All textbooks/Collection of problems in analytical geometry and linear algebra/3 · Quadratic curves

Section 9

General theory of second-order curves

[22/73%]

Show

LaTeX

Show LaTeX

On this page

9.1 · Reducing the equation of a second-order curve to canonical form (9.1-9.10)

9.2 · Second-order curves in a general Cartesian coordinate system (9.11-9.22)

Subsection 9.1Reducing the equation of a second-order curve to canonical form (9.1-9.10)[10/90%]

Problem 9.1

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)

Determine the type of the curve of the second order, find its canonical equation, and find the canonical coordinate system:

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

$6x^{2} + 6y^{2} + 6x - 2y - 1 = 0$

(2)

$9x^{2} - 16y^{2} - 6x + 8y - 144 = 0$

(3)

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

(4)

$12x^{2} - 12x - 32y - 29 = 0$

(5)

$9y^{2} - 7y - 16 = 0$

(6)

$2x^{2} + y^{2} + 4x - 6y + 11 = 0$

(7)

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

(8)

$4x^{2} - 25y^{2} - 2x - 75y + 44 = 0$

(9)

$25x^{2} - 30x + 9 = 0$

(10)

$45x^{2} - 36y^{2} - 90x - 24y + 41 = 0$ .

Problem 9.2

(1)(2)

Under what necessary and sufficient condition does the equation $Ax^{2} + By^{2} + 2Cx + 2Dy + E = 0$ define:

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

an ellipse

(2)

a hyperbola?

Problem 9.3

(1)(2)(3)(4)(5)(6)(7)(8)

Determine the type of the curve of the second order, find its canonical equation, and find the canonical coordinate system:

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

$2x^{2} + 6xy + 10y^{2} - 121 = 0$

(2)

$9xy + 4 = 0$

(3)

$2x^{2} - 2\sqrt{3} xy + 9 = 0$

(4)

$18x^{2} + 24xy + 11y^{2} - 3 = 0$

(5)

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

(6)

$9x^{2} - 6xy + y^{2} - 10 = 0$

(7)

$81x^{2} - 36xy + 4y^{2} = 0$

(8)

$3x^{2} - 4\sqrt{5} xy + 4y^{2} = 0$ .

Problem 9.4

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)

Determine the type of the curve of the second order, find its canonical equation, and find the canonical coordinate system:

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

$2x^{2} - 4xy + 5y^{2} + 8x - 2y + 9 = 0$

(2)

(р) $4xy - 3y^{2} - 4x + 10y - 6 = 0$

(3)

$9x^{2} - 24xy + 16y^{2} - 8x + 19y + 4 = 0$

(4)

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

(5)

$xy + 2x + y = 0$

(6)

$x^{2} - 2xy + y^{2} - 10x - 6y + 25 = 0$

(7)

$5x^{2} + 12xy + 10y^{2} - 6x + 4y - 1 = 0$

(8)

$8x^{2} + 34xy + 8y^{2} + 18x - 18y - 17 = 0$

(9)

$25x^{2} - 30xy + 9y^{2} + 68x + 19 = 0$

(10)

$8x^{2} + 6xy + 6x + 3y + 1 = 0$

(11)

$4x^{2} + 12xy + 9y^{2} - 8x - 12y - 5 = 0$

(12)

$225x^{2} - 240xy + 64y^{2} + 30x - 16y + 1 = 0$

(13)

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

(14)

$5x^{2} - 6xy + 5y^{2} + 2x - 14y + 13 = 0$

(15)

$x^{2} - 2xy + y^{2} + 8x - 8y + 22 = 0$

(16)

$15x^{2} + 24xy + 15y^{2} + 30x - 24y - 20 = 0$

(17)

$15x^{2} - 16xy - 15y^{2} - 62x - 44y - 13 = 0$ .

Problem 9.5

Prove that the curve of the second order given by the equation $34x^{2} + 24xy + 41y^{2} - 44x + 58y + 1 = 0$ is an ellipse. Find the lengths of the semi-axes and the eccentricity of this ellipse, the coordinates of the center and foci, and find the equations of the axes and directrices.

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

Problem 9.6

Prove that the curve of the second order given by the equation $7x^{2} + 48xy - 7y^{2} - 62x - 34y + 98 = 0$ is a hyperbola. Find the lengths of the semi-axes and the eccentricity of this hyperbola, the coordinates of the center and foci, and find the equations of the axes, directrices, and asymptotes.

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

Problem 9.7

Prove that the curve of the second order given by the equation $x^{2} + 2xy + y^{2} + x = 0$ is a parabola. Find the parameter of this parabola, the coordinates of the vertex and focus, and find the equations of the axis and directrix.

