In this video I go over another example on parabolas, and this time go over a very useful application of parabolas, which is the property that lines extending from the focus of a parabola to a point on the parabola get protruded in a horizontal line. This property makes it very useful to design headlights and telescopes in a paraboloid shape; which is a 3D parabola formed by rotating a parabola about its central axis. In this particular video I prove that this is the case by proving the angle between a line tangent to point on the parabola and a line extending to that point from the focus is equivalent to the angle between the tangent line and a horizontal line protruding from that point on the parabola. Many cars and large-scale mirror telescopes use this design property of the parabola, and thus it is pretty cool to see the math behind just why it works! This is a very interesting video in it is a more advanced example, part of the Problems Plus section of my Calculus book, as well as its real world applications so make sure to watch this video!
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Example
Let P(x1 , y1) be a point on the parabola y2 = 4px with focus F(p , 0).
Let α be the angle between the parabola and the line segment FP, and let β be the angle between the horizontal line y = y1 and the parabola as in the figure.
Prove that α = β.
Thus, by a principle of geometrical optics, light from a source placed at F will be reflected along a line parallel to the x-axis.
This explains why paraboloids, the surfaces obtained by rotating parabolas about their axes, are used as the shape of some automobile headlights and mirrors for telescopes.
https://www.hotrodhotline.com/headlights-part-2-lowhigh-beams#.WVNPoYjyvic
Retrieved: 27 June 2017
Archive: https://archive.is/05CmKHeadlights Part 2: Low/High Beams
This is a side view of typical lens optics. Light is dispersed vertically (shown) and horizontally (not shown).
http://www.mathedpage.org/parabolas/geometry/
Retrieved: 28 June 2017
Archive: https://archive.is/8PsjAGeometry of the Parabola (2D)
Author: Henri Picciotto
This property is of course the basis of many applications (headlights, flashlights, satellite dishes, radar...) For example, here is a diagram of how this works in a reflector telescope:
The primary mirror is parabolic, reflecting the parallel rays to the focus. The secondary (flat) mirror redirects this towards the eyepiece.
