In Exercises 19–24, find the standard form of the equation of each ellipse and give the location of its foci.

Key Concepts

Standard Form of an Ellipse

The standard form of the equation of an ellipse is given by (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center, a is the semi-major axis, and b is the semi-minor axis. This form helps identify the orientation of the ellipse (horizontal or vertical) and its dimensions based on the values of a and b.

Foci of an Ellipse

The foci of an ellipse are two fixed points located along the major axis, which are crucial for defining the shape of the ellipse. The distance from the center to each focus is denoted as c, where c² = a² - b². The foci are essential for understanding the properties of the ellipse, such as its eccentricity.

Vertices of an Ellipse

The vertices of an ellipse are the points where the ellipse intersects its major and minor axes. For a horizontally oriented ellipse, the vertices are located at (h ± a, k) and for a vertically oriented ellipse at (h, k ± b). Identifying the vertices is important for sketching the ellipse and understanding its dimensions.