Find the standard form of the equation of the ellipse satisfying the given conditions. Foci: (-4,0), (4,0); Vertices: (-5,0) (5,0)
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Ellipse Definition
An ellipse is a set of points in a plane where the sum of the distances from two fixed points, called foci, is constant. The standard form of an ellipse's equation varies based on its orientation, either horizontal or vertical, and is crucial for identifying its properties such as foci, vertices, and axes.
The standard form of the equation of an ellipse centered at the origin is given by (x²/a²) + (y²/b²) = 1 for a horizontal ellipse, where 'a' is the distance from the center to the vertices along the x-axis, and 'b' is the distance along the y-axis. For a vertical ellipse, the form is (x²/b²) + (y²/a²) = 1. Understanding this form is essential for deriving the equation from given foci and vertices.
In an ellipse, the distance between the foci and the vertices is related to the semi-major axis (a) and the semi-minor axis (b). The distance between the foci is 2c, where c is the distance from the center to each focus, and it is calculated using the relationship c² = a² - b². This relationship helps in determining the values of 'a' and 'b' necessary for writing the standard form of the ellipse's equation.