Conic Sections
      Definition and construction of the hyperbola
         Construction of the hyperbola
      Equation of the hyperbola
         Properties of the hyperbola
         Examining equation of the hyperbola
Definition and construction of the hyperbola
The set of points in the plane whose distance from two fixed points (foci, F1 and F2 ) has a constant difference 2a is called the hyperbola.
Therefore, for every point of the hyperbola
| F1P - F2P | = 2a
with the focal distance |F1F2| = 2c, so that a < c.
The ratio   e = c/ae > 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
r1 = A1' R,
so that    r1 > a + c or  r1 > 2a + (c - a),
one centered at F1 and another at F2.
Then, these arcs intersect by two arcs of the radius
   r2 = A2' R = r1 - 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 x-axis is the transverse or the real axis, the other which lies on the y-axis 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
 F1(-c, 0) and F2(c, 0).
For every point P(x, y) of the hyperbola, according to  
definition      | r1 - r2 | = 2a,
using the formula for the distance of two points,
after squaring and reducing
By squaring again and grouping
(c2 - a2) x2 - a2y2 = a2 (c2 - a2).
Substituting c2 - a2  = b2 or   obtained is b2x2 - a2y2 = a2b2 equation of the hyperbola,
and after division by  a2b2 ,   the standard equation of the hyperbola.
Examining equation of the hyperbola
The hyperbola is determined by parameters, a and b, where a is the semi-major axis (or transverse semi-axis) and b is the semi-minor axis (or conjugate semi-axis). 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 x-axis at vertices
A1(-a, 0) and A2(a, 0).
The segment A1A2 = 2 is called the transverse axis.
By setting,    x = 0  =>   y =  bi  there are no intercepts with y-axis.
Perpendicular to the transverse axis at the midpoint is the conjugate axis, whose length is  B1B2 = 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 (-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 semi-latus 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.
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