Consider an ellipse to be the set of points in a plane whose...

Consider an ellipse to be the set of points in a plane whose distances from two fixed points have a constant sum $2 a$. Derive the equation of an ellipse. Assume the two fixed points are on the $x$ -axis equidistant from the origin.

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Key Concepts

Ellipse

An ellipse is a set of points in a plane such that the sum of the distances from each point to two fixed points (called foci) is constant. This definition highlights the unique geometric property that distinguishes ellipses from other conic sections, like parabolas and hyperbolas.

Foci

The foci are two fixed points located along the major axis of the ellipse. Their positions determine the shape and eccentricity of the ellipse since every point on the ellipse maintains a constant sum of distances to these foci.

Major and Minor Axes

The major axis is the longest line segment that passes through the center of the ellipse and the two foci, while the minor axis is the line segment perpendicular to the major axis at the center. Their lengths are related by the equation b² = a² - c², where 'a' is the length of the semi?major axis, 'b' is the length of the semi?minor axis, and 'c' is the distance from the center to a focus.

Standard Equation of Ellipse

The standard equation of an ellipse, given by x²/a² + y²/b² = 1, represents an ellipse centered at the origin with its major axis aligned with the x-axis (and the foci along it). This equation encapsulates the geometric features of the ellipse, linking its algebraic form to its defining properties such as the lengths of the axes and the constant sum of distances to the foci.

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