Conic Sections
     Hyperbola
      Equilateral or rectangular hyperbola
      Translated hyperbola
      Equation of the hyperbola in vertex form
      Parametric equations of the hyperbola
         Examples of hyperbola
Equilateral or rectangular hyperbola
The hyperbola whose semi-axes are equal, i.e., a = b
has the equation x2 - y2 = a2.  
Its asymptotes  y  x are perpendicular and inclined to the x-axis at an angle of 45.
Foci of the equilateral hyperbola, 
F1(-2 a, 0) and F2(2 a, 0),
and the eccentricity  e = c/a = 2.
Translated hyperbola
The equation of a hyperbola translated from standard position so that its center is at S(x0, y0)  is given by
b2(x - x0)2 - a2(y - y0)2 = a2b2
 or  
and after expanding and substituting constants obtained is
Ax2 + By2 + Cx + Dy + F = 0.
An equation of that form represents the hyperbola if
A B < 0 
that is, if coefficients of the square terms have different signs.
Equation of the hyperbola in vertex form
By translating the hyperbola, centered at (0, 0), in the 
negative direction of the
x-axis by x0 = -a, so that new
position of the center
S(-a, 0) then its equation is
  b2(x + a)2 - a2y2 = a2b2.  After squaring and reducing,
b2x2 + 2ab2x - a2y2 = 0 or
since obtained is
the equation of the hyperbola in vertex form.
Parametric equation of the hyperbola
In the construction of the hyperbola, shown in the below figure, circles of radii a and b are intersected by an arbitrary line through the origin at points M and N. Tangents to the circles at M and N intersect the x-axis at R and S. On the perpendicular through S, to the x-axis, mark the line segment SP of length MR to get the point P of the hyperbola. We can prove that P is a point of the hyperbola.
In the right triangles ONS and OMR,
by replacing OS = x and MR = SP  
and substituting
by dividing by b2,
therefore, P(x, y) is the point of the hyperbola.
The coordinates of the point P(x, y) can also be expressed by the angle t common to both mentioned 
triangles, so that    is the parametric equation of the hyperbola.
By substituting these parametrically expressed coordinates into equation of the hyperbola
that is,  known trigonometric identity.
Examples of hyperbola
Example:  Given is the hyperbola  4x2 - 9y2 = 36,  determine the semi-axes, equations of the asymptotes,
coordinates of foci, the eccentricity and the semi-latus rectum.
Solution:   Put the equation in the standard form to 
determine the semi-axes, thus
4x2 - 9y2 = 36 | 36
Asymptotes,
Applying,
coordinates of foci,  F1(-13, 0) and F2(13, 0).
The eccentricity, and the semi-latus rectum,
Example:  Write equation of a hyperbola with the focus at F2(5, 0) and whose asymptotes are,
Solution:
Therefore, the equation of the hyperbola,  
Example:  Find the angle subtended by the focal radii r1 and r2 at a point A(8, y > 0) of the hyperbola 
9x2 - 16y2 = 144.
Solution:   We determine the ordinate of the point A by plugging its abscissa into equation of the hyperbola,
   x = 8   =>   9x2 - 16y2 = 144
                  9 82 - 16y2 = 144,
                               16y2 = 432    =>    y2 = 27,    
 y1,2 = 33  so that  A(8, 33).
The equations of the lines of the radii r1 and r2, we     write using the formula of a line through two               points. Since,    
r1::   AF1 and F1(-c, 0) ,  and
then
                  
Therefore, the angle between the focal radii r1 and r2 at the point A of the hyperbola, as
Example:  The hyperbola is given by equation  4x2 - 9y2 + 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 x0 and y0 we can find by using the method of completing the square rewriting the equation in
the standard form,
Thus,           4x2 + 32x - 9y2 + 54y - 53 = 0,
                   4(x2 + 8x) - 9(y2 - 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   a2 = 9a = 3,   b2 = 4b = 2,  and the center of the hyperbola at  S(x0, y0)  or  S(-4, 3).
Half the focal distance the eccentricity
and the foci,  F1(x0 - c, 0)  so  F1(-4 - 13, 0)  and  F2(x0 + c, 0),   F1(-4 + 13, 0).
Equations of the asymptotes of a translated hyperbola
therefore, the asymptotes of the given hyperbola,
Example:   Write the equation of the hyperbola  9x2 - 25y2 = 225 in the vertex form.
Solution:   Using parallel shifting we should place the center of the hyperbola at S(-a, 0).
Rewrite    9x2 - 25y2 = 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
Intermediate algebra contents
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