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

step 1 We are writing the equation of an ellipse, and the center of this ellipse is at negative 2, 3. S

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

Key Concepts

Orientation

The orientation of an ellipse refers to the direction in which its major axis lies. If the major axis is vertical, the larger denominator (a^2) is associated with the y-term, and if it is horizontal, it is associated with the x-term. This determines how the ellipse is stretched along the x and y axes.

Center of an Ellipse

The center of an ellipse is the midpoint of both the major and minor axes, and it serves as the point around which the ellipse is symmetric. In the equation of the ellipse, the center is represented by (h, k), which translates the ellipse from the origin to the point (h, k).

Major and Minor Axes

The major axis is the longest diameter of the ellipse, and its length is twice the semi-major axis (a). Similarly, the minor axis is the shortest diameter, with length twice the semi-minor axis (b). The values of a and b are determined by halving the given lengths of the major and minor axes respectively.

Standard Form Equation for an Ellipse

The standard form of an ellipse's equation depends on the orientation of its major axis. For an ellipse centered at (h, k) with a horizontal major axis, the equation is ((x - h)^2)/(a^2) + ((y - k)^2)/(b^2) = 1, and for a vertical major axis it is ((x - h)^2)/(b^2) + ((y - k)^2)/(a^2) = 1, where a is the semi-major axis and b is the semi-minor axis.

Ellipse

An ellipse is a type of conic section that represents the set of all points for which the sum of the distances to two fixed points (the foci) is constant. It is characterized by having two principal axes, one being the longer (major axis) and the other being the shorter (minor axis).

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