Suppose we have a number of rays that all start the same distance away from [math]F_2[/math], which we can call [math]d[/math].[br][br]By the reflective property of hyperbolas, each ray will strike the hyperbola at some point [math]P[/math] and reflect to [math]F_1[/math].[br][br]How far will they travel?[br][br]Each ray was supposed to travel [math]d[/math] units to [math]F_2[/math], but that was cut short by a distance of [math]PF_2[/math]. So each ray will have traveled [math]d-PF_2[/math] units when it hits the hyperbola. It will then travel a further [math]PF_1[/math] units to arrive at [math]F_1[/math].[br][br]Total distance traveled by the ray: [math]d-PF_2+PF_1=d+PF_1-PF_2[/math][br][br]But, by the definition of a hyperbola, the differences of distances from any point on the hyperbola is a constant, which we can call [math]2a[/math] (the length of the transverse axis).[br][br][br]Therefore, every ray will travel a distance of [math]d+\left(PF_1-PF_2\right)=d+2a[/math] units.[br][br]This means the 14 rays, which all started an equal distance from [math]F_2[/math], but are coming from different directions will all travel the same distance by the time they reach [math]F_1[/math]. If they are moving at the same speed, they will arrive simultaneously. Animate the diagram to see this happen.