Hyperbola
Hyperbola.
Focuses.
Equation of hyperbola.
Focal length.
Real and imaginary axes of
hyperbola. Eccentricity.
Asymptotes of hyperbola. Equation
of tangent line to hyperbola.
Tangency condition of straight
line and hyperbola.
A
hyperbola
(
Fig.1
) is
called a locus of points, a modulus of difference of distances from which to the
two given points
F_{1}
and F_{2
}
,
called focuses of hyperbola, is a constant value.
An
equation of hyperbola
(
Fig.1
) is
:
Here the origin of coordinates is a
center of symmetry of hyperbola, and the coordinate axes are its axes of
symmetry.
The segment F_{1}F_{2}
= 2
ñ,
where
is
called
a focal length.
The segment AB = 2
a
is called
a real axis
of hyperbola, the segment CD
= 2
b
is called an imaginary
axis of hyperbola. The number e = c / a , e
> 1 is called an
eccentricity of
hyperbola. The straight lines
y
=
±
(
b
/
a
)
x are
called asymptotes of hyperbola.
Let
Ð
(
õ_{1}
,
ó_{
1}
) be
a point of hyperbola, then an equation of tangent line to hyperbola
in
this point is:
A
tangency
condition
of
a
straight line
y
=
m
x
+
k
and
a
hyperbola
õ
^{2
}/^{
}a
^{2
} – ó^{
2}
/ b^{
2
}= 1
:
k^{ 2
} =
m ^{2 }a
^{2}
–
b^{ 2}
.
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