General theory of surfaces of the second order

$R = 4$

$R = 3$

$R = 2$

$R = 1$

$r = 3$

$r = 2$

$r = 1$.

paraboloids and parabolic cylinders

surfaces consisting of planes

"imaginary" surfaces: "imaginary ellipsoids", "imaginary cones", "imaginary elliptic cylinders", "pairs of imaginary intersecting planes", "pairs of imaginary parallel planes".

Which of the "imaginary" surfaces of the second order (see problem $11. 3, 3)$) have no real points? Characterize by means of invariants this group of surfaces.

Which of the "imaginary" surfaces of the second order have real points, and what do these surfaces look like? Characterize by means of invariants this group of surfaces.

two families of rectilinear generators

one family of rectilinear generators.

$\xi = \xi^{\prime } + \mathbf{b}$

$\xi = S\xi^{\prime } + \mathbf{b}$.

Verify that there exist second-degree equations in which, by passing to a new Cartesian coordinate system, it is impossible to eliminate all terms with first powers of the variables. List all types of such surfaces and characterize them by means of invariants.

Verify that there exist second-degree equations in which, by passing to a new Cartesian coordinate system, it is possible to eliminate all terms of degree below the second. List all types of such surfaces and characterize them by means of invariants.

Which real surfaces of the second order have a center of symmetry?

How many centers of symmetry can a surface of the second order have?

Prove that for a surface of the second order only the following possibilities can occur. The surface:

(1) has no center of symmetry;

(2) has a unique center of symmetry;

(3) has a line consisting of centers of symmetry;

(4) has a plane consisting of centers of symmetry.

having no center

having a unique center

having infinitely many centers.

if a surface of the second order has a center, and it is located at the origin, then the equation of the surface contains no linear terms

if the equation of a surface of the second order contains no linear terms, then the surface has a center located at the origin.

Using the results of problems 11. 8 and 11. 12, obtain the system of equations for the center of a surface of the second order (i.e., equation (10) from the introduction to § 11).

How will the constant term of equation (2) change if the origin is placed at the center of the surface of the second order?

The equation of a surface of the second order is written in expanded form (1) and in matrix form (2). All coefficients of the expanded equation are multiplied by a number $\mu \neq 0$. What happens to the matrix $A$? How do the roots of the characteristic equation $\left|A - \lambda E\right| = 0$ change as a result?

The equation of a surface of the second order is written in a rectangular coordinate system, and a passage to another rectangular system is made in it. Prove that the characteristic equation $\left|A - \lambda E\right| = 0$ does not change as a result, and hence its roots do not change. Does $\operatorname {det} A$ change?

A second-order equation is given in matrix form (2). Express the matrix $B$ of the large quadratic form of the surface given by this equation in terms of $A, \mathbf{a}, k$.

In the equation of the surface a change of coordinates $\xi = S\xi^{\prime } + \mathbf{b}$ is made. Write out the matrix $T$ by which the large quadratic form of the surface is transformed, and prove that, under a passage from one rectangular coordinate system to another, $\operatorname {det} B$ does not change.

Prove that, under an orthogonal change of coordinates that leaves the origin fixed, the characteristic equation $\left|B - \lambda E\right| = 0$ does not change, and hence its coefficients and roots do not change.

$(x + y + z)(x - y + 125z) = 1$

$(x + y)(x + y + 1) = 1$

$(x + y)(x + y + 1) = x - y$

$(x + y + z + 1)(x - y + z) = x + z + 1$

$(x + y + z + 1)(x - y + z) = x + 2z + 1$

$(x + y)(x - y) = z$

$(x + y + z)(x - y + z) = 0$

$(x + 2y)(x + 2y + 1) = z$

$(x + y)(x - 75y) = z^{2}$

$(x + y)^{2} = 3x + z$

$(x + 2y + 3z)(2x + 3y + 4z) + (3x + 4y + 75z)^{2} = 1$

$(x + 2y + 3z)(2x + 3y + 4z) - (3x + 4y + 75z)^{2} = 1$

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

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

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

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

$(x + y + z)^{2} + (x + y)^{2} + (y + z)^{2} = 1987$

$(x + y + z)^{2} + (x + y)^{2} = 1987$

$(x + y + z)^{2} + (x + y)^{2} + (2x + 2y + z)^{2} = 1$

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

$x_{1}^{2} + x_{2}^{2} + 4x_{3}^{2} + 4x_{1} x_{3} + 2x_{2} x_{3} + 2x_{1} + 2x_{2} + 5x_{3} + k = 0$

