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Ellipse, hyperbola, parabola and their properties.

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Curves of the second order in plane: theorem on canonic forms (case B ¹ 0). Degenerate curves.

Let an orthonormal system of coordinates and some curve be given on a plane.

A curve is called an algebraic curve of the second order if its equation in a given system of coordinates has the form:

where numbers and are not equal to zero simultaneously (), and and are the coordinates of the radius-vector of a point lying in the curve .

Introduce the following notation: .

Theorem 1. For every curve of the second order there exists an orthonormal system of coordinates in which an equation of this curve has (for ) one of the following nine (called canonic) forms:

Type of curve
Empty sets  
Points    
Coinciding lines    
Non-coinciding lines  
Curves Ellipse Hyperbola Parabola

Proof: 1. Beforehand we notice that without loss of generality we can assume the following: and . Indeed, if then we can change the signs of all coefficients in the equation (*). If then by transiting to a new orthonormal system of coordinates for which we obtain the desired equation since at this transition the equalities hold. Observe also that .

4. If then we transit to a new system of coordinates obtained from the original one by turning around the point on angle such that the coefficient at the product will be equal to zero. Let's deduce a choice rule of this angle. Consider a turning:

with and consequently .

Substituting expressions for “old” coordinates by “new”, we obtain the equation (*) in the form:

or .

Then we have

Ellipse, hyperbola, parabola and their properties.

A curve of which the equation in some orthonormal system of coordinates is , , is called an ellipse. The number is the eccentricity of an ellipse. The points are the focuses of an ellipse. The lines are the directrices of an ellipse. The number is the focal parameter of an ellipse.

Properties of an ellipse: 1. An ellipse is a restricted curve: and that follows from the record of canonic equation in the form: .

2. An ellipse has axial symmetry regarding to the axes and and also central symmetry regarding to the origin of coordinates. This follows from: .

Denote by the distance between geometric objects and , and denote by and the angles between the tangent and focal radiuses and .

A curve of which the equation in some orthonormal system of coordinates is ; , , is called a hyperbola. The number is the eccentricity of a hyperbola. The points are the focuses of a hyperbola. The lines are the directrices of a hyperbola. The number is the focal parameter of a hyperbola.

Properties of a hyperbola: 1. A hyperbola is a unrestricted curve existing for that follows from the record of canonic equation in the form: .

2. A hyperbola has axial symmetry regarding to the axes and and also central symmetry regarding to the origin of coordinates. This follows from: .

Denote by and the angles between the tangent and focal radiuses.

A curve of which the equation in some orthonormal system of coordinates is ; , is called a parabola. The point are the focus of a parabola. The line are the directrix of a parabola. The number is the focal parameter of a parabola.

Denote by the angle between the tangent and focal radius and by – the angle between the tangent and positive direction of the abscissa axis.

Properties of a parabola: 1. A parabola is a unrestricted curve existing for every ;

2. A parabola has axial symmetry regarding to the axis that follows from:

.

 


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Интересные клинические случаи| Surfaces of the second order in the space: cylindrical and conic surfaces, surfaces of rotation.

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