In this video I go over an example on Ellipses that looks at the amazing reflective property of ellipses! This property is such that the angle between a line drawn from one focus to a point on the ellipse is equal to the angle from that same point to the other focus. This means that emitting light or sound from one focus gets reflected onto the other focus! This is an amazing property of ellipses and is used in many real world applications such as Lithotripsy to treat kidney stones, and in Whispering Galleries. I will go over these applications in a later video so stay tuned!
In proving this property, I use the formula I derived from my earlier video on the angle between two intersecting lines in terms of the slopes of the lines. Using that formula twice, one for each angle, and determining the slope of the tangent line at the point on the ellipse, I show how the tan value of both angles are equivalent, and thus the angles themselves. Although this is a very tedious and detailed example, the methodical steps I go through provide a great understanding of the algebra and equation manipulating behind the derivation, so make sure to watch this video!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhvlZADcp8Lq3bF-tUw
View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/video-notes-conic-sections-ellipses-example-3-amazing-reflective-property
Conic Sections: Ellipses: Example 2: Vertical Ellipse: https://youtu.be/cK0z2ZMC1fU
Conic Sections: Ellipses: Example 1: https://youtu.be/m9FXGv3kMkk
Conic Sections: Ellipses: Definition and Derivation of Formula (Including Circles): https://youtu.be/9dETsJ2tz_M
Conic Sections: Parabolas: Definition and Formula: https://youtu.be/kCJjXuuIqbE
Angle Between Two Lines: Formula in terms of Slopes: https://youtu.be/O8M4ZErxE-M .
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