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Today in Precalculus

Today in Precalculus. Go over homework Notes: Ellipses Completing the square Applications Homework. Ellipses. Prove that the graph of x 2 + 4y 2 + 2x + 8y + 1 = 0 is an ellipse. Find the center, vertices and foci. Then graph the ellipse by hand.

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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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