Math Problems Vault

Problem 542

Set up the canonical equations of the curves, if their invariants are given:

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(1)

$I_{1} = 0, I_{2} = -25, I_{3} = 125$

(2)

$I_{1} = 2, I_{2} = -15, I_{3} = -30$ ;

(3)

$I_{1} = 7, I_{2} = 10, I_{3} = -20$ ;

(4)

$I_{1} = 5, I_{2} = 0, I_{3} = -50$ ;

(5)

$I_{1} = 0, I_{2} = -9, I_{3} = 0$ ;

(6)

$I_{1} = 9, I_{2} = 18, I_{3} = 36$ ;

(7)

$I_{1} = 3, I_{2} = 3, I_{3} = 3$ ;

(8)

$I_{1} = 1, I_{2} = 0, I_{3} = 1$ .

Problem 543

Prove that the conditions $I_{1}^{2} = 4 I_{2}, I_{1} I_{3} < 0$ are necessary and sufficient for the equation of a second-order curve to define a real circle.

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Problem 544

Express, in terms of the invariants $I_{1}, I_{2}, I_{3}$ , the conditions necessary and sufficient for the second-order equation $F(x, y) = 0$ to define a real ellipse. Express, in terms of $I_{1}, I_{2}, I_{3}$ , the lengths $a$ and $b$ of the semi-axes of the ellipse, and find its area $S$ .

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Problem 545

Using invariants, express the necessary and sufficient conditions for the equation $F(x, y) = 0$ to define a hyperbola lying in the acute angle formed by its asymptotes.

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Problem 546

Express, in terms of the invariants $I_{1}, I_{2}, I_{3}$ , the conditions necessary and sufficient for the quadric $F(x, y) = 0$ to split into a pair of perpendicular lines.

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Problem 547

Prove that the equation $F(x, y) = 0$ defines a rectangular (equilateral) hyperbola if and only if $I_{1} = 0$ and $I_{3} \neq 0$ .

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Problem 548

The equation $F(x, y) = 0$ defines a hyperbola. Using orthogonal invariants, write the equation of the pair of its asymptotes.

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Problem 549

Express, in terms of the invariants $I_{1}, I_{2}, I_{3}$ , the necessary and sufficient conditions for the curve $F(x, y) = 0$ to be a parabola. Express, in terms of $I_{1}, I_{2}, I_{3}$ , its focal parameter $p$ .

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Problem 550

The equation $F(x, y) = 0$ defines a pair of intersecting lines. Express, in terms of orthogonal invariants, the angle between them.

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Problem 551

(Semi-invariant.) Prove that the quantity

I_{2}^{*} =\left|\begin{smallmatrix} a_{22} & a_{2} \\ a_{2} & a \end{smallmatrix}\right|+\left|\begin{smallmatrix} a_{11} & a_{1} \\ a_{1} & a \end{smallmatrix}\right|

is an invariant of equation (6.1) under homogeneous orthogonal transformations (i.e., those under which the origin does not change).

Prove that if $I_{2} = I_{3} = 0$ , then $I_{2}^{*}$ is preserved under any orthogonal transformations.

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Problem 552

A second-order equation $F(x, y) = 0$ defines two parallel lines. Express the distance $d$ between them in terms of the invariants and the semi-invariant of the polynomial $F(x, y)$ .

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Problem 553

A second-order curve, given by the equation $F(x, y) = 0$ , splits into a pair of intersecting, not mutually perpendicular lines. Find the necessary and sufficient condition for a given point $M_{0}\left(x_{0}, y_{0}\right)$ to lie in the acute angle formed by these lines.

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Problem 554

A second-order curve, given by the equation $F(x, y) = 0$ , splits into a pair of parallel lines. Under what necessary and sufficient condition does a given point $M_{0}\left(x_{0}, y_{0}\right)$ lie between them?

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Problem 555

The equation $a_{11} x^{2}+2 a_{12} x y+a_{22} y^{2} = 0$ defines a pair of intersecting lines $(a_{11} a_{22}-a_{12}^{2} < 0)$ . Prove that if the coordinate system is rectangular, then the pair of bisectors between these lines can be given by the equation

\left|\begin{smallmatrix} a_{11} x+a_{12} y & a_{12} x+a_{22} y \\ x & y \end{smallmatrix}\right| = 0

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Problem 556

The equations of two hyperbolas have the form

\begin{aligned} & a_{11} x^{2}+2 a_{12} x y+a_{22} y^{2}+2 a_{1} x+2 a_{2} y+a = 0 \\ & a_{11} x^{2}+2 a_{12} x y+a_{22} y^{2}+2 a_{1} x+2 a_{2} y+b = 0 \end{aligned}

Under what necessary and sufficient condition will they lie in different vertical angles formed by their common asymptotes?

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