What do #a# and #b# represent in the equation of a hyperbola?

1 Answer
Feb 3, 2016

In the general equation of a hyperbola
#color(white)("XXX")a # represents the distance from the vertex to the center
#color(white)("XXX")b # represents the distance perpendicular to the transverse axis from the vertex to the asymptote line(s).

Explanation:

For a hyperbola with a horizontal transverse axis,
the general formula is:
#color(white)("XXX")(x^2)/(a^2)-(y^2)/(b^2)=1#

For a hyperbola with a vertical transverse axis,
the general formula is:
#color(white)("XXX")(y^2)/(a^2)-(x^2)/(b^2)=1#

Note that the #(a^2)# always goes with the positive of #x^2# or #y^2#

The significance of #a# and #b# can (hopefully) be seen by the diagrams below:
enter image source here
(the #color(red)("red lines")# represent the asymptotes and are not part of the hyperbolae)

For a hyperbola with a horizontal transverse axis,
the slopes of the two asymptotes are #b/a# and #-(b/a)#

For a hyperbola with a vertical transverse axis
the slopes of the two asymptotes are #a/b# and #-a/b#

{I hope the reason for this is clear from the above diagrams and the definition of slope.]