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, faux -sai, lengths of the major and minor axes, for

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

Key Concepts

Graphing an Ellipse

Graphing an ellipse involves plotting its center, drawing the major and minor axes using the lengths 2a and 2b, respectively, and marking the vertices and foci. The resulting graph is a smooth, closed curve that reflects the symmetry and key distances derived from the ellipse's equation.

Ellipse

An ellipse is the set of all points in a plane for which the sum of the distances to two fixed points (called the foci) is constant. It is one of the conic sections, and its shape is oval, exhibiting symmetry about both of its axes.

Standard Form Equation

The standard form of the ellipse's equation is written as (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center of the ellipse, and a and b are the lengths of the semi-axes along the x-axis and y-axis, respectively. The equation may need to be manipulated algebraically to fit this standard form.

Center of an Ellipse

The center of an ellipse is the midpoint of the line segment joining its two foci and is the point about which the ellipse is symmetric. In the standard form, the center is given by the coordinates (h, k), and it provides a reference for determining the positions of the vertices and foci.

Foci of an Ellipse

The foci are two fixed points located along the major axis of the ellipse, and their positions are determined using the relationship c² = |a² - b²|, where c is the distance from the center to each focus. The foci are crucial for defining the ellipse, as the sum of the distances from any point on the ellipse to these foci is constant.

Major Axis

The major axis of an ellipse is the longest diameter and passes through its center and both foci. Its total length is 2a if a is the semi-major axis, and it defines the widest extent of the ellipse in the direction where the ellipse is most stretched.

Minor Axis

The minor axis is the shortest diameter of the ellipse and is perpendicular to the major axis at the center. Its length is 2b if b is the semi-minor axis, indicating the narrowest span of the ellipse.

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