
Conic
Sections 


Hyperbola

Definition and construction
of the hyperbola

Construction of the hyperbola

Equation of the hyperbola

Properties of the hyperbola

Examining equation of the hyperbola

Examples of hyperbola 





Definition and construction
of the hyperbola

The set of points in the plane whose distance from two fixed points (foci,
F_{1}
and
F_{2}
) has a constant difference 2a
is called the hyperbola. 
Therefore, for every point of the
hyperbola 
 F_{1}P

F_{2}P

= 2a 
with the focal distance
F_{1}F_{2}
= 2c, so that
a
< c. 
The ratio
e =
c/a,
e
> 1 
is called eccentricity of the hyperbola.

The hyperbola can also be defined as the locus of points
the ratio of whose distances from the focus, to a vertical
line known as the directrix, is a constant e, where






Construction of the hyperbola

Given are values,
a
and c. Draw two equal arcs of an
arbitrary radius r_{1
}=
A_{1}'
R,

so that
r_{1}
>
a + c
or r_{1}
>
2a +
(c

a),

one centered at
F_{1}
and another at
F_{2}. 
Then, these arcs intersect by two arcs of the radius

r_{2} =
A_{2}'
R =
r_{1}

2a,
drawn from both foci.

Thus, obtained are four points of the hyperbola.

Repeating this procedure by changing radii of arcs, we can get enough points of the hyperbola.

It is obvious from the construction that the hyperbola
has two axes of symmetry which intersects at the center of the hyperbola. The axis which coincides with the
xaxis
is the transverse or the
real axis, the other which lies on the
yaxis is the
conjugate or the
imaginary
axis.





Equation of the hyperbola

If in the direction of the axes we introduce a coordinate system so that the center of the hyperbola
coincides with the origin, then coordinates of foci are 
F_{1}(c,
0)
and
F_{2}(c,
0). 
For every point
P(x, y)
of the hyperbola, according to 
definition  r_{1}

r_{2}

= 2a, 
using the formula for the distance of two points,


after squaring
and reducing


By squaring again and grouping

(c^{2} 
a^{2}) · x^{2} 
a^{2}y^{2}
= a^{2}
·
(c^{2} 
a^{2}).




Substituting
c^{2} 
a^{2}
= b^{2
}or 

obtained is 
b^{2}x^{2}

a^{2}y^{2}
= a^{2}b^{2} 
equation of the hyperbola, 

and after division by
a^{2}b^{2} ,


the standard equation of the hyperbola. 


Examining equation of the hyperbola

The hyperbola is determined by parameters,
a
and
b, where
a
is the semimajor axis (or transverse
semiaxis) and b
is the
semiminor axis (or conjugate
semiaxis). The intersection points of the hyperbola with coordinate axes we determine from its equation, 

by
setting, y
= 0
=>
x
= ± a 
the
hyperbola has intercepts with the
xaxis
at vertices

A_{1}(a,
0)
and
A_{2}(a,
0). 
The
segment A_{1}A_{2}
= 2a _{
}is called the
transverse axis.

By
setting, x
= 0
=> y
= ± b·i
there are no
intercepts with yaxis.

Perpendicular to the transverse axis at the midpoint is
the
conjugate axis, whose length is
B_{1}B_{2}
= 2b.

The hyperbola
consists of two branches.

If we solve the equation of the hyperbola for
y,





it follows that the value of the square root will be real if
 x 
>
a. That is, if the absolute value of
x
is less then a,
then y
is an imaginary number, so that in the interval of x
Î
(a,
a) there are no points of the hyperbola. 
As the hyperbola is symmetric to both coordinate axes, we can examine behavior of only part of the curve
located at the first quadrant for values of x
>
a. 
Rewrite the above equation so that it represents only the
positive values of the curve in this interval of x, so 

Since the value of
x increases and tends to infinity,
x
®
oo
, then the term 

tends to zero, so that 

value of the square root tends to
1
and the equation of the hyperbola changes to 



This equation shows that points
of the hyperbola become closer and closer to this line for large values of
x. 
From the above equation of the hyperbola we see that for every value
x
> a, the value of the square root is
less then 1, it means that the ordinate
y
of every point of the hyperbole is less then the ordinate
y’ of the
corresponding point of the line. 
We also see that as
x
increases, the difference of ordinates, y
and y’ becomes smaller, what means that points
of the hyperbola become closer to this line. 

The
lines, 

are the asymptotes of the hyperbola. 

The asymptotes of a hyperbola coincide with the diagonals of the rectangle whose center is the center of the
curve, and whose sides are parallel and equal to the length 2a and
2b, of the axes of the curve, as
shows the above figure. 
From
the equation of the hyperbola 

for
x
= ± c,
obtained is 


the
semilatus rectum p. 
The length of
p can also be calculated from the right triangle with legs
p and
2c,
whose hypotenuse is p + 2a, according to the definition of the hyperbola. 

The latus rectum 



are the chords perpendicular to the transverse axis and passing through the foci. 
The hyperbola which has for its transverse and conjugate axes the transverse and conjugate axes of another 
hyperbola, is said to be the
conjugate
hyperbola. So the hyperbolas, 

are 

conjugate
hyperbolas of each other. 
A given hyperbola and its conjugate are constructed on the same reference rectangle.
Thus, they have the common asymptotes and their foci lie on a circle. 

Examples
of hyperbola 
Example:
Given is the hyperbola
4x^{2} 
9y^{2} = 36,
determine the semiaxes, equations of the asymptotes,
coordinates of foci, the eccentricity and the semilatus rectum.

Solution:
Put the equation in the standard form to
determine the semiaxes, thus

4x^{2}

9y^{2} = 36  ¸
36


Asymptotes, 


Applying, 


coordinates of foci, F_{1}(Ö13,
0) and
F_{2}(Ö13,
0).




The eccentricity, 

and the
semilatus rectum,




Example:
Write equation of a hyperbola with the focus at
F_{2}(5,
0) and whose asymptotes are, 




Therefore,
the equation of the hyperbola,





Example:
The hyperbola is given by equation
4x^{2}

9y^{2} + 32x + 54y 
53 =
0. 
Find coordinates of the
center, the foci, the eccentricity and the asymptotes of the hyperbola. 
Solution:
The given hyperbola is translated in the direction of the coordinate axes so the values of
translations x_{0} and
y_{0}
we can find by using the method of completing the square
rewriting the equation in 
the standard
form, 


Thus,
4x^{2} + 32x

9y^{2} + 54y 
53 =
0,

4(x^{2} + 8x) 
9(y^{2} 
6y) 
53 =
0

4[(x +
4)^{2} 
16] 
9[(y 
3)^{2} 
9] 
53 =
0

4(x + 4)^{2} 
9(y 
3)^{2} = 36  ¸
36

Therefore,






it
follows that a^{2} =
9, a
= 3,
b^{2} = 4,
b
= 2,
and the center of the hyperbola at S(x_{0}, y_{0})
or S(4,
3). 
Half
the focal distance 

the
eccentricity 


and the foci, F_{1}(x_{0
}
c,
0) so
F_{1}(4_{
}
Ö13,
0) and
F_{2}(x_{0
} + c,
0),
F_{1}(4_{
} + Ö13,
0). 
Equations of the asymptotes of a translated hyperbola 

therefore, the asymptotes of the given
hyperbola, 










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