Difference of Focal Distances

The defining property of a hyperbola: the absolute difference of distances from any point on the hyperbola to the two foci is constant, equal to the transverse axis length 2a. Points on the right branch satisfy r₂ − r₁ = 2a; on the left branch r₁ − r₂ = 2a.

For the right branch ($x_1 > 0$): $r_1 = ex_1 + a$, $r_2 = ex_1 - a$.

$$r_1 - r_2 = (ex_1+a) - (ex_1-a) = 2a$$

For the left branch ($x_1 < 0$): $r_1 = -ex_1-a$, $r_2 = -ex_1+a$.

$$r_2 - r_1 = (-ex_1+a) - (-ex_1-a) = 2a$$

In both cases $|r_1 - r_2| = 2a$.

Contrast with ellipse: For the ellipse, $r_1 + r_2 = 2a$ (sum is constant). For the hyperbola, $|r_1 - r_2| = 2a$ (difference is constant). Both are $2a$, the length of the primary axis.

Physical construction: Fix two pins (the foci) farther apart than $2a$. With a rigid rod of length $2a$ and a string, the endpoints of the rod trace a hyperbola as the rod pivots — the difference of distances from the two pins to any point on the rod is always $2a$.