In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (-5, 0), (5, 0); vertices: (-8, 0), (8,0)

Key Concepts

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.

Standard Form of an Ellipse

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.

Distance Between Foci and Vertices

The distance between the foci and the vertices of an ellipse is related to its semi-major axis (a) and semi-minor axis (b). The distance 'c' from the center to each focus is calculated using the relationship c² = a² - b². This relationship helps in determining the lengths of the axes and ultimately in forming the standard equation of the ellipse.