Exer. 1-14: Find the vertices and foci of the ellipse. Sketch its graph, showing the foci. ((x-3

Exer. 1-14: Find the vertices and foci of the ellipse....

Key Concepts

Graphical Representation of an Ellipse

Graphing an ellipse involves plotting its center, vertices, and foci, then sketching the smooth, rounded curve that connects them. This representation visually communicates the ellipse's symmetry, orientation, and spatial relationships among its key features.

Determining the Focal Distance (c)

The focal distance is calculated using the equation c = ?(a² - b²) for a horizontal ellipse (and similarly adjusted for a vertical ellipse). This value, when added to and subtracted from the center’s coordinate along the major axis, gives the locations of the foci. It plays a crucial role in distinguishing the shape of the ellipse.

Foci

The foci are two fixed points located along the major axis of the ellipse. The defining property of an ellipse is that the sum of the distances from any point on the ellipse to these foci is constant. The positions of the foci are determined by the focal distance, which is essential for understanding the ellipse’s geometric properties.

Vertices

Vertices are the points on the ellipse that are farthest from the center along the major axis. For an ellipse with a horizontal major axis, the vertices have coordinates (h ± a, k); for a vertical major axis, they are (h, k ± b). They describe the extreme boundaries of the ellipse along its longest direction.

Standard Ellipse Equation

The standard form of an ellipse is given by (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center of the ellipse, and a and b are the distances from the center to the vertices along the horizontal and vertical axes, respectively. This form is the basis for identifying the ellipse’s key attributes.