The equation of the hyperbola and its derivation

Definition of the hyperbola

In the lesson on the equation of the ellipse and of the circle, the ellipse was defined as the set of points whose sum of distances to two foci is constant. The hyperbola arises from replacing the sum with the difference.

Let $F_1$ and $F_2$ be two distinct points of the plane, the foci, at a distance $2c$ from each other, and let $a$ be a number with $0 < a < c$. The hyperbola with foci $F_1$ and $F_2$ is the set of points $P$ of the plane whose distances to the foci differ by exactly $2a$, whichever focus is closer.

In symbols, if $d_1$ and $d_2$ are the distances from $P$ to $F_1$ and to $F_2$, the point $P$ belongs to the hyperbola when it satisfies one of the two equations below; the points that satisfy the first are closer to $F_2$ and form a branch, and those that satisfy the second, closer to $F_1$, form the other:

The condition $0 < a < c$ is not arbitrary. In the triangle with vertices $P$, $F_1$ and $F_2$, the difference between two sides is always less than the third, which measures $2c$; it follows that no point off the line through the foci can have a difference of distances equal to or greater than $2c$. If $a = c$, the only points that satisfy the condition are those of the two rays that start at the foci and point outward, and if $a > c$, there are none; the first of these statements follows from the equality case of the triangle inequality, proved in the lesson on vectors in the plane: applied to $\vec{u} = \vec{PF_2}$ and $\vec{v} = \vec{F_2F_1}$, the equality $\operatorname{d}(P, F_1) = \operatorname{d}(P, F_2) + 2c$ requires one of these vectors to be the product of the other by a nonnegative number, that is, $P$ must lie on the line through the foci and $F_2$ must lie between $P$ and $F_1$ (or coincide with $P$); if $a = 0$, the condition says that $P$ is equidistant from the foci and describes the perpendicular bisector of the line segment joining the foci. None of these sets is a hyperbola, and that is why they are excluded.

The midpoint of the line segment joining the foci is the center of the hyperbola, and the line through the foci is its focal axis. The two points of the hyperbola lying on the focal axis are its vertices; the distance between them is $2a$, so each vertex is at a distance $a$ from the center.