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Presentation transcript:

Concept

Write an equation for the ellipse. Write an Equation Given Vertices and Foci Write an equation for the ellipse. Step 1 Find the center. The foci are equidistant from the center. The center is at (0, 0). Step 2 Find the value of a. We know that the length of the major axis of any ellipse is 2a units. In this ellipse, the length of the major axis is the distance between the points at (0, 5) and (0, –5). This distance is 10 units. Example 1

2a = 10 Length of major axis = 10 Write an Equation Given Vertices and Foci 2a = 10 Length of major axis = 10 a = 5 Divide each side by 2. Step 3 Find the value of b. The foci are located at (0, 4) and (0, –4), so c = 4. We can use the relationship between a, b, and c to determine the value of b. c2 = a2 – b2 Equation relating a, b, and c 16 = 25 – b2 c = 4 and a = 5 b2 = 9 Solve for b2. Example 1

Answer: An equation of the ellipse is . Write an Equation Given Vertices and Foci Step 4 Write the equation. Since the major axis is vertical, substitute 25 for a2 and 9 for b2 in the form . Answer: An equation of the ellipse is . Example 1

Concept

The center of the ellipse is at Write an Equation Given the Lengths of the Axes Write an equation for the ellipse with vertices at (–6, –2) and (4, –2) and co-vertices at (–1, –4) and (–1, 0). The y-coordinate is the same for both vertices, so the ellipse is horizontal. The center of the ellipse is at The length of the major axis is 4 – (–6) or 10 units, so a = 5. Example 2

The length of the minor axis is 0 – (–4) or 4 units, so b = 2. Write an Equation Given the Lengths of the Axes The length of the minor axis is 0 – (–4) or 4 units, so b = 2. The equation for the ellipse is . a2 = 25, b2 = 4 Answer: Example 2

Write an Equation for an Ellipse SOUND A listener is standing in an elliptical room 150 feet wide and 320 feet long. When a speaker stands at one focus and whispers, the best place for the listener to stand is at the other focus. Write an equation to model this ellipse, assuming the major axis is horizontal and the center is at the origin. Understand We need to determine an equation representing the elliptical room. Example 3

Solve Find the value of a. 2a = 320 Length of major axis = 320 Write an Equation for an Ellipse Plan The length of the major axis is 320 feet. The length of the minor axis is 150 feet. Use this information to determine the values of a and b. Solve Find the value of a. 2a = 320 Length of major axis = 320 a = 160 Divide each side by 2. Example 3

2b = 150 Length of minor axis = 150 b = 75 Divide each side by 2. Write an Equation for an Ellipse Find the value of b. 2b = 150 Length of minor axis = 150 b = 75 Divide each side by 2. Substitute a = 160 and b = 75 into the form . Answer: An equation for the ellipse is Example 3