The length of a hyperbola's transverse axis is equal to the _____ the distances from any point on the hyperbola to each focus.

The length of a hyperbola's transverse axis is equal to the difference of the distances from any point on the hyperbola to each focus.

In a hyperbola, the foci are located along the transverse axis, and the key characteristic of a hyperbola is that the absolute difference between the distances to the two foci is constant and equal to the length of the transverse axis. Specifically, if you have a point $P$ on the hyperbola and two foci $F_1$ and $F_2$, then:

$|d(P, F_1) - d(P, F_2)| = 2a$

where $a$ is the distance from the center of the hyperbola to a vertex along the transverse axis, and thus the full length of the transverse axis is $2a$.