Write an equation for each ellipse. See Example 2. Foci at (0,4),(0,-4) ; sum of distances from

step 1 Okay, our goal is to find the equation of the ellipse and we have some information. I'm going to

Write an equation for each ellipse. See Example 2. Foci at...

Key Concepts

Ellipse

An ellipse is a type of conic section defined as the set of all points in a plane such that the sum of the distances from these points to two fixed points (the foci) is constant. This definition leads to unique geometric properties and is used to derive the standard equation of the ellipse.

Foci

The foci are the two fixed points used in the definition of an ellipse. Their location relative to the center of the ellipse helps determine the ellipse’s shape. The distance from the center to each focus, denoted as c, is related to the lengths of the semi-major axis (a) and the semi-minor axis (b) through the equation c² = a² - b².

Major and Minor Axes

The major axis of an ellipse is the longest diameter that passes through its center and both foci, while the minor axis is the shortest diameter that is perpendicular to the major axis at the center. The lengths of these axes, given by 2a and 2b respectively, define the size and shape of the ellipse.

Standard Equation of an Ellipse

The standard equation of an ellipse is derived from its geometric definition and varies depending on its orientation. When the ellipse has a horizontal major axis, the equation is written as (x-h)²/a² + (y-k)²/b² = 1, and when it has a vertical major axis, it is written as (x-h)²/b² + (y-k)²/a² = 1, where (h, k) represents the center of the ellipse.

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