Open the “Ellipse” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Interactive Figures) or at bit.ly/3raFUGB.

(a) Uncheck the box “Equation of Ellipse.” Grab the point F₁, which represents a focus of an ellipse, and move the point F 1 to (−2, 2). Grab the point F₂, which represents the other focus of an ellipse, and move the point F₂ to (6, 4). What is the center of the ellipse? Write your answer as an ordered pair. Confirm the center of the ellipse is at the midpoint of the two foci.

(b) Click “Show Trace.” Grab Point A and move it around the Cartesian plane. Complete a trace of the entire ellipse. What is the sum of the distances from the foci to any point on the ellipse?

(c) Hit the reset button on the interactive figure(upper-right corner). Uncheck the box “Equation of Ellipse.” Move the point F₁ to (−4, 0) and move the point F₂ to (4, 0). Click “Show Ellipse.” The value of a is defined as the distance from the center of the ellipse to either vertex (V₁ or V₂). What is the value of a ?

(d) From the question in part (c), notice that the sum of the distances from the foci to any point on the ellipse is 2 a. What is the value of c , the distance from the center to either focus?

(e) The major axis is the line drawn through the foci of the ellipse. We define b as the distance from the center to points on the ellipse directly above and below the center (when the major axis is parallel to, or on, the x -axis). What is the value of b?

(f) H ow are a, b, and c related to each other?

(g) Click “Equation of Ellipse.” Notice the value of a² is the denominator of the x² -term and the value of b² is the denominator of the y² -term. Now move F₁ to (0, 3) and F₂ to (0, −3). What is the value of a ? What is the value of c ? What is the value of b ?

(h) What is the equation of the ellipse drawn in part (g)? What do you notice about the denominators of the x² -term and y² -term?

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