$2x_{1}^{2} + kx_{2}^{2} + 8x_{1} x_{2} + 4x_{1} x_{3} - 4x_{1} - 8x_{2} - 4x_{3} = 0$

$3x_{1}^{2} + 2x_{2}^{2} + 2x_{3}^{2} + 4x_{1} x_{2} - 2x_{1} x_{3} + 4x_{1} + 4x_{2} + 4x_{3} + k = 0$

$kx_{1}^{2} + 8x_{2}^{2} + x_{3}^{2} + 16x_{1} x_{2} + 4x_{1} x_{3} + 4x_{2} x_{3} - 4x_{1} - 4x_{2} + 2x_{3} = 0$

$3x_{2}^{2} + x_{3}^{2} + 6x_{1} x_{2} + 2x_{1} x_{3} + 4x_{2} x_{3} + kx_{2} + x_{3} + 1 = 0$

(р) $x_{1}^{2} - 2x_{2}^{2} - 3x_{3}^{2} - 4x_{1} x_{2} - 6x_{1} x_{3} - 2x_{1} + 4x_{2} + 6x_{3} + k = 0$.

$2x^{2} + 9y^{2} + 2z^{2} - 4xy + 4yz - 1 = 0$

$4y^{2} - 3z^{2} + 4xy - 4xz + 8yz = 0$

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

$x^{2} + y^{2} + z^{2} - xy + xz + yz + 3x + 3y - 3z = 0$

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

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

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

$x^{2} + 2y^{2} + 5z^{2} + 4yz + 20y + 20z - 10 = 0$

$- x^{2} + 5y^{2} + 5z^{2} + 8yz + 2x + 12y + 24z + 36 = 0$

$2x^{2} + 5y^{2} + 5z^{2} + 6yz + 4x + 16y + 16z + 10 = 0$

$4x^{2} + 4y^{2} - 4xy - 12x - 12y - 5z + 1 = 0$

$x^{2} + y^{2} + z^{2} + 2xy - 12x + 4y + 6z - 3 = 0$

$4x^{2} + 9y^{2} - 12xy + 2x + 10y + 1 = 0$

$6xy - 8y^{2} - z^{2} + 60y + 2z + 89 = 0$

$5x^{2} + 8y^{2} + 4xy + 2x + 44y - 36z + 65 = 0$

(р) $- x^{2} + y^{2} + z^{2} - 2yz + 2x + 3y - 5z + 1 = 0$

$9y^{2} + 16z^{2} + 24yz + 5x + 10y + 5z + 11 = 0$

$16x^{2} + 9y^{2} - z^{2} - 24xy - 9x - 12y + 4z + 71 = 0$

$2x^{2} + 2y^{2} + z^{2} - 10xy + 20x - 8y + 29 = 0$

$- x^{2} + 7y^{2} - 24yz + 2x + 120y = 0$

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

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

$- x^{2} - 9y^{2} + 6xy + 50x - 50y - 15z - 100 = 0$

(р) $4x^{2} + y^{2} + 9z^{2} + 4xy - 12xz - 6yz + 2x + 6y - 6z - 5 = 0$.

$5x^{2} + 5y^{2} + 3z^{2} + 2xy + 2\sqrt{2} xz + 2\sqrt{2} yz + 26x + 34y + 10\sqrt{2} z + 49 = 0$;

$2x^{2} + 9y^{2} + 2z^{2} - 4xy + 4yz + 4x + 2y - 4z - 1 = 0$

$4y^{2} - 3z^{2} + 4xy - 4xz + 8yz + 4x - 2z + k = 0$;

(1) $k = - 1$;

(2) $k = 5$;

(3) $k = 11$.

