Standard Equation and Properties of a Hyperbola

A hyperbola is defined as the locus of a point $P$ that moves so that the absolute difference of its distances from two fixed points, the foci $F_1$ and $F_2$, is always the same constant, $2a$ (with $2a$ less than the distance $F_1F_2=2c$ between the foci -- the reverse inequality from the ellipse, since a difference of two sides of a triangle is always less than the third side).

Standard equation. Placing the foci symmetrically on the $x$-axis at $F_1(-c,0)$ and $F_2(c,0)$, the condition $|PF_1-PF_2|=2a$ works out, after simplification, to the standard equation of a hyperbola:

Unlike the ellipse, $a$ and $b$ carry no size ordering here ($b$ can be larger, smaller, or equal to $a$) -- what matters is only that $c^2=a^2+b^2$, so $c$ is always the largest of the three.

Eccentricity. The eccentricity is again $e=\dfrac{c}{a}$, but since $c>a$ for a genuine hyperbola (the defining inequality $2a<2c$), $e>1$ always -- the opposite range from the ellipse's $0<e<1$. Rearranging $b^2=c^2-a^2$ using $c=ae$ gives

Key terms. The foci are $F_1(-c,0)$ and $F_2(c,0)$ with $c=ae$; the vertices are $(\pm a,0)$, the points where the curve actually crosses the transverse axis; the transverse axis (joining the vertices) has length $2a$, and the conjugate axis (along the $y$-axis, not touching the curve) has length $2b$; the centre is the origin. The latus rectum through either focus has length

the same formula as the ellipse's.

Asymptotes. A hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ has two straight lines, $y=\pm\dfrac{b}{a}x$, called asymptotes -- lines that the two branches of the curve approach arbitrarily closely at large distance from the centre, but never actually touch. No other conic in this chapter has asymptotes; their presence is a distinguishing feature of the hyperbola alone, arising because the curve is unbounded in a way the ellipse (bounded) and parabola (unbounded but single-branched, with no straight-line limiting direction) are not. …