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

In Exercises $25-36,$ find the standard form of the equation of each ellipse satisfying the given conditions.
Endpoints of major axis: $(2,2)$ and $(8,2)$ Endpoints of minor axis: $(5,3)$ and $(5,1)$

Key Concepts

Ellipse

An ellipse is a closed curve defined as the set of all points for which the sum of the distances to two fixed points (the foci) is constant. It is a central conic section with many distinctive geometric properties and symmetries.

Standard Form of the Ellipse Equation

The standard form of an ellipse's equation provides a normalized representation of its geometric shape, usually expressed as ((x-h)²/a² + (y-k)²/b² = 1) for a horizontally oriented ellipse or ((x-h)²/b² + (y-k)²/a² = 1) for a vertically oriented ellipse, where (h, k) represents the center, and 'a' and 'b' are the lengths of the semi-major and semi-minor axes, respectively.

Major and Minor Axes

The major axis of an ellipse is its longest diameter, passing through both foci, while the minor axis is the shortest diameter perpendicular to the major axis. The endpoints of these axes are crucial in determining the overall dimensions and orientation of the ellipse.

Finding the Center

The center of an ellipse is located at the midpoint of either pair of endpoints of the major or minor axis. It serves as the origin in the translated coordinate system for the standard form equation, simplifying the expression of the ellipse's geometry.

Calculating Semi-Axis Lengths

The semi-major axis (denoted as 'a') and the semi-minor axis (denoted as 'b') are half the lengths of the major and minor axes, respectively. These distances are used in the standard form equation to express the ellipse's size and shape, directly influencing its stretching along the coordinate axes.

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