Find an equation for the ellipse that satisfies the given conditions. Eccentricity: (1)/(9), foc

step 1 So let's start with the foci. If the foci are at 0, 2, and 0 negative 2, then we know the center

Find an equation for the ellipse that satisfies the given...

Key Concepts

Ellipse

An ellipse is a conic section defined as the set of points for which the sum of the distances to two fixed points (the foci) is constant. It is a fundamental shape in geometry and appears in various physical contexts, such as planetary orbits. The ellipse’s shape is determined by its major and minor axes and its overall symmetry about its center.

Foci

The foci of an ellipse are two fixed points located along the major axis such that the sum of the distances from any point on the ellipse to the foci is constant. The position of the foci relative to the center of the ellipse helps determine the shape of the ellipse; a closer pair of foci results in a shape closer to a circle, while a larger separation makes it more elongated.

Eccentricity

Eccentricity is a measure of how much an ellipse deviates from being a circle. It is defined as the ratio of the distance from the center to a focus (c) to the length of the semi-major axis (a), represented by e = c/a. A smaller eccentricity indicates a shape nearly circular, while an eccentricity closer to 1 indicates a more elongated ellipse.

Standard Equation of an Ellipse

The standard form of the equation of an ellipse depends on the orientation of its major axis. For an ellipse centered at the origin with a horizontal major axis, the equation is written as (x^2/a^2) + (y^2/b^2) = 1, and for one with a vertical major axis, as (x^2/b^2) + (y^2/a^2) = 1. This form simplifies the process of identifying the lengths of the semi-major and semi-minor axes which define the ellipse’s shape.

Relationship Between Parameters

In an ellipse, the relationship between the semi-major axis (a), the semi-minor axis (b), and the focal distance (c) is given by the equation c^2 = a^2 - b^2. This relationship connects the geometric features of the ellipse with its eccentricity, since the eccentricity is defined as e = c/a. Understanding this relation is crucial in deriving the equation of an ellipse when certain properties, such as the location of the foci and the eccentricity, are known.

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