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 is equals to 12 and length of its

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

Key Concepts

Orientation of the Ellipse

The orientation of an ellipse refers to the alignment of its axes. If the major axis is horizontal, then a is associated with the x-direction and is placed under the (x - h)² term in the equation. This distinction is crucial for accurately representing the ellipse’s shape and position relative to the coordinate axes.

Relationship Between Axis Lengths and a, b

In the standard form equation, the parameters a and b represent the semi-axes lengths of the ellipse. Specifically, the length of the major axis is 2a and the length of the minor axis is 2b. Knowing this relationship allows one to correctly calculate a and b from the given axis lengths and plug them into the equation.

Standard Form of an Ellipse

The standard form of an ellipse with center (h, k) is given by ((x - h)²)/(a²) + ((y - k)²)/(b²) = 1. This form is central to understanding ellipses in coordinate geometry, as it explicitly shows the relationship of any point (x, y) on the ellipse to its center and the lengths of its axes.

Major and Minor Axes

The major and minor axes of an ellipse are the longest and shortest diameters, respectively, that pass through the center of the ellipse. The major axis determines the ellipse’s overall width, while the minor axis determines its height. Identifying which is which is essential for correctly setting up the parameters in the standard form equation.

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