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

Problem 9.8

Let $M(x, y) = Ax^{2} + 2Bxy + Cy^{2} + 2Dx + 2Ey + F$ be a second-degree polynomial in the coordinates $(x, y)$ of a point in a rectangular coordinate system. Denote

S = A + C, \quad \delta =\left[\begin{smallmatrix} A & B \\ B & C\end{smallmatrix}\right], \quad \Delta =\left[\begin{smallmatrix} A & B & D \\ B & C & E \\ D & E & F\end{smallmatrix}\right].

Prove that the quantities $S, \delta , \Delta$ do not change under a passage to another rectangular coordinate system (i.e., are orthogonal invariants of the polynomial $M(x, y))$ .

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

Problem 9.9

(1)(2)(3)(4)

A curve of the second order in a rectangular coordinate system is given by the equation $Ax^{2} + 2Bxy + Cy^{2} + 2Dx + 2Ey + F = 0$ . Prove the following statements (for the notation, see problem 9. 8):

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

The roots $\lambda_{1}$ and $\lambda_{2}$ of the characteristic equation $\lambda^{2} - S\lambda + \delta = 0$ are real, and at least one of them is nonzero.

(2)

The curve is a hyperbola if and only if $\delta < 0$ and $\Delta \neq 0$ . Express the semi-axes of the hyperbola in terms of $\delta , \Delta , \lambda_{1}, \lambda_{2}$ .

(3)

The curve is an ellipse if and only if $\delta > 0$ , and $S\Delta < 0$ . Express the semi-axes of the ellipse in terms of $\delta , \Delta , \lambda_{1}, \lambda_{2}$ .

(4)

The curve is a parabola if and only if $\delta = 0$ , and $\Delta \neq 0$ . Express the parameter of the parabola in terms of $S$ and $\Delta$ .

Problem 9.10

(1)(2)(3)

Using the orthogonal invariants (problems 9. 8 and 9. 9), determine the type and find the canonical equation of the curve:

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

$x^{2} + 3xy - 3y^{2} + 5x - 7y + 1 = 0$

(2)

$5x^{2} + 2xy + 5y^{2} - 12x + 20y + 32 = 0$

(3)

$x^{2} - 4xy + 4y^{2} + 2x + 13 = 0$ .

Subsection 9.2Second-order curves in a general Cartesian coordinate system (9.11-9.22)[12/58%]

Problem 9.11

(1)(2)(3)(4)(5)(6)(7)(8)(9)

Prove that:

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

the curve given by the equation $x^{2} + y^{2} = 1$ is an ellipse

(2)

the curve given by the equation $x^{2} - y^{2} = 1$ is a hyperbola

(3)

the curve given by the equation $y^{2} = x$ is a parabola

(4)

the equation $x^{2} - y^{2} = 0$ defines a pair of intersecting lines

(5)

the equation $x^{2} + y^{2} = 0$ defines a single point

(6)

the equation $y^{2} - 1 = 0$ defines a pair of parallel lines

(7)

the equation $y^{2} = 0$ defines a single line (a pair of coincident lines)

(8)

the curve given by the equation $xy = 1$ is a hyperbola

(9)

the curve given by the equation $y = x + \frac{1}{x}$ is a hyperbola.

Problem 9.12

Prove that the curve given, in a general Cartesian coordinate system, by the equation $Ax^{2} + 2Bxy + Cy^{2} + 2Dx + 2Ey + F = 0$ , is a hyperbola if and only if $\delta < 0$ and $\Delta \neq 0$ , an ellipse if and only if $\delta > 0$ and $S\Delta < 0$ , a parabola if and only if $\delta = 0$ and $\Delta \neq 0$ .

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

Problem 9.13

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)

Determine the type of the curve of the second order given by the equation:

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

$(3x - 4y)^{2} - 5(x + 2y - 1)^{2} = 1$

(2)

$(12x - 17y - 6)^{2} + (17y + 5x + 1)^{2} = 1$

(3)

$(x - y - 3)(x + y + 3) = 4$

(4)

$(4x + 3y - 1)^{2} + (4x + 3y + 2)^{2} = 5$

(5)

$17x^{2} - 2xy + y^{2} - 3x - y - 3 = 0$

(6)

$4x^{2} + 28xy + 49y^{2} - 3x - 15y + 2 = 0$

(7)

$4x^{2} - 12xy + 8y^{2} - 15x + 25y + 14 = 0$

(8)

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

(9)

$2x^{2} - 5xy - 3y^{2} + 9x + y + 4 = 0$

(10)

$x^{2} + 10xy + 25y^{2} + 2x + 10y - 3 = 0$

(11)

$5x^{2} - 16xy + 13y^{2} + 6x - 10y + 2 = 0$

(12)

$x^{2} - 4xy + 4y^{2} + 4x - 8y + 5 = 0$

(13)

$x^{2} - 8xy + 16y^{2} + 6x - 24y + 9 = 0$ .

