For the following exercises, find the area of the ellipse. The area of an ellipse is given by th

step 1 So we want to find the area of an ellipse given the standard equation. So we know that the area

For the following exercises, find the area of the ellipse....

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Key Concepts

Axes of an Ellipse (Semi-Major and Semi-Minor Axes)

The lengths a and b in the standard form of an ellipse determine the dimensions of the ellipse's axes. The larger value between a and b corresponds to the semi-major axis, representing the longest radius, while the smaller value corresponds to the semi-minor axis, representing the shortest radius. These axes are crucial in understanding the shape and size of the ellipse.

Standard Form of an Ellipse

An ellipse is represented in standard form as (x - h)^2/a^2 + (y - k)^2/b^2 = 1, where (h, k) represents the center of the ellipse and a and b are the distances from the center to the ellipse along the x and y axes, respectively. This form allows one to easily identify the geometric properties of the ellipse.

Center of an Ellipse

The center of an ellipse is a key concept that denotes the point about which the ellipse is symmetric. It is directly derived from the standard form where (h, k) indicates how the ellipse is translated from the origin in the coordinate plane.

Area of an Ellipse

The area of an ellipse is given by the formula Area = ?ab, where a and b are the lengths of the semi-major and semi-minor axes, respectively. This formula shows how the ellipse's dimensions directly influence its total area, and it is a fundamental result in the geometry of ellipses.

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