ\varphi between the generators of the cone z2+2xy+2xz+2yz=0z^{2} + 2 x y+2 x z+2 y z = 0, as well as the angles α,β,γ\alpha , \beta , \gamma that its axis makes with the coordinate axes.

?
Problem 725

Express, using invariants, the necessary and sufficient conditions for a general second-degree equation to define a paraboloid of revolution, and find its parameter.

?
Problem 726

Using invariants, express the necessary and sufficient conditions for a general second-degree equation to define a hyperbolic paraboloid with equal parameters of the principal sections, and find these parameters.

?
Problem 727

Using invariants, find the necessary and sufficient condition for a general second-degree equation in three unknowns to define a one-sheeted or two-sheeted hyperboloid with equal semi-axes.

Using the invariants, write the canonical equation of these hyperboloids.

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Problem 728

A second-order surface is given

x2+y2+z2+2axy+2axz+2ayz=1 x^{2}+y^{2}+z^{2}+2 a x y+2 a x z+2 a y z = 1

Prove that for all values of the parameter aa this surface is a surface of revolution; find the direction vector of its axis of rotation.

For which values of the parameter aa will the surface be an ellipsoid?

?
Problem 729

Determine kk so that the surface x2−2xy+kz2=0x^{2}-2 x y+k z^{2} = 0 is a cone of revolution, and find the axis of rotation.

?
Problem 730

Find the equation of a second-order cone intersecting the plane OyzO y z in the circle y2+z2=2ry,x=0y^{2}+z^{2} = 2 r y, x = 0, and the plane OxzO x z in the parabola z2=2px,y=0z^{2} = 2 p x, y = 0.

?
Problem 731

Find the equation of a second-order cone on which the circles

y2+z2−2by=0,x=0;x2+z2−2ax=0,y=0 y^{2}+z^{2}-2 b y = 0, x = 0 ; \quad x^{2}+z^{2}-2 a x = 0, y = 0

lie.

?
Problem 732

Find the equation of a second-order surface intersecting the coordinate planes in the hyperbolas

yz=a22,x=0;xz=b22,y=0;xy=c22,z=0 y z = \frac{a^{2}}{2}, x = 0 ; \quad x z = \frac{b^{2}}{2}, \quad y = 0 ; \quad x y = \frac{c^{2}}{2}, z = 0

Determine the type of this surface. Reduce its equation to canonical form when a=b=c=1a = b = c = 1.

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Problem 733

Find the equation of a paraboloid passing through the lines x=0,z=2x = 0, z = 2 and y=0,z=−2y = 0, z = -2 and through the two points (0,1,−1),(1,−1,0)(0,1,-1),(1,-1,0).

?
Problem 734

Prove that the sum of the reciprocals of the squares of the lengths of any three pairwise perpendicular radii of the ellipsoid

x2a2+y2b2+z2c2=1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1

is constant and equals

1r12+1r22+1r32=1a2+1b2+1c2 \frac{1}{r_{1}^{2}}+\frac{1}{r_{2}^{2}}+\frac{1}{r_{3}^{2}} = \frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}
?
Problem 735

Prove that all planes passing through the endpoints of three pairwise perpendicular radii of an ellipsoid are tangent to the sphere inscribed in the cube inscribed in the ellipsoid.

?
Problem 736

Prove that the locus of the feet of the perpendiculars dropped from the center of an ellipsoid onto the planes passing through the endpoints of three pairwise perpendicular diameters of the ellipsoid is the sphere inscribed in the cube which, in turn, is inscribed in the given ellipsoid.

?
Problem 737

Find the locus of vertices of a right trihedral angle whose edges are tangent to the ellipsoid

x2a2+y2a2+z2a2=1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{a^{2}} = 1
?
Section 7.5Tangent and Diametral Planes. Rectilinear Generators[41/88%]
Problem 738

Find the necessary and sufficient conditions for a non-degenerate second-order surface

a11x2+a22y2+a33z2+2a23yz+2a13zx+2a12xy++2a1x+2a2y+2a3z+a=0 \begin{aligned} a_{11} x^{2}+a_{22} y^{2}+a_{33} z^{2}+2 a_{23} y z+2 a_{13} z x & +2 a_{12} x y+ \\ & +2 a_{1} x+2 a_{2} y+2 a_{3} z+a = 0 \end{aligned}

to be tangent to the plane OxyO x y at the origin.

