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

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).
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:
(the ##color(red)(“red lines”)## represent the 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.]