the reflective property of conics
Why is a satellite dish shaped like a parabola, a flashlight reflector parabolic, and a whispering gallery elliptical? Because each conic has a magical reflection rule built into its geometry: a ray of light or sound that obeys the ordinary 'angle in equals angle out' law of bouncing does something special with the focus. This is the reflective (or optical) property, and it is what makes conics the workhorses of optics and acoustics.
Each conic does it differently. Parabola: every ray arriving parallel to the axis reflects off the curve and passes exactly through the focus — and in reverse, a source placed at the focus sends out a perfectly parallel beam. (That is why a dish gathers faint distant signals to a single receiver at the focus, and a headlight throws a straight beam.) Ellipse: a ray leaving one focus reflects off the curve and passes through the other focus, every time — so all sound from one focus reconverges at the other, the 'whispering gallery' effect, and a lithotripter focuses shock waves from one focus onto a kidney stone at the other. Hyperbola: a ray aimed toward the far (virtual) focus reflects toward the near focus, a property used to fold light paths in Cassegrain telescopes. In each case the underlying reason is a tangent-line bisection: the tangent at any point makes equal angles with the focal lines (or with the axis direction), which is exactly the law of reflection.
These properties are not coincidences but theorems flowing straight from the focal definitions, and they are why conics, discovered as abstract slices of a cone two thousand years before any application, turned out to be precisely the right shapes for telescopes, antennas, headlamps, and medical devices.
Stand at one focus of an elliptical 'whispering gallery' and murmur; a friend at the other focus hears you clearly though people between you hear nothing — your sound rays all reflect off the wall and reconverge on the second focus. A parabolic dish does the reverse for radio: parallel waves from a distant satellite all bounce to the single focus, where the receiver sits.
Parabola: parallel-in to focus. Ellipse: focus to focus. Hyperbola: toward virtual focus to near focus.
The reflection obeys the same 'angle in = angle out' law as a flat mirror; the conic's magic is only in its shape, which steers those equal-angle bounces to the focus. The property is exact for the ideal curve — real dishes lose a little to manufacturing imperfections.