?
Problem 739

Prove that a tangent plane to a second-order surface always intersects the surface in a pair of lines (real, imaginary, or coincident).

?
Problem 740

Write the equation of the tangent plane to the ellipsoid

x232+y218+z25=1 \frac{x^{2}}{32}+\frac{y^{2}}{18}+\frac{z^{2}}{5} = 1

passing through the point (12,−3,−1)(12,-3,-1) and parallel to the axis OzO z.

?
Problem 741

Write the equation of the plane passing through the line

x−22=y−3−1=z−20 \frac{x-2}{2} = \frac{y-3}{-1} = \frac{z-2}{0}

and tangent to the ellipsoid

x216+y212+z24=1 \frac{x^{2}}{16}+\frac{y^{2}}{12}+\frac{z^{2}}{4} = 1
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Problem 742

Find the equation of the plane passing through the line

x−150=y2=z−11−1 \frac{x-15}{0} = \frac{y}{2} = \frac{z-11}{-1}

and tangent to the hyperbolic paraboloid

x29−y24=2z \frac{x^{2}}{9}-\frac{y^{2}}{4} = 2 z
?
Problem 743

Find the equation of the tangent plane to the surface

2x2+5y2+2z2−2xy+6yz−4x−y−2z=0 2 x^{2}+5 y^{2}+2 z^{2}-2 x y+6 y z-4 x-y-2 z = 0

passing through the line 4x−5y=0,z−1=04 x-5 y = 0, z-1 = 0.

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Problem 744

Find the tangent plane to the surface

4x2+6y2+4z2+4xz−8y−4z+3=0 4 x^{2}+6 y^{2}+4 z^{2}+4 x z-8 y-4 z+3 = 0

parallel to the plane x+2y+2=0x+2 y+2 = 0.

?
Problem 745

Find the tangent plane to the one-sheeted hyperboloid

x2a2+y2b2−z2c2=1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} = 1

cutting off on the axes OxO x and OyO y segments equal to aa and bb respectively, and find the lines along which this plane intersects the hyperboloid.

?
Problem 746

Write the equation of the tangent plane to the elliptic paraboloid

x2p+y2q=2z \frac{x^{2}}{p}+\frac{y^{2}}{q} = 2 z

cutting off on the axes OxO x and OyO y segments equal to pp and qq respectively.

?
Problem 747

Given the hyperbolic paraboloid

x22−y28=2z \frac{x^{2}}{2}-\frac{y^{2}}{8} = 2 z

and the plane 2x+3y−z=02 x+3 y-z = 0. Write the equation of the plane parallel to the given one and intersecting the paraboloid in a pair of lines; find these lines.

?
Problem 748

Find the point of intersection of the rectilinear generators of the one-sheeted hyperboloid x2+y2−z2=1x^{2}+y^{2}-z^{2} = 1 along which it is intersected by the plane parallel to the plane x+y−z=0x+y-z = 0, and determine the angle between these generators.

?
Problem 749

Find the point of intersection of the rectilinear generators of the hyperbolic paraboloid x2−y2=2zx^{2}-y^{2} = 2 z along which it is intersected by the plane parallel to the plane x−y+z+1=0x-y+z+1 = 0, and determine the angle between these generators.

?
Problem 750

Find the angle φ\varphi between the rectilinear generators of the one-sheeted hyperboloid

x2+y2−z24=1 x^{2}+y^{2}-\frac{z^{2}}{4} = 1

passing through the point (1,4,8)(1,4,8), taking on these generators the rays directed from the given point toward the throat ellipse.

?
Problem 751

Given the one-sheeted hyperboloid

x24+y29−z216=1 \frac{x^{2}}{4}+\frac{y^{2}}{9}-\frac{z^{2}}{16} = 1

Through its generator

x−20=y3=z4 \frac{x-2}{0} = \frac{y}{3} = \frac{z}{4}

and the point (0,3,0)(0,3,0) a plane is drawn. Find the second line of the intersection of this plane with the hyperboloid.

