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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
F1
and F2
,
called focuses of ellipse, is a constant value.
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
ÎÕ (
Fig.1 ), at a < b
focuses of ellipse are
placed on axis ÎY
, 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 F1F2
= 2
ñ,
where
is called a
focal length. The
segment AB = 2
a
is called a
large axis of
ellipse, the segment CD
= 2
b
is called a small axis of ellipse. The number e = c
/ a, e < 1 is called an eccentricity of ellipse.
Let
Ð
(
õ1
,
ó
1
) be
a point of ellipse, then an equation of tangent line to
ellipse
in
this point is
A tangency condition of a
straight line
y
=
m
x
+ k and
an ellipse
õ
2
/
a
2
+ ó
2
/ b
2
= 1
:
k
2
=
m 2
a
2
+
b
2
.
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