An ellipse is given. Find the center, the foci, the length of the major axis, and the length of

step 1 In this problem, we wish to find the center, Fosai, lengths of the major and minor axes, for the

An ellipse is given. Find the center, the foci, the length...

Key Concepts

Ellipse

An ellipse is a type of conic section defined as the set of all points in a plane for which the sum of the distances to two fixed points (the foci) is constant. Understanding ellipses involves analyzing their geometric properties such as axes, foci, and center.

Standard Form of an Ellipse

The standard form of an ellipse's equation is typically written as (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center, and a and b are the lengths of the semi-axes. Converting an ellipse's equation to this form is crucial for identifying its key attributes.

Completing the Square

Completing the square is an algebraic technique used to rewrite a quadratic equation in a form that reveals its geometric properties. In the context of conic sections, it is especially useful for converting a general quadratic equation into the standard form of an ellipse or other conics.

Center

The center of an ellipse is the midpoint of both the major and minor axes and serves as the symmetry point for the ellipse’s geometry. In the standard form, the center is represented by the coordinates (h, k).

Foci

Foci are two distinct points located along the major axis of an ellipse. The defining property of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is constant, and their positions depend on the distances between the center and the vertices along the major axis.

Major and Minor Axes

The major axis is the longest diameter of the ellipse, running through the foci, while the minor axis is the shortest diameter, perpendicular to the major axis. The lengths of these axes, determined by the values of a and b in the standard form, are essential for understanding the size and shape of the ellipse.

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