Math Problems Vault

Problem 720

Using the invariants

\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

\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

(1)(2)(3)

(Orthogonal semi-invariants). Prove that:

?

(1)

the functions

\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

\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)

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

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)

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

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 $I_{1}, I_{2}, I_{3}, I_{4}$ and semi-invariants $I_{2}^{*}, I_{3}^{*}$ , 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)

$3 x^{2}+3 y^{2}+4 x y+2 x z-2 y z-6 x-14 y+9 = 0$

(2)

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

(3)

$3 x^{2}+3 y^{2}+4 x y+2 x z-2 y z-2 x-8 y+2 z+5 = 0$ ;

(4)

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

(5)

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

(6)

$2 x^{2}+2 y^{2}+z^{2}+x y+x z-y z-3 x-2 y-3 z+5 = 0$ ;

(7)

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

Problem 724

Find the largest angle $\varphi$ between the generators of the cone $z^{2} + 2 x y+2 x z+2 y z = 0$ , as well as the angles $\alpha , \beta , \gamma$ 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.

?

Problem 728

A second-order surface is given

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 $a$ this surface is a surface of revolution; find the direction vector of its axis of rotation.

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

?

Problem 729

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

?

Problem 730

Find the equation of a second-order cone intersecting the plane $O y z$ in the circle $y^{2}+z^{2} = 2 r y, x = 0$ , and the plane $O x z$ in the parabola $z^{2} = 2 p x, y = 0$ .

?

Problem 731

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

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

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 $a = b = c = 1$ .

?

Problem 733

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

?

Problem 734

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

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

is constant and equals

\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

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+\frac{z^{2}}{a^{2}} = 1

?