Hyperbolas: Reflective Property and Distance

Suppose we have a number of rays that all start the same distance away from , which we can call . By the reflective property of hyperbolas, each ray will strike the hyperbola at some point and reflect to . How far will they travel? Each ray was supposed to travel units to , but that was cut short by a distance of . So each ray will have traveled units when it hits the hyperbola. It will then travel a further units to arrive at . Total distance traveled by the ray: But, by the definition of a hyperbola, the differences of distances from any point on the hyperbola is a constant, which we can call (the length of the transverse axis). Therefore, every ray will travel a distance of units. This means the 14 rays, which all started an equal distance from , but are coming from different directions will all travel the same distance by the time they reach . If they are moving at the same speed, they will arrive simultaneously. Animate the diagram to see this happen.