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 an ellipse's equation 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 and its dimensions, which are crucial for graphing and understanding its properties.

Foci of an Ellipse

The foci of an ellipse are two fixed points located along the major axis, equidistant from the center. The distance from the center to each focus is denoted as 'c', where c² = a² - b². The foci play a significant role in defining the ellipse's shape and are essential for applications in physics and engineering.

Graphing Ellipses

Graphing an ellipse involves plotting its center, determining the lengths of the semi-major and semi-minor axes, and marking the foci. Understanding the coordinate plane and how to interpret the graph is vital for visualizing the ellipse's properties and solving related problems effectively.