In Exercises 25-36, find the standard form of the equation of each ellipse satisfying the given

step 1 In this question we have given that the length of major axis and which is vertical is 20 and len

In Exercises 25-36, find the standard form of the equation...

Key Concepts

Ellipse

An ellipse is a set of points in the plane for which the sum of the distances from any point on the ellipse to two fixed points (foci) is constant. It is a type of conic section characterized by its oval shape and various properties such as its center, axes, and foci.

Standard Form Equation

The standard form of the equation for an ellipse depends on its orientation. For an ellipse centered at (h, k) with a horizontal major axis, the equation is ((x - h)² / a²) + ((y - k)² / b²) = 1, whereas for a vertical major axis the equation is ((x - h)² / b²) + ((y - k)² / a²) = 1. This form clearly displays the lengths of the semi-axes and the center of the ellipse.

Center

The center of an ellipse is the point (h, k) around which the ellipse is symmetrically positioned. It is a fundamental component in the standard equation since it helps determine the ellipse's location in the coordinate plane.

Major Axis

The major axis of an ellipse is its longest axis and passes through the center and both foci. Its length is given as 2a when a is the semi-major axis. In this context, if the major axis is vertical, a is associated with the y-coordinate, reflecting the ellipse’s vertical elongation.

Minor Axis

The minor axis is the shortest axis of an ellipse and is perpendicular to the major axis at the center. Its length is 2b where b is the semi-minor axis. This axis determines the width of the ellipse in the direction perpendicular to the major axis.

Semi-Axes

The semi-major axis (a) and semi-minor axis (b) are half the lengths of the major and minor axes respectively. They are used in the standard equation of the ellipse to express its dimensions. Knowing the lengths of the major and minor axes allows you to calculate these semi-axes, which are crucial in writing the ellipse's equation.

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