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, foci, lengths of the major and minor axis for the g

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

Key Concepts

Foci of an Ellipse

The foci (singular: focus) of an ellipse are two distinct points located along the major axis, on either side of the center. Their positions are determined by the distance c from the center, where c is found using the relation c² = a² - b². The unique property of the ellipse is that for any point on the ellipse, the sum of its distances to these two foci is constant.

Sketching an Ellipse

Sketching an ellipse typically involves first identifying and plotting its center, then marking the vertices along the major and minor axes based on the distances a and b. With these key points, the smooth, symmetric oval shape of the ellipse can be sketched, incorporating the placement of the foci along the major axis to underline its geometric properties.

Major and Minor Axes

The major axis of an ellipse is the longest line segment that can be drawn through the center and touches the ellipse at its furthest points, while the minor axis is the shortest. In the standard form, the values 'a' and 'b' represent the lengths of the semi-major and semi-minor axes respectively, and the overall lengths are double these values.

Standard Form Equation of an Ellipse

The standard form of an ellipse’s equation is written as (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center of the ellipse. The constants a and b represent the semi-major and semi-minor axes respectively, determining the lengths of the ellipse along its principal directions.

Ellipse

An ellipse is a type of conic section defined as the set of all points in a plane where the sum of the distances from two fixed points, known as the foci, remains constant. This property gives the ellipse its characteristic oval shape and distinguishes it from other conic sections such as parabolas and hyperbolas.

Center of an Ellipse

The center of an ellipse is the point that lies equidistant from all its vertices and is given by the coordinates (h, k) in the standard equation. It serves as the point of symmetry for the ellipse and is key for determining other features like the axes and foci.

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