Hyperbola - Reflections and Tangent Lines

Reflections and Tangent Lines

The ancient Greek geometers recognized a reflection property of hyperbolas. If a ray of light emerges from one focus and is reflected from the hyperbola, the light-ray appears to have come from the other focus. Equivalently, by reversing the direction of the light, rays directed at one of the foci from the exterior of the hyperbola are reflected towards the other focus. This property is analogous to the property of ellipses that a ray emerging from one focus is reflected from the ellipse directly towards the other focus (rather than away as in the hyperbola). Expressed mathematically, lines drawn from each focus to the same point on the hyperbola intersect it at equal angles; the tangent line to a hyperbola at a point P bisects the angle formed with the two foci, F₁PF₂.

Tangent lines to a hyperbola have another remarkable geometrical property. If a tangent line at a point T intersects the asymptotes at two points K and L, then T bisects the line segment KL, and the product of distances to the hyperbola's center, OK×OL is a constant.