?
Problem 752

Prove that the orthogonal projections of the rectilinear generators of the one-sheeted hyperboloid

x2a2+y2b2−z2c2=1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} = 1

and the hyperbolic paraboloid

x2p−y2q=2z \frac{x^{2}}{p}-\frac{y^{2}}{q} = 2 z

onto the coordinate planes are tangent to the sections of these surfaces by the coordinate planes (for the hyperbolic paraboloid only the projections onto the planes OyzO y z and OxzO x z are considered).

?
Problem 753

Find the rectilinear generators of the surface

x2+y2+z2+2xy−2xz−yz+4x+3y−5z+4=0 x^{2}+y^{2}+z^{2}+2 x y-2 x z-y z+4 x+3 y-5 z+4 = 0

passing through the point (−1,−1,1)(-1,-1,1).

?
Problem 754

Given the hyperbolic paraboloid

x216−y29=2z \frac{x^{2}}{16}-\frac{y^{2}}{9} = 2 z

Through its generator

x4=y3=z0 \frac{x}{4} = \frac{y}{3} = \frac{z}{0}

and the point (1,1,1)(1,1,1) a plane is drawn. Find the second line of the intersection of the paraboloid with this plane.

?
Problem 755

Prove that if the equation

a11x2+a22y2+a33z2+2a23yz+2a13zx+2a12xy++2a1x+2a2y+2a3z+a=0 \begin{aligned} a_{11} x^{2}+a_{22} y^{2}+a_{33} z^{2}+2 a_{23} y z+2 a_{13} z x & +2 a_{12} x y+ \\ & +2 a_{1} x+2 a_{2} y+2 a_{3} z+a = 0 \end{aligned}

defines a paraboloid, then the equation

2a1x+2a2y+2a3z+a=0 2 a_{1} x+2 a_{2} y+2 a_{3} z+a = 0

is the equation of a tangent plane to this paraboloid.

?
Problem 756
?
(1)

Find the coordinates of the direction vectors of the rectilinear generators of the one-sheeted hyperboloid

x2a2+y2b2−z2c2=1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} = 1

passing through the point (x0,y0,z0)\left(x_{0}, y_{0}, z_{0}\right) of the hyperboloid.

(2)

Consider the special case where the point belongs to the throat ellipse of the surface.

Problem 757

Find the direction vectors of the rectilinear generators of the hyperbolic paraboloid

x2p−y2q=2z \frac{x^{2}}{p}-\frac{y^{2}}{q} = 2 z

passing through the point (x0,y0,z0)\left(x_{0}, y_{0}, z_{0}\right) of the paraboloid.

?
Problem 758

Find the locus of points lying on the hyperbolic paraboloid

x2p−y2q=2z \frac{x^{2}}{p}-\frac{y^{2}}{q} = 2 z

through each of which two mutually perpendicular generators pass.

?
Problem 759

Write the equation of a second-order surface containing the three lines:

x=1/2,y=z;y=−1,z=2x;y=1,z=−2x x = 1 / 2, \quad y = z ; \quad y = -1, z = 2 x ; \quad y = 1, z = -2 x
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Problem 760

Under what necessary and sufficient condition is the plane Ax+By+Cz+D=0A x+B y+C z+D = 0 tangent to the non-degenerate second-order surface

a11x2+a22y2+a33z2+2a23yz+2a13zx+2a12xy++2a1x+2a2y+2a3z+a=0? \begin{aligned} & a_{11} x^{2}+a_{22} y^{2}+a_{33} z^{2}+2 a_{23} y z+2 a_{13} z x+2 a_{12} x y+ \\ & +2 a_{1} x+2 a_{2} y+2 a_{3} z+a = 0? \end{aligned}
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Problem 761

Prove that any plane intersects any one-sheeted hyperboloid, hyperbolic paraboloid, and second-order cone.

