
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



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. 









Intermediate
algebra contents 



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