Conics in Polar Coordinates: Unified Theorem: Hyperbola Proof
In this video I go over further into the Unified the Theorem for Conics and this time prove it is applicable for hyperbolas. Recall that the theorem states that a conic is formed when the ratio of the distance to the focus divided by the distance to the directrix is a constant e, called the eccentricity. In my earlier videos I proved the case for parabolas (e = 1) and ellipses (e is less than 1), and in this video I prove the case for when e is greater than 1, i.e. for hyperbolas. But to save time I carry off from the proof of ellipses in which I first derived a formula for the conic for the general case of e is not equal to 1. The resulting formula in Cartesian or Rectangular coordinates looks very much like that for ellipses and hyperbolas, but the different values of e decides which conic is described. For the case that e is greater than 1, I show that indeed the unified theorem describes a hyperbola, and in fact it is a shifted horizontally shifted horizontal hyperbola. And just like for the ellipses case, the focus defined by the unified theorem is the exact same as that defined by the conventional theorem, albeit having two foci.
This is a very important video to understand how different theorems can describe the same curve but for better or worse across different coordinate systems. I will be going over some examples in later videos to better illustrate this theorem and its very simple formulation in polar coordinates so stay tuned!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh5pCF992AdJKQ8h1Dw
View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/conics-in-polar-coordinates-unified-theorem-hyperbola-proof
Related Videos:
Conics in Polar Coordinates Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0H4OJpJ2gslXVLT8mP-SgJP
Conic Sections Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FEUsuxP3KS5DRbidSSGBPL
Polar Coordinates Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUFiPLsYw5_Ljd5riOUzjP
Conics in Polar Coordinates: Unified Theorem: Ellipse Proof:
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▶️ 3Speak
Hi! I am a robot. I just upvoted you! I found similar content that readers might be interested in:
https://www.youtube.com/watch?v=0QpA9t0UlsMI don’t always prove the unified theorem for conic sections is applicable for hyperbolas but when I do I usually carry off from my ellipses proof to save time from doing the algebra over again ;)
View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/conics-in-polar-coordinates-unified-theorem-hyperbola-proof
UPDATE: At the 15:33 mark of the video, I made an error because the Directrix is in the wrong location! Please see my later video (See Question/Statement 10 of the Quiz) explaining why the Directrix should be located in between the Focus and Center.: https://peakd.com/mathematics/@mes/parametric-equations-and-polar-coordinates-true-false-quiz-and-solutions