Conic Sections: Ellipses: Definition and Derivation of Formula (Including Circles)

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    mes

    Published on May 10, 2021
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    In this video I look further into conic sections, or conics, and this time go over the definition of an ellipse and derive it’s basic formula. Note though that I have made videos on ellipses several years ago, but I am revisiting them so that they better tie in with the conics video series that I am covering now. The definition of an ellipse is the set of points in a plane the sum of whose distances from two fixed points, known as the foci (plural for focus), are constant. Graphically viewing this definition shows clearly the “ellipse” shape that we are most familiar with. Using Pythagoras to determine the sum of the distances and setting it equal to a constant 2a, we can then do some detailed algebra to eventually get the formula for an ellipse: x2/a2 + y2/b2 = 1, in which a is greater than b and the major axis (or longer side of the ellipse) is on the horizontal axis. I also show how if we wanted a “vertical ellipse” we can switch the x and y variables to obtain x2/b2 + y2/a2 = 1, and thus the major axis is on the vertical axis. When a = b = r, we obtain the common formula for a circle: x2 + y2 = r2, thus essentially this is video is a prove for the formula of an ellipse and a circle, which is just a more symmetric ellipse by definition.

    I also go over a brief history overview of Johannes Kepler, the 16th/17th century German mathematician and astronomer, which is famous for Kepler’s law that states the orbits of the planets in the solar system are ellipses with the Sun at one focus.

    This is a very extensive and detailed derivation of ellipses, but if you go through it all you will no doubt become an expert in the math behind ellipses, so make sure to watch this whole video!

    Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhvkMmGx-M9SHmQbXvA

    View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/video-notes-conic-sections-ellipses-definition-and-derivation-of-formula-including-circles

    Related Videos:

    Conic Sections: Parabolas: Definition and Formula: https://youtu.be/kCJjXuuIqbE
    Ellipses: Definition and Proof of Equation: http://youtu.be/tvfNgT4CrFs
    Inverted Ellipses: http://youtu.be/NQQFj66-bxA
    Equation of a Circle and it's proof: http://youtu.be/xMXYJ9UeF4I .


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