Conic Sections

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## Sunbeams and Satellite DishesRecall that a point It follows that the construction of a point on the parabola with a compass and straightedge is relatively straightforward. Given a point Each diameter of a parabola is parallel to its axis. To see this, let
Furthermore,
On the other hand,
Similarly,
It follows that Based on the picture at the top of the page, you may have guessed that the altitude of the isosceles triangle This leads to one of the most famous and important properties of the parabola, namely, that the diameter and the focal chord to the endpoint of the diameter form congruent
angles with the tangent at the point. This is because the altitude of an isosceles triangle bisects the angle, while vertical angles have equal measure. So the angle made by the diameter and the tangent is equal to the angle made by This is of course the principle behind satellite dishes. A satellite dish is in the shape of a paraboloid of revolution, obtained by revolving a parabola about its axis. Waves traveling parallel to the axis from a distant source strike the surface of the dish and bounce off, concentrating the signal at the focus, which is where the feed antenna is placed. Click here for a demonstration. There is an ancient legend that the Greek mathematician Archimedes designed parabolic reflectors during the siege of Syracuse, to focus sunlight and set the ships of the enemy on fire. This is now thought apocryphal, but the idea is sound enough. The reflectors behind car headlights are shaped like paraboloids with the light source at the focus; this directs the light beams straight ahead of the car. Telescope reflectors are also in the shape of paraboloids. Liquid mirror telescopes employ a rotating circular vat of liquid metal: under the influences of gravity and centrifugal force, the liquid climbs the sides of the vat, forming a parabolic surface. Such telescopes can only be directed upward, of course. The same principle can be used to concentrate sound waves. If two satellite dishes are positioned vertically, facing each other, then a person whispering into one dish can be heard at the focus of the other. When I was a freshman in high school, my algebra teacher asked me to find him a pair of dishes, having seen the demonstration at the Exploratorium in San Francisco. Much to his surprise, I was able to acquire a large fiberglass dish in a short amount of time, thanks to my father's military contacts. He wasn't ready for it, so it languished in the backyard for three years. I had the same teacher for calculus when I was a senior and he asked if I still had the dish. My father and I disassembled the dish into two pieces and I took it to school in my truck. Then our class stood the halves up on desks at opposite ends of the library and whispered to one another. It worked, and everyone was amazed. I also once encountered a pair of receivers like this in a park. Apparently no one knew what they were for, and a small tree had even grown up between them. But they still worked, the noise of the busy playground notwithstanding. We have seen how to construct points on the parabola using only a compass and straightedge. We can also construct the parabola as an This is what admits the |

**Michael Ortiz, Ph.D.
Assistant Professor of Mathematics
Department of Natural and Behavioral Sciences**

Sul Ross State University Rio Grande College

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Copyright © 2013 Michael Luis Ortiz. All rights reserved.