?
Problem 762

Under what necessary and sufficient condition does the plane Ax+By+Cz+D=0A x+B y+C z+D = 0 intersect the following surfaces:

?
(1)

x2a2+y2b2+z2c2=1\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1;

(2)

x2a2+y2b2−z2c2=−1\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} = -1;

(3)

x2p+y2q=2z(p>0,q>0)\frac{x^{2}}{p}+\frac{y^{2}}{q} = 2 z(p > 0, q > 0)?

Problem 763

Under what necessary and sufficient condition is the plane Ax+By+Cz+D=0A x+B y+C z+D = 0 tangent to the following surfaces:

?
(1)

x2a2+y2b2+z2c2=1\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1;

(2)

x2a2+y2b2−z2c2=1\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} = 1;

(3)

x2a2+y2b2−z2c2=−1\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} = -1;

(4)

x2p+y2q=2z\frac{x^{2}}{p}+\frac{y^{2}}{q} = 2 z;

(5)

x2p−y2q=2z\frac{x^{2}}{p}-\frac{y^{2}}{q} = 2 z?

Problem 764

Write the equation of a second-order surface passing through the point (0,0,1)(0,0,1), having its center at the point (0,0,−1)(0,0,-1), and intersecting the plane OxyO x y in the line 3x2−4xy−3=0,z=03 x^{2}-4 x y-3 = 0, z = 0.

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Problem 765

Given the one-sheeted hyperboloid

x24+y29−z216=1 \frac{x^{2}}{4}+\frac{y^{2}}{9}-\frac{z^{2}}{16} = 1

and the plane 6x+4y−3z−12=06 x+4 y-3 z-12 = 0. Determine the direction of the chords conjugate to the diametral plane parallel to the given plane.

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Problem 766

Write the equation of the diametral plane of the hyperbolic paraboloid

x26−y29=2z \frac{x^{2}}{6}-\frac{y^{2}}{9} = 2 z

passing through the line x=y,z=1x = y, z = 1, and find the direction of the chords conjugate to this plane.

?
Problem 767

Given the elliptic paraboloid

x28+y218=2z \frac{x^{2}}{8}+\frac{y^{2}}{18} = 2 z

and the two points (3,0,5)(3,0,5) and (0,4,7)(0,4,7). Write the equation of the diametral plane of the paraboloid passing through the given points, and determine the direction of the chords conjugate to it.

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Problem 768

Given the paraboloid of revolution x2+y2=2pzx^{2}+y^{2} = 2 p z and the circular cylinder (x−a)2+z2=r2(x-a)^{2}+z^{2} = r^{2}. Write the equation of a plane that would be a diametral plane for both the paraboloid and the cylinder.

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Problem 769

Write the equation of the plane intersecting the one-sheeted hyperboloid

x22+y23−z25=1 \frac{x^{2}}{2}+\frac{y^{2}}{3}-\frac{z^{2}}{5} = 1

in a line whose center is at the point (6,6,5)(6,6,5).

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Problem 770

Prove that if three non-coplanar chords of a second-order surface pass through a single point and are bisected by it, then this point is the center of the surface.

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Problem 771

Prove that if an arbitrary line not having an asymptotic direction of a second-order surface is taken as the axis OzO z, and the diametral plane conjugate to the direction of this line is taken as the plane Oxy, then the equation of the surface will contain no terms with the products xzx z and yzy z.

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Problem 772

How will the equation of a one-sheeted hyperboloid be written if a point of the surface is taken as the origin, the rectilinear generators passing through it are taken as the axes OxO x and OyO y, and the diameter passing through it is taken as the axis Oz\mathrm{Oz}?

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Problem 773

Prove that every plane parallel to the axis of symmetry of an elliptic paraboloid is a diametral plane conjugate to a uniquely determined non-asymptotic direction. Is this true for a hyperbolic paraboloid?

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Problem 774

Write the equation of a hyperbolic paraboloid, taking an arbitrary point OO of the surface as the origin, the rectilinear generators passing through this point as the axes OxO x and OyO y, the diameter passing through it as the axis OzO z, and an arbitrary point of the surface distinct from the point OO as the unit point EE.

