
Conic
Sections 


Hyperbola

Examples of hyperbola 
Equilateral or rectangular hyperbola with the coordinate axes as its
asymptote

Translation
of equilateral or rectangular hyperbola with the coordinate axes as its
asymptote 





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:
Find the angle subtended by the focal radii
r_{1}
and
r_{2}
at a point A(8, y
> 0) of the hyperbola

9x^{2}

16y^{2} = 144. 
Solution:
We determine the ordinate of the point
A
by plugging its abscissa into equation of the hyperbola,

x
= 8 =>
9x^{2}

16y^{2} = 144

9 · 8^{2}

16y^{2} = 144,

16y^{2} = 432 =>
y^{2} =
27,

y_{1,2} = ±3Ö3
so that A(8,
3Ö3).

The equations of the lines of the radii r_{1}
and
r_{2}, we
write
using the formula of a line through two
points. Since,

r_{1}::
AF_{1} and
F_{1}(c,
0) ,
and 


then 







Therefore, the angle between the focal radii r_{1}
and
r_{2}
at the point A
of the hyperbola, as



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, 



Example:
Write the equation of the hyperbola 9x^{2}

25y^{2} = 225
in the vertex form. 
Solution:
Using parallel shifting we should place the center of the hyperbola at
S(a,
0). 
Rewrite
9x^{2}

25y^{2} = 225  ¸
225 


therefore,
a = 5 and
b = 3,
so that S(5,
0). 

Then, the translated
hyperbola with the center at S(5,
0) has the equation



Equilateral or rectangular hyperbola with the coordinate axes as its
asymptote

The
graph of the reciprocal function y
= 1/x or y
= k/x is a rectangular
(or right) hyperbola of which asymptotes are the coordinate axes. 
If k
> 0 then, the
function is decreasing from zero to negative infinity
and from positive infinity to zero, i.e., the graph of
the rectangular hyperbola opening in the first and third
quadrants as is shown in the right figure. 
The vertices, 







Translation
of equilateral or rectangular hyperbola with the coordinate axes as its
asymptote 
The rational function 

by dividing the
numerator by denominator, 

can be
rewritten into 

where, 


is the constant, 

are the vertical and
the horizontal asymptote respectively. 

Therefore,
the values of the vertical and the horizontal asymptotes correspond to
the coordinates of the horizontal and the vertical translation of the
reciprocal function y
= k/x as is shown in
the figure below. 









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