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

Feb 3, 2016

In the general equation of a hyperbola
$\textcolor{w h i t e}{\text{XXX}} a$ represents the distance from the vertex to the center
$\textcolor{w h i t e}{\text{XXX}} b$ 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:
$\textcolor{w h i t e}{\text{XXX}} \frac{{x}^{2}}{{a}^{2}} - \frac{{y}^{2}}{{b}^{2}} = 1$

For a hyperbola with a vertical transverse axis,
the general formula is:
$\textcolor{w h i t e}{\text{XXX}} \frac{{y}^{2}}{{a}^{2}} - \frac{{x}^{2}}{{b}^{2}} = 1$

Note that the $\left({a}^{2}\right)$ 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 $\textcolor{red}{\text{red lines}}$ 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 $\frac{b}{a}$ and $- \left(\frac{b}{a}\right)$

For a hyperbola with a vertical transverse axis
the slopes of the two asymptotes are $\frac{a}{b}$ and $- \frac{a}{b}$

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