Conic Sections
Hyperbola
Examples of hyperbola Translation of equilateral or rectangular hyperbola with the coordinate axes as its asymptote
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(-Ö13, 0) and F2(Ö13, 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 = ±3Ö3  so that  A(8, 3Ö3).
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 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 