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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. |