In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (0, -4), (0, 4); vertices: (0, −7), (0, 7)

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, which is determined by the placement of its foci and vertices.

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 for a horizontal ellipse, and (x-h)²/b² + (y-k)²/a² = 1 for a vertical ellipse. Here, (h, k) is the center of the ellipse, 'a' is the distance from the center to the vertices, and 'b' is the distance from the center to the co-vertices.

Foci and Vertices Relationship

In an ellipse, the distance from the center to each focus is denoted as 'c', and it relates to 'a' and 'b' through the equation c² = a² - b². The vertices are located at a distance 'a' from the center along the major axis, while the foci are located at a distance 'c' from the center along the same axis, which helps in determining the overall shape and dimensions of the ellipse.