Find an equation of an ellipse that satisfies the given conditions. Vertices (-1, ±3) and foci (

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

Key Concepts

Relationship between a, b, and c

In an ellipse, the semi-major axis (a), the semi-minor axis (b), and the distance from the center to either focus (c) are related by the equation a² = b² + c² when a represents the semi-major axis. This relationship is fundamental as it links the geometric properties of the ellipse, ensuring the sum-of-distances property holds true.

Standard Form Equation of an Ellipse

The standard form equation of an ellipse is (x - h)²/b² + (y - k)²/a² = 1 for a vertically oriented ellipse and (x - h)²/a² + (y - k)²/b² = 1 for a horizontally oriented ellipse, where (h, k) is the center, a is the length of the semi-major axis, and b is the length of the semi-minor axis. This form makes it straightforward to identify these critical components of the ellipse.

Center

The center of an ellipse is the midpoint between its vertices (and also between its foci). It acts as a reference point for the ellipse and is used in the standard form of the ellipse's equation, defining the coordinates (h, k) in translations of the ellipse.

Foci

Foci (singular: focus) are two fixed points located along the major axis within an ellipse. The defining property of an ellipse is that for any point on the ellipse, the sum of its distances to the two foci is constant. The distance from the center to a focus (denoted as c) is a key parameter in defining the ellipse’s shape.

Vertices

Vertices are the points on the ellipse that lie on the endpoints of the major axis. They represent the farthest points from the center along the direction of the major axis and are essential for determining the size and orientation of the ellipse.

Ellipse

An ellipse is a set of points in a plane such that the sum of the distances to two fixed points, called the foci, is constant. It is a type of conic section that exhibits symmetry about both its major and minor axes, and its shape is determined by the relative distances between its center, vertices, and foci.

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