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Problem 775

Write the equation of a second-order cone, taking its generators as the axes OxO x and OyO y, the line conjugate to the plane passing through these generators as the axis OzO z, and an arbitrary point of the cone distinct from its vertex as the unit point EE.

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Problem 776

How will the equation of a second-order surface be written if a point OO of the surface is taken as the origin, the diameter passing through this point as the axis Oz\mathrm{Oz}, and lines lying in the tangent plane and having conjugate directions with respect to the given surface as the axes OxO x and OyO y?

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Problem 777

Write the equation of an ellipsoid, taking its center as the origin, pairwise conjugate diameters as the coordinate axes, and the points of intersection of these diameters with the ellipsoid as the unit points of the coordinate axes.

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Problem 778

Two planes, each of which is parallel to the direction conjugate to the other with respect to a second-order surface, are called conjugate with respect to this surface. Find the necessary and sufficient condition for the two planes

A1x+B1y+C1z+D1=0,A2x+B2y+C2z+D2=0 A_{1} x+B_{1} y+C_{1} z+D_{1} = 0, \quad A_{2} x+B_{2} y+C_{2} z+D_{2} = 0

to be conjugate with respect to the second-order surface

a11x2+a22y2+a33z2+2a23yz+2a13zx+2a12xy++2a1x+2a2y+2a3z+a=0 \begin{aligned} a_{11} x^{2}+a_{22} y^{2}+a_{33} z^{2}+2 a_{23} y z+2 a_{13} z x & +2 a_{12} x y+ \\ & +2 a_{1} x+2 a_{2} y+2 a_{3} z+a = 0 \end{aligned}
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Section 7.6Plane Sections of Second-Order Surfaces[33/94%]
Problem 779

Determine whether the plane 2x+2y+z−3=02 x+2 y+z-3 = 0 intersects the ellipsoid x2+y2+z24=1x^{2}+y^{2}+\frac{z^{2}}{4} = 1.

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Problem 780

Along what curve does the plane x+y−z+3=0x+y-z+3 = 0 intersect the two-sheeted hyperboloid x2+y2−z2=−4x^{2}+y^{2}-z^{2} = -4?

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Problem 781

Prove that if p′p^{\prime } and q′q^{\prime } are the parameters of the parabolas obtained in the section of the elliptic paraboloid

x2p+y2q=2z(p>0,q>0) \frac{x^{2}}{p}+\frac{y^{2}}{q} = 2 z \quad (p > 0, q > 0)

by two of its conjugate diametral planes, then p′+q′=p+qp^{\prime }+q^{\prime } = p+q.

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Problem 782

Prove that if p′p^{\prime } and q′q^{\prime } are the parameters of the parabolas obtained in the section of the hyperbolic paraboloid

x2p−y2q=2z(p>0,q>0) \frac{x^{2}}{p}-\frac{y^{2}}{q} = 2 z \quad (p > 0, q > 0)

by two of its conjugate diametral planes, then p′−q′=p−qp^{\prime }-q^{\prime } = p-q.

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Problem 783

Find the equations of the planes passing through the point (a,0,0)(a, 0,0) and intersecting the one-sheeted hyperboloid

x2a2+y2b2−z2c2=1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} = 1

in two parallel lines.

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Problem 784

Find the line of intersection of the one-sheeted hyperboloid x2+y2−z2=1x^{2}+y^{2}-z^{2} = 1 and the plane 3x+4y−5z=03 x+4 y-5 z = 0.

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Problem 785

Prove that the plane

xp−yq+D=0 \frac{x}{\sqrt{p}}-\frac{y}{\sqrt{q}}+D = 0

intersects the hyperbolic paraboloid

x2p−y2q=2z(p>0,q>0) \frac{x^{2}}{p}-\frac{y^{2}}{q} = 2 z \quad (p > 0, q > 0)

in a line, and find its equations.

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Problem 786

Along what curve does a tangent plane to a one-sheeted hyperboloid cut its asymptotic cone?

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Problem 787

Along what curve is a one-sheeted hyperboloid cut by a tangent plane to its asymptotic cone?

