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Description
An ellipse is the set of all points in a plane such that, for any point P of the set, the sum of the distances to two fixed points, called the foci, is a constant - specifically 2a, the length of the major axis of the ellipse. Like circles, parabolas and hyperbolas, ellipses are conic sections, i.e. curves obtained by slicing a right circular cone with a plane. Specifically, we obtain an ellipse when the plane intersects the lateral surface of the cone at an angle to its axis of rotation different from 90° but doesn’t intersect its base.
The foci of an ellipse are two fixed points in a plane about which the ellipse is constructed. They’re usually denoted by F1 and F2, and lie on the major axis of the ellipse, symmetrically on either side of the centre. For any point P on the ellipse, the sum of the distances from P to each of the two foci is always equal to 2a, where a is the length of the semi-major axis. If the ellipse is oriented horizontally, the coordinates of the two foci are (-c, 0) and (c, 0) respectively, where c is the focal distance.
The major axis and the minor axis of an ellipse are its two axes of symmetry, which intersect perpendicularly at the centre of the ellipse. They define the ellipse’s shape and size. For instance, the longer the major axis, the more elongated the ellipse becomes. In an ellipse, the major axis is typically longer than the minor axis. If the major axis and the minor axis are equal in length, the ellipse becomes a circle, and the two foci coincide at the centre of the circle.
The vertices of an ellipse are the two points where the ellipse intersects its major axis, reaches its maximum distance from the centre, and turns around. If the ellipse is oriented horizontally, the coordinates of the two vertices are (-a, 0) and (a, 0), respectively, where a is the length of the semi-major axis.
The co-vertices of an ellipse are the two points where the ellipse intersects its minor axis, reaches its minimum distance from the centre, and turns around. If the ellipse is oriented horizontally, the coordinates of the two co-vertices are (0, b) and (0, -b), respectively, where b is the length of the semi-minor axis.
Lastly, the centre of an ellipse is the midpoint of both the major axis and the minor axis. It is also the midpoint between the two foci. An ellipse can be placed anywhere in the plane, so its centre can have any coordinates (h, k). However, for simplicity, ellipses are often positioned with their centre at the origin (0, 0) of the Cartesian coordinate system. An ellipse is symmetric with respect to its centre. This means that if you rotate the ellipse around its centre, it will look the same.
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Playlists
1) Geometry: [ Ссылка ]
2) Trigonometry: [ Ссылка ]
3) Arithmetic: [ Ссылка ]
4) Algebra: [ Ссылка ]
5) Differential calculus: [ Ссылка ]
6) Conic sections: [ Ссылка ]
7) Triangles: [ Ссылка ]
8) Quadrilaterals: [ Ссылка ]
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Lastly, a disclaimer:
1) My YouTube videos are intended to serve as guides to understanding mathematics, the English language, linguistics, etc.
2) The voice that narrates my YouTube videos is not mine. It is an AI algorithm that converted my text into speech. English is my second language, and speech is not my forte, so I’ve chosen to use an AI algorithm to convert my text into speech.
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