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: $(7,9)$ and $(7,3)$ Endpoints of minor axis: $(5,6)$ and $(9,6)$

Key Concepts

Major and Minor Axes

The major axis of an ellipse is its longest diameter and the minor axis is the shortest. These axes determine the ellipse's overall shape and orientation, as well as providing the necessary measurements for calculating the lengths of the semi-axes used in the standard form equation.

Center of an Ellipse

The center of an ellipse is the midpoint of both the major and minor axes, and serves as the reference point in the standard equation. It can be found by averaging the coordinates of any pair of opposite endpoints of the axes, which is essential for repositioning the ellipse within the coordinate system.

Semi-major and Semi-minor Axis Lengths

The semi-major axis is half the length of the major axis and the semi-minor axis is half the length of the minor axis. These values are critical parameters in the ellipse’s standard form equation, impacting the width and height of the ellipse on a graph.

Ellipse

An ellipse is a conic section defined as the set of all points in a plane such that the sum of the distances from two fixed points (the foci) is constant. It is a fundamental shape in analytic geometry with distinct properties regarding symmetry and curvature.

Standard Form of Ellipse Equation

The standard form of an ellipse's equation is used to express the ellipse in a simple and canonical way, typically written as (x-h)²/a² + (y-k)²/b² = 1, where (h, k) represents the center of the ellipse and a and b are the lengths of the semi-major and semi-minor axes, respectively. This form facilitates analysis and graphing of the ellipse.

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