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Problem 788

A plane intersects the cone

x2a2+y2b2−z2c2=0,a⩾b \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} = 0, \quad a \geqslant b

in a hyperbola. Within what limits can the eccentricity of the hyperbola vary?

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Problem 789

Write the equation of the plane passing through the origin and intersecting the elliptic cylinder

x2a2+y2b2=1,a>b \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1, \quad a > b

in a circle.

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Problem 790

Find the equation of the plane intersecting the hyperbolic cylinder

x2a2−y2b2=1,a≠b \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1, \quad a \neq b

in a rectangular hyperbola and passing through the axis:

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(1)

OxO x;

(2)

OyOy. Find the semi-axes of these hyperbolas.

Problem 791

Find the equation of the plane intersecting the hyperbolic paraboloid

x2p−y2q=2z,p>0,q>0 \frac{x^{2}}{p}-\frac{y^{2}}{q} = 2 z, \quad p > 0, q > 0

in a rectangular hyperbola and passing through the axis:

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(1)

OxO x;

(2)

OyO y. Find the semi-axes of these hyperbolas.

Problem 792

In each of the following cases, determine the type of the line of intersection of the surface with the plane and determine the position of the line relative to the original coordinate system:

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(1)

x2+y2−z2=0,x−z+1=0x^{2}+y^{2}-z^{2} = 0, x-z+1 = 0;

(2)

x29+y24+z21=1,x+z+453=0\frac{x^{2}}{9}+\frac{y^{2}}{4}+\frac{z^{2}}{1} = 1, x+z+\frac{4 \sqrt{5}}{3} = 0;

(3)

x2+2y2+z2+4xy−2xz−4yz+2x−6z=0,x−z=0x^{2}+2 y^{2}+z^{2}+4 x y-2 x z-4 y z+2 x-6 z = 0, x-z = 0.

Problem 793

Determine the type of the line along which the plane x+y+z−1=0x+y+z-1 = 0 intersects the parabolic cylinder y2=2xy^{2} = 2 x. Write the canonical equation of this line and find the canonical coordinate system.

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Problem 794

Determine the type of the line of intersection of the cone x2+y2−z2=0x^{2}+y^{2}-z^{2} = 0 with the plane 4x−3y−5z+4=04 x-3 y-5 z+4 = 0. Write the canonical equation of this line and find the canonical coordinate system.

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Problem 795

Determine the type of the line of intersection of the one-sheeted hyperboloid x2+y2−z2=1x^{2}+y^{2}-z^{2} = 1 and the plane 2x+2y+z−1=02 x+2 y+z-1 = 0.

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Problem 796

Find the foci of the ellipse obtained by intersecting the cylinder x2+y2=36x^{2}+y^{2} = 36 with the plane 3x+4y+12z=03 x+4 y+12 z = 0.

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Problem 797

Find all values of the parameter kk for which the plane of the pencil 2x+y+2+k(y+z)=02 x+y+2+k(y+z) = 0 intersects the cone x2+y2−−z2=0x^{2}+y^{2}- -z^{2} = 0 in an ellipse.

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Problem 798

Through the line 2x=2y=z2 x = 2 y = z draw a plane intersecting the hyperbolic paraboloid 4x2−y2+z=04 x^{2}-y^{2}+z = 0 in a rectangular hyperbola.

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Problem 799

Along curves of what type can a plane intersect each of the following surfaces:

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(1)

an ellipsoid;

(2)

a one-sheeted hyperboloid;

(3)

a two-sheeted hyperboloid;

(4)

a cone;

(5)

an elliptic paraboloid;

(6)

a hyperbolic paraboloid?

Problem 800

Find all planes intersecting each of the following surfaces in a circle:

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(1)

x2a2+y2b2+z2c2=1,a>b>c\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1, a > b > c;

(2)

x2a2+y2b2−z2c2=1,a>b\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} = 1, a > b;

(3)

x2a2+y2b2−z2c2=−1,a>b\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} = -1, a > b;

(4)

x2a2+y2b2−z2c2=0,a>b\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} = 0, a > b;

(5)

x2p+y2q=2z,p>q>0\frac{x^{2}}{p}+\frac{y^{2}}{q} = 2 z, p > q > 0;

(6)

x2a2+y2b2=1,a>b\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1, a > b.

