Degenerate curves of the second order
1.
Non-coinciding lines. The
equation
determines a pair of intersecting lines in the system of coordinates
.
And the equation
for
determines a pair of parallel lines.
Ellipse and its properties
A
curve of which the equation in some orthonormal system of coordinates
is
,
,
is called anellipse.
The number
is theeccentricity
of an ellipse. The points
are the
focuses of
an ellipse. The lines
are thedirectrices
of an ellipse. The number
is thefocal
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
.
Theorem.
Let
be a point belonging to an ellipse
given by a canonic equation. Then the following holds: 1.
;
2.
;
3.
;
4.
where
is orthogonal to the axis
;
5.
.
Hyperbola and its properties
A
curve of which the equation in some orthonormal system of coordinates
is
;
,
,
is called ahyperbola.
The number
is theeccentricity
of a hyperbola. The points
are the
focuses of
a hyperbola. The lines
are thedirectrices
of a hyperbola. The number
is thefocal
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.
Definition.
A line
is an asymptote for line
for
if
and
3.
A hyperbola has asymptotes
.
4.
;
5.
.
6.
.
The
canonic equation for hyperbola
studied in course of elementary mathematics is obtained by the
following changing of coordinates:
.
Parabola and its properties
A
curve of which the equation in some orthonormal system of coordinates
is
;
,
is called aparabola.
The point
are the
focus of
a parabola. The line
are thedirectrix
of a parabola. The number
is thefocal
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:
.
Theorem.
Let
be a point belonging to a parabola
given by a canonic equation. Then the following holds: 1.
;
2.
;
3.
;
4.
.
The
canonic equation for parabola
studied in course of elementary mathematics is obtained by mutual
renaming of the coordinate variables.