Question

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

Answer 1

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:

(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.]