In this video I go over further into the Unified Theorem for Conics, and its simple implementation in polar coordinates, and this time prove that it is indeed applicable for ellipses. Recall that the unified theorem for conics states that the ratio of the distance from the conic to the focus over that of the distance from the directrix is a constant e and called the eccentricity. In this proof I show that when e is less than 1, then the conic being described is an ellipse. I prove this by first developing a polar equation to describe the unified theorem. Then by squaring the polar equation and converting it Cartesian or Rectangular Coordinates, and a LOT OF ALGEBRA later, we can write a formula that resembles that of the conventional theorem for Ellipses. In fact I show that when e is less than 1, the ellipse described by the unified theorem is a shifted ellipse with even the focus having the same meaning as the foci in the conventional theorem. This is a very important video in both understanding how careful derivations are performed and the beauty of mathematically connecting two different theorems from two different coordinate systems; so make sure to watch this video!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh5osiwr43qjNKSMwcg
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/conics-in-polar-coordinates-unified-theorem-ellipse-proof
Related Videos:
Conics in Polar Coordinates: Unified Theorem: Parabola Proof:
Conics in Polar Coordinates: Variations in Polar Equations Theorem:
Conics in Polar Coordinates: Unified Theorem for Conic Sections:
Conic Sections: Hyperbola: Definition and Formula:
Conic Sections: Ellipses: Definition and Derivation of Formula (Including Circles):
Conic Sections: Parabolas: Definition and Formula:
Polar Coordinates: Cartesian Connection:
Polar Coordinates: Infinite Representations:
Polar Coordinates:
Completing the Square: .
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I don’t always prove that the unified theorem for conics is applicable for ellipses but when I do I usually go over some EXTENSIVE algebra in the derivation ;)
View Video Notes: https://steemit.com/mathematics/@mes/conics-in-polar-coordinates-unified-theorem-ellipse-proof
A cone is more difficult for me hihi; the cylinder I find much easier for such maths. This just tells me HOW far I am in your world of numbers. Stay on course, I am always your fan here.
Haha thanks! You can always keep up by watching my math videos!! =D
Wow great ..
I have just read conic section in my class ..
And it was helpful for me .. but it was also a bit complex to me because i am new to it .
I felt sorry that i just watch 1st video only .. but i will make sure to come back and watch the left all.
Sweet! Glad I can help. I go through conics in detail so make sure to watch my other videos! :)