Conics in Polar Coordinates: Unified Theorem: Ellipse Proof

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!

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Related Videos:

Conics in Polar Coordinates: Unified Theorem: Parabola Proof: https://youtu.be/LBTfGQeDBGQ
Conics in Polar Coordinates: Variations in Polar Equations Theorem: https://youtu.be/ud5f4C4pkpk
Conics in Polar Coordinates: Unified Theorem for Conic Sections: https://youtu.be/eUvzyxCfJCw
Conic Sections: Hyperbola: Definition and Formula: https://youtu.be/UBIHovXNV9U
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
Polar Coordinates: Cartesian Connection: https://youtu.be/HcaTYrpmGaU
Polar Coordinates: Infinite Representations: https://youtu.be/QJYbnO7NzCk
Polar Coordinates: https://youtu.be/-KAdZL-N4ok
Completing the Square: http://youtu.be/5uxV6-f_qc0 .

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