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

$3x^{2} + 3y^{2} + 3z^{2} - 2xy - 2xz - 2yz - 8x + 8y + k = 0$

(1) $k = 4$;

(2) $k = 8$.

$x^{2} + y^{2} + 4z^{2} - 2xy + 4xz - 4yz - 12x + 12y - 24z + 6 = 0$

$x^{2} + y^{2} + 4z^{2} - 2xy + 4xz - 4yz + 24y - 24z + 12 = 0$

$x^{2} + y^{2} + 4z^{2} - 2xy + 4xz - 4yz + 5x + y - 2z = 0$

$2x^{2} + 2y^{2} + 2z^{2} - 2xy + 2xz + 2yz - 6x - 6y - 12z + k = 0$

(1) $k = 15$;

(2) $k = 18$.

$2x^{2} + 2y^{2} + 2z^{2} - 2xy + 2xz + 2yz + 18z + 18 = 0$;

$x^{2} + 2y^{2} + z^{2} - 2xy - 2yz + 6x - 6y + k = 0$

(1) $k = 8$;

(2) $k = 9$.

$x^{2} + 2y^{2} + z^{2} - 2xy - 2yz + 18x - 6y + 6z - 18 = 0$

$3x^{2} - 7y^{2} + 3z^{2} + 8xy - 8yz - 8xz - 4x + 6y + 8z + k = 0$;

(1) $k = - 12$;

(2) $k = - 3$;

(3) $k = 6$.

$2y^{2} - 3z^{2} - 2\sqrt{3} xy - 4xz + 4\sqrt{3} yz + 50z + k = 0$;

(1) $k = - 75$;

(2) $k = - 70$;

(3) $k = - 80$.

$2x^{2} + 2y^{2} + 2z^{2} + 8xy + 8xz - 8yz - 8x - 4y + 8z + k = 0$;

(1) $k = - 4$;

(2) $k = 2$;

(3) $k = 8$.

$4x^{2} + 4y^{2} - 2z^{2} + 4xy - 8xz + 8yz + 12x - 12y + 24z + k = 0$;

(1) $k = - 42$;

(2) $k = - 36$;

(3) $k = - 30$.

$8y^{2} + 4xy + 2xz + 4yz + 4x + 8y + k = 0$

(1) $k = 0$;

(2) $k = - 9$.

$8y^{2} + 4xy + 2xz + 4yz + 8x + 6y + 4z + 6 = 0$

$4x^{2} + y^{2} + 9z^{2} + 4xy - 12xz - 6yz + 11x + 3y - z + 1 = 0$

$4x^{2} + y^{2} + 9z^{2} + 4xy - 12xz - 6yz + 6x - 2y - 6z + 11 = 0$

$4x^{2} + y^{2} + 9z^{2} + 4xy - 12xz - 6yz + 4x + 2y - 6z + k = 0$;

(1) $k = 1$;

(2) $k = - 13$.

$- x^{2} + 10y^{2} - z^{2} - 8xy + 6xz + 8yz + 24x - 8y - 16z + k = 0$;

(1) $k = - 26$;

(2) $k = - 14$;

(3) $k = - 2$.

$2x^{2} - y^{2} + 2z^{2} + 4xy - 2xz + 4yz + 10x - 2y - 2z + k = 0$;

(1) $k = 2$;

(2) $k = 5$;

(3) $k = 8$.

$x^{2} + y^{2} - 2z^{2} + 10xy + 4xz - 4yz + 13x + 11y + 2z + 6 = 0$.

$x^{2} + y^{2} - 2z^{2} + 10xy + 4xz - 4yz + 24x - 12z + k = 0$;

(1) $k = - 12$;

(2) $k = - 6$.