Problem 801

Find the radius of the circular section of the ellipsoid

x2a2+y2b2+z2c2=1,a>b>c \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1, \quad a > b > c

and the one-sheeted hyperboloid

x2a2+y2b2−z2c2=1,a>b \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} = 1, \quad a > b

by a plane passing through the center of the surface.

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Problem 802

Find the necessary and sufficient condition for the plane

Ax+By+Cz+D=0(7.3) A x+B y+C z+D = 0 \tag {7.3}

to intersect the second-order surface

a11x2+a22y2+a33z2+2a23yz+2a13zx+2a12xy++2a1x+2a2y+2a3z+a=0 \begin{align} a_{11} x^{2}+a_{22} y^{2}+a_{33} z^{2}+2 a_{23} y z & +2 a_{13} z x+2 a_{12} x y+ \\ & +2 a_{1} x+2 a_{2} y+2 a_{3} z+a = 0 \end{align}

in a central line. Determine the coordinates of the center of the line of intersection of the surface (7.4) with the plane (7.3).

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Problem 803

Through the given point (x0,y0,z0)\left(x_{0}, y_{0}, z_{0}\right) draw a plane so that it intersects the surface

a11x2+a22y2+a33z2+2a23yz+2a13zx+2a12xy++2a1x+2a2y+2a3z+a=0 \begin{aligned} a_{11} x^{2}+a_{22} y^{2}+a_{33} z^{2}+2 a_{23} y z+2 a_{13} z x & +2 a_{12} x y+ \\ & +2 a_{1} x+2 a_{2} y+2 a_{3} z+a = 0 \end{aligned}

in a central curve centered at this point, given that the latter is not the center of the given surface.

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Problem 804

Find the locus of centers of the sections (real or imaginary) of the ellipsoid

x2a2+y2b2+z2c2=1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1

whose planes pass through a given point (x0,y0,z0)(x_{0}, y_{0}, z_{0}) outside the ellipsoid.

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Problem 805

Using the previous problem, find the second-order cone whose generators are tangent to the ellipsoid

x2a2+y2b2+z2c2=1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1

and whose center lies at the point (x0,y0,z0)(x_{0}, y_{0}, z_{0}) outside the ellipsoid.

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Problem 806

Find the boundary of the shadow cast by the ellipsoid

x24+y24+z21=1 \frac{x^{2}}{4}+\frac{y^{2}}{4}+\frac{z^{2}}{1} = 1

onto the plane z=−1z = -1, if the light source is located at the point (2,2,2)(2,2,2).

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Problem 807

Given the circular cone

x2r2+y2r2=z2 \frac{x^{2}}{r^{2}}+\frac{y^{2}}{r^{2}} = z^{2}

Through the point (x0,y0,z0)\left(x_{0}, y_{0}, z_{0}\right) draw a plane intersecting the given cone in a non-degenerate second-order curve, one of whose foci is at the indicated point.

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Problem 808

Find the foci of the section of the cone x2+y2+z2−2yz−2zx−−2xy=0x^{2}+y^{2}+z^{2}-2 y z-2 z x- -2 x y = 0 by the plane x+2y+2z=24x+2 y+2 z = 24.

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Problem 809

Find the locus of foci of the parabolas obtained by sectioning the circular cone

x2r2+y2r2=z2 \frac{x^{2}}{r^{2}}+\frac{y^{2}}{r^{2}} = z^{2}
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Problem 810

Find the locus of centers of the rectangular hyperbolas lying on the hyperbolic paraboloid

x2p−y2q=2z,p≠q \frac{x^{2}}{p}-\frac{y^{2}}{q} = 2 z, \quad p \neq q
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Problem 811

Find the necessary and sufficient condition under which the plane Ax+By+Cz+D=0A x+B y+C z+D = 0 intersects the parabolic cylinder y2=2pxy^{2} = 2 p x in a parabola with parameter pp.

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