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Today in Precalculus

Explore the characteristics of ellipses, from graphing by hand to finding the center, vertices, and foci. Practice proving equations of ellipses and solving related problems. Complete precalculus homework on ellipses. Study notes on ellipses, completing the square, and practical applications.

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Today in Precalculus

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  1. Today in Precalculus • Go over homework • Notes: Ellipses Completing the square Applications • Homework

  2. Ellipses Prove that the graph of x2 + 4y2 + 2x + 8y + 1 = 0 is an ellipse. Find the center, vertices and foci. Then graph the ellipse by hand. x2 + 2x + 4y2 + 8y = -1 x2 + 2x + 4(y2 + 2y) = -1 x2 + 2x + 1 + 4(y2 + 2y + 1) = -1 + 1 + 4 (x + 1)2 + 4(y + 1)2 = 4

  3. Ellipses Find the center, vertices and foci. Then graph the ellipse by hand. Center: (-1, -1) a = ±2 Vertices: (-3, -1), (1, -1) c2 = 4 – 1 = 3 c = ±1.7 Foci: (-2.7, -1), (0.7, -1)

  4. Ellipses Prove that the graph of 9x2 + 4y2 + 18x – 16y – 11 = 0 is an ellipse. Find the center, vertices and foci. Then graph the ellipse by hand. 9x2 + 18x + 4y2 – 16y = 11 9(x2 + 2x) + 4(y2 – 4y) = 11 9(x2 + 2x + 1) + 4(y2 – 4y + 4) = 11 + 9 + 16 9(x + 1)2 + 4(y – 2)2 = 36

  5. Ellipses Find the center, vertices and foci. Then graph the ellipse by hand. Center: (-1, 2) a = ± 3 Vertices: (-1, -1), (-1, 5) c2 = 9 – 4 = 5 c = ±2.2 Foci: (-1, -0.2), (-1, 4.2)

  6. Orbit and Eccentricity Where a is the semimajor axis and b is the semiminor axis Find the eccentricity for examples 1 & 2. Example 1: Example 2:

  7. Applications - example The ellipse used to generate the ellipsoid of a Lithotripter has a major axis of 12 ft. and a minor axis of 5 ft. How far apart are the foci? 2a = 12 a = 6 2b = 5 b = 2.5 c2 = 62 – 2.52=29.75 c = ±5.454 So the foci are 2(5.454) or 10.908ft apart.

  8. Homework Page 654: 45-48, 59, 60

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