Math Problems Vault

Problem 10.1

(1)(2)

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(1)

What is an algebraic surface of the first order?

(2)

Give an example of an algebraic surface of the third order and depict it in a rectangular Cartesian coordinate system.

Problem 10.2

Can an algebraic surface of the second order be a line? A plane? The empty set? Give examples.

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Problem 10.3

A family of surfaces is given in a rectangular coordinate system by an equation containing an arbitrary parameter $\lambda$ . Determine the type of the surface for all possible values of $\lambda$ :

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(1)

$x^2 + y^2 + z^2 = \lambda$

(2)

$\lambda x^2 + y^2 + z^2 = 1$

(3)

$\lambda x^{2} + y^{2} + z^{2} = \lambda$

(4)

$x^{2} + y^{2} - z^{2} = \lambda$

(5)

$x^{2} - y^{2} - z^{2} = \lambda$

(6)

$x^{2} + \lambda \left(y^{2} + z^{2}\right) = 1$

(7)

$x^{2} + \lambda \left(y^{2} + z^{2}\right) = \lambda$

(8)

$x^{2} + y^{2} = \lambda z$

(9)

$\lambda x^{2} + y^{2} = z$

(10)

$\lambda \left(x^{2} + y^{2}\right) = z$

(11)

$x^{2} + \lambda y^{2} = \lambda z$

(12)

$x^{2} + \lambda y^{2} = \lambda z + 1$

(13)

$x^{2} + y^{2} = \lambda$

(14)

$x^{2} - y^{2} = \lambda$ .

Problem 10.4

(1)(2)

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(1)

Indicate the types of surfaces of the second order that do not contain any surface of revolution.

(2)

List the types of surfaces of the second order to which some surfaces of revolution belong.

Problem 10.5

(1)(2)

Write the equation of the sphere:

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(1)

with center at the point $C(1, 1, 1)$ and radius $\sqrt{3}$

(2)

with center at the point $C(1, 2, 3)$ and radius 1.

Problem 10.6

(1)(2)

Find the coordinates of the center $C$ and the radius $R$ of the sphere:

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(1)

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

(2)

$2x^{2} + 2y^{2} + 2z^{2} + 4x + 8y + 12z + 3 = 0$ .

Problem 10.7

(1)(2)

Find the coordinates of the center of the surface, its semi-axes, and the equations of the planes of symmetry, and depict the surface in the original coordinate system:

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(1)

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

(2)

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

Problem 10.8

(1)(2)

Find the coordinates of the center of the surface, its vertices, the equations of the axis of symmetry and the planes of symmetry, and depict the surface in the original coordinate system:

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(1)

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

(2)

$2x^{2} + 3y^{2} - 4z^{2} + 4x - 8z + 10 = 0$ .

Problem 10.9

Determine the type of the surface:

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(1)

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

(2)

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

(3)

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

(4)

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

(5)

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

(6)

$2x^{2} + z^{2} + 2x + z = 0$ .

Problem 10.10

Determine the type of the surface, depict the surface in the original coordinate system:

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(1)

$xy = 0$

(2)

$xy = 1$

(3)

$xy = - 1$

(4)

$2xy + z = 0$

(5)

$2xy - z = 0$ .

Problem 10.11

Find the axis of revolution of the surface, depict the surface $x^{2} + z^{2} + x = 0$ .

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Problem 10.12

(1)(2)

Determine the type of the surface, find the axis of revolution, the coordinates of the vertices, and depict the surface:

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(1)

$x^2 + z^2 + 2y = 1$

(2)

$z^2 = 2xy$

Problem 10.13

Find the coordinates of the center of the surface, the equations of the axis of revolution and of the throat circle, determine the radius of the throat circle, and depict the surface $x^{2} + 2yz = 1$ .

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Problem 10.14

(1)(2)(3)

Find the points of intersection of the surface $x^{2} + y^{2} = z$ and the line:

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(1)

$x = y = t, \quad z = 4t$

(2)

$x = y = z + 1$

(3)

$\frac{x - 1}{1} = \frac{y + 1}{1} = \frac{z + 6}{8}$ .

Problem 10.15

How many common points can a line and a surface of the second order have? Give examples.

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Problem 10.16

Determine whether the point $M(1, 1, 1)$ lies inside or outside the ellipsoid $x^{2} + 2y^{2} + 3z^{2} = 4$ .

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Problem 10.17

The axis $Oz$ points upward. Determine whether the point $M(1, 1, 1)$ lies above or below the paraboloid $x^{2} + 2y^{2} = 2z$ .

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