Problem 9.14

(1)(2)(3)(4)(5)(6)

Find the equation and determine the type of the curve of the second order passing through 5 points given by their coordinates:

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

$(- 1, - 1), (1, 0), (0, 1), (3, 2), (2, 3)$

(2)

$(1, 1), (1, 0), (0, 1), (3, 2), (2, 3)$

(3)

$(- 1, 0), (1, 0), (0, 1), (3, 2), (2, 3)$

(4)

$(- 3, 0), (1, 0), (0, 1), (3, 2), (2, 3)$

(5)

$(- 1, 1), (0, 1), (2, 3), (- 2, - 1), (3, 4)$

(6)

$(1, 0), (0, 1), (1 / 4, 1 / 4), (4 / 9, 1 / 9), (1 / 9, 4 / 9)$ .

Problem 9.15

(1)(2)(3)(4)

Investigate the dependence of the type of the curve of the second order on the parameter:

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

$4x^{2} + 2\lambda xy + y^{2} = 1$

(2)

$\lambda \left(x^{2} + y^{2}\right) - 10xy + x + y + 4 = 0$

(3)

$x^{2} - 2xy + y^{2}(\lambda - 1) + 2\lambda (x - y + 1) = 0$

(4)

$\lambda x^{2} - 2xy + 2y^{2} - 2x + 2y - 1 = 0$ .

Problem 9.16

(1)(2)(3)(4)(5)

What types of curves of the second order can be given by the equation:

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

$\left(A_{1} x + B_{1} y + C_{1}\right)^{2} = A_{2} x + B_{2} y + C_{2}$

(2)

$\left(A_{1} x + B_{1} y + C_{1}\right)^{2} + \left(A_{2} x + B_{2} y + C_{2}\right)^{2} = 1$

(3)

$\left(A_{1} x + B_{1} y + C_{1}\right)^{2} - \left(A_{2} x + B_{2} y + C_{2}\right)^{2} = 1$

(4)

$\left(A_{1} x + B_{1} y + C_{1}\right)\left(A_{2} x + B_{2} y + C_{2}\right) = 1$

(5)

$\left(A_{1} x + B_{1} y + C_{1}\right)\left(A_{2} x + B_{2} y + C_{2}\right) = 0$ ?

Problem 9.17

(1)(2)

Find the equations of the asymptotes of the hyperbola (It is assumed that $A_{1} B_{2} - A_{2} B_{1} \neq 0$ ).

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

$\left(A_{1} x + B_{1} y + C_{1}\right)^{2} - \left(A_{2} x + B_{2} y + C_{2}\right)^{2} = 1$

(2)

$\left(A_{1} x + B_{1} y + C_{1}\right)\left(A_{2} x + B_{2} y + C_{2}\right) = 1$ .

Problem 9.18

Without using equations (6) from the introduction to the present chapter, prove that the origin is a center of symmetry of a curve of the second order if and only if the equation of the curve contains no terms with first powers of the variables $x$ and $y$ . Based on this statement, derive equations (6) for the coordinates of the center of a curve of the second order.

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

Problem 9.19

(1)(2)(3)

Verify that the given curve of the second order is central. Find the coordinates of the center and eliminate, in the equation, the terms of the first degree by shifting the origin to the center

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

$x^{2} - 8xy + 17y^{2} + 8x - 38y + 24 = 0$

(2)

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

(3)

$8x^{2} - 24xy + 16y^{2} + 3x - 7y - 2 = 0$ .

Problem 9.20

Prove that the curve $Ax^{2} + 2Bxy + Cy^{2} + 2Dx + 2Ey + F = 0$ has a unique center of symmetry if and only if $\delta \neq 0$ .

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

Problem 9.21

Prove that the set of centers of symmetry of a curve of the second order is either empty, or consists of a single point, or is a straight line.

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

Problem 9.22

(1)(2)(3)

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(1)

Prove that the set of centers of symmetry of an algebraic curve is either empty, or consists of a single point, or is a straight line.

(2)

Prove that the set of centers of symmetry of an arbitrary set of points in the plane is either empty, or consists of a single point, or is infinite.

(3)

Give an example of a continuous curve whose set of centers of symmetry is infinite, but is not a straight line.