In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Major axis vertical with length 10; length of minor axis = 4; center: (-2, 3)

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse's equation is given by (x-h)²/b² + (y-k)²/a² = 1, where (h, k) is the center, 'a' is half the length of the major axis, and 'b' is half the length of the minor axis. For ellipses with a vertical major axis, the equation is rearranged to (y-k)²/a² + (x-h)²/b² = 1.

Major and Minor Axes

The major axis of an ellipse is the longest diameter, while the minor axis is the shortest. The lengths of these axes are crucial for determining the values of 'a' and 'b' in the standard form equation. In this case, the major axis length is 10, so 'a' is 5, and the minor axis length is 4, making 'b' equal to 2.

Center of the Ellipse

The center of an ellipse is the midpoint of both axes and is represented by the coordinates (h, k) in the standard form equation. For the given problem, the center is specified as (-2, 3), which directly influences the placement of the ellipse on the coordinate plane and is essential for writing the equation correctly.