Reflective Property of a Hyperbola

Reflective property. Exactly as for the ellipse, the two lines from a hyperbola's foci to a point $P$ on it make equal angles with the tangent at $P$ (Fig. 5.66) — so a wave directed at one focus is reflected towards the other focus.

This underlies ship/aircraft location systems: two receiving stations act as the two foci, and the difference in arrival time (hence distance) from a distress signal restricts the source to one branch of a hyperbola (Example 5.40 works this exact scenario, with two coast-guard stations $600\,\text{km}$ apart). It is also exploited in two-mirror telescopes: a parabolic primary mirror and a hyperbolic secondary mirror are positioned so that they share one focus — light reflected by the parabola towards its focus (§5.7.4) is intercepted by the hyperbola and, by the hyperbola's own reflective property, redir …