Some Useful Terms of the Hyperbola

A few structural facts about the hyperbola that distinguish it from the ellipse:

• The hyperbola intersects the $X$-axis at its two vertices, $A(a,0)$ and $A'(-a,0)$ — but it never meets the $Y$-axis at all (setting $x=0$ in $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ gives $-\frac{y^2}{b^2}=1$, which has no real solution). This is a sharp contrast with the ellipse, which meets both axes.
• The segment $AA'$, of length $2a$, is called the transverse axis. The segment $BB'$ (a segment perpendicular to it through the centre, of length $2b$, though $B,B'$ themselves are not on the curve) is called the conjugate axis. Together these are the hyperbola's principal axes — in the standard form, the $X$-axis is the transverse axis and the $Y$-axis is the conjugate axis. …