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Parabola: Reflective Property

In the diagram below, you can move to be any point on the parabola.
  1. First, consider a ray that is parallel to the axis of symmetry of the parabola and coming in to point . [Show the ray] How will it be reflected off of the surface of the parabola? We can find the angles of incidence and reflection by using the tangent line to the parabola at . [Show the tangent]
  2. Now we can also look at the two segments and . By the definition of a parabola they are congruent. Also, notice that . [Show segments from ]
  3. Consider where is the y-intercept of the tangent line. Notice that
  4. So we have
  5. This means . Why?
  6. Also we can conclude that . Why?
  7. In conclusion, because , we know that the ray to will be reflected along to . [Show the arrows]
  8. You can drag around to see that this will always be true at any point on the parabola.
Theorem Any ray parallel to the axis of symmetry will reflect off of the surface to the parabola to the focus. Any ray emitted from the focus of the parabola will reflect off the surface such that its image is parallel to the axis of symmetry. [Show the arrows and reverse them to see this property]