Q.Points $A$ and $B$ are $10\,\text{km}$ apart and it is determined from the sound of an explosion heard at those points at different times that the location of the explosion is $6\,\text{km}$ closer to $A$ than $B$. Show that the location of the explosion is restricted to a particular curve and find an equation of it.

Each conic's reflective (or orbital, or structural) property explains why it is the shape engineers and nature actually choose:

Parabola. A ray from the focus reflects parallel to the axis, and conversely — headlight bulbs and dish receivers sit at the focus (car headlights, satellite dishes, field microphones, solar cookers). Parabolic arches (river bridges, the Eiffel Tower) are chosen additionally for structural stability.

Ellipse. By Kepler's laws, planets orbit the Sun on ellipses with the Sun at one focus (comets like Halley's, $e\approx0.97$, too). A ray from one focus reflects to the other focus — whispering-gallery ceilings, and lithotripters (a shock wave from one focus is focused onto a kidney stone at the other, avoiding surgery). Earth itself is an oblate spheroid, an ellipse revolved about its minor axis.

Hyperbola. Some comets travel on hyperbolic (non-returning) paths. A ray directed at one focus reflects to the other — used to locate a ship or aircraft from time-difference-of-arrival signals at two stations (one branch of a hyperbola with the stations as foci), and in two-mirror telescopes, where a parabolic primary and hyperbolic secondary share a focus so light is relayed to the hyperbola's second focus behind the primary mirror. …