Today in Precalculus

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 2.5229.75
• c 5.454
• So the foci are 2(5.454) or 10.908ft apart.

8
Homework

• Page 654 45-48, 59, 60