Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of an Ellipse
The standard form of the equation of an ellipse is given by (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center, a is the semi-major axis, and b is the semi-minor axis. This form allows for easy identification of the ellipse's dimensions and orientation, which is crucial for graphing and understanding its properties.
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Foci of an Ellipse
The foci of an ellipse are two fixed points located along the major axis, which are essential for defining the shape of the ellipse. The distance from the center to each focus is denoted as c, where c² = a² - b². The foci play a significant role in the geometric definition of the ellipse, as the sum of the distances from any point on the ellipse to the two foci is constant.
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Foci and Vertices of an Ellipse
Vertices of an Ellipse
The vertices of an ellipse are the points where the ellipse intersects its major and minor axes. For a horizontally oriented ellipse, the vertices are located at (h ± a, k) and for a vertically oriented ellipse at (h, k ± b). Identifying the vertices is crucial for accurately sketching the ellipse and understanding its dimensions and orientation.
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Foci and Vertices of an Ellipse