For any point $P$ on a hyperbola with foci $F_1$ and $F_2$ (transverse-axis length $2a$), the absolute difference $|PF_1 - PF_2|$ equals:

**Definition of a hyperbola**: the locus of points $P$ such that the *absolute difference* of distances to the two foci is a constant — exactly $2a$, the length of the transverse axis. $$|PF_1 - PF_2| = 2a$$ (For comparison, the ellipse uses *sum* of distances $= 2a$ — same $2a$, different operator. Both are the load-bearing definitional identity for their conic.) **Why option C ($2c$) is tempting**: $2c$ is the *distance between the foci*, not the locus's invariant. The points where this difference equals $2c$ lie *on* the focal segment, not on the hyperbola.