**
Ellipse
**

*
Ellipse. Focuses. Equation of
ellipse.
*
*
Focal length.
*

*
Large and small axes of ellipse.
Eccentricity.
*

*
Equation of tangent line to
ellipse.
*

*
Tangency condition of straight
line and ellipse.
*

**
An
ellipse
**
(
Fig.1
) is
called a locus of points, a sum of distances from which to the two given points

*F*

_{ 1 }and

*F*

_{ 2 }, called

**of ellipse, is a constant value.**

*focuses*

**
An
equation of ellipse
**
(
Fig.1
) is
:

Here the origin of coordinates is a
center of symmetry of ellipse, and the coordinate axes are its axes of symmetry.
At
*
a
*
>
*
b
*
focuses of ellipse are placed on axis
*
*
*
ну
*
(
Fig.1 ), at
*
a
*
<
*
b
*
focuses of ellipse are
placed on axis
*
*
*
н
Y
*
, and at
*
*
*
a
*
=
*
b
*
an ellipse becomes a circle
( in this case focuses of
ellipse coincide with a center of circle ). Thus,
*
a circle
*
*
is a particular case of an
ellipse
*
.
The segment
*
F
*
_{
1
}
*
F
*
_{
2
}
= 2
*
Я
*
,
where
is called
**
**

**. The segment**

*a focal length**AB*= 2

*a*is called

**of ellipse, the segment**

*a large axis*^{ }

*CD*= 2

*b*is called

**of ellipse. The number**

*a small axis**e*=

*c*/

*a*,

*e*< 1 is called

**of ellipse.**

*an eccentricity*

Let
*
п
*
(
*
У
*
_{
1
}
,
*
С
*
_{
1
}
) be
a point of ellipse, then
*
an equation
*

**in this point is**

*of tangent line to ellipse*

**
A tangency condition of a
straight line
**

*y*=

*m x*+

*k*

*and an ellipse**У*

^{ 2 }/

^{ }

*a*

^{ 2 }+

*С*

^{ 2 }/

*b*

^{ 2 }= 1 :

*
k
*
^{
2
}
=
*
m
*
^{
2
}
*
a
*
^{
2
}
*
*
+
*
*
*
b
*
^{
2
}
.