In Exercises 37-50, graph each ellipse and give the location of its foci. (x+3)^2+4(y-2)^2=16

Name: Problem 39, Chapter 10: Algebra and Trigonometry
Uploaded: 2020-06-08T04:47:26-08:00
Duration: 1 min 57 s
Channel: Ankit Gupta
Description: In Exercises 37-50, graph each ellipse and give the location of its foci. (x+3)^2+4(y-2)^2=16

step 1 Here we have given that x plus 3 whole square plus 4 y minus 2 whole square is equals to 16. Hen

In Exercises 37-50, graph each ellipse and give the location...

Key Concepts

Center and Axes Identification

Identifying the center and the lengths of the axes is crucial for graphing an ellipse. The center (h, k) indicates the midpoint of the ellipse, while the values of a^2 and b^2 (obtained from the denominators) reveal the distances from the center to the vertices along the major and minor axes. This information is used to determine the overall shape and placement of the ellipse on the coordinate plane.

Graphing an Ellipse

Graphing an ellipse involves plotting its center, marking the vertices along the major axis at distances of a units from the center, and marking the co-vertices along the minor axis at distances of b units. Connecting these points with a smooth, oval curve results in the complete graph of the ellipse. Grid sketching or a step-by-step plotting method is commonly used to accurately represent the ellipse.

Foci Determination in an Ellipse

The foci of an ellipse are two fixed points along the major axis, and their positions are determined using the relationship c^2 = a^2 - b^2, where c is the distance from the center to each focus. Once the value of c is computed, the foci are located at (h ± c, k) for a horizontal major axis or (h, k ± c) for a vertical major axis. These foci are essential in defining the ellipse as the set of points for which the sum of distances to the foci is constant.

Standard Form of an Ellipse

The standard form of an ellipse is expressed as (x - h)^2/a^2 + (y - k)^2/b^2 = 1, where (h, k) represents the center of the ellipse and a and b are the lengths of the semi-major and semi-minor axes, respectively. This form facilitates identifying key features of the ellipse, such as its center, vertices, and orientation along the coordinate axes.

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