Today in Precalculus

1
Today in Precalculus

• Notes Conic Sections - Ellipses
• Homework
• Go over quiz

2
Ellipses

• Definition An ellipse is the set of all points
  in a plane whose distances from two fixed points
  have a constant sum. The fixed points are the
  foci (F). The line through the foci is the focal
  axis. The point on the focal axis midway between
  the two foci is the center (C). The points on
  the ellipse that intersect with the focal axis
  are the vertices (V).

F2
V
F1
C
V
focal axis
3
Ellipses

• Standard form for the equation of an ellipse
  centered at the origin with the x-axis as its
  focal axis is
• There is a pythagorean relationship between a,b,
  and c
• c2 a2 b2 (Note the change in the sign
  between a and b)

(0,b)
F1(-c,0)
F2(c,0)
(-a,0)
(a,0)
C(0,0)
(0,-b)
4

• A line segment with endpoints on an ellipse is a
  chord of the ellipse.
• The chord lying on the focal axis is the major
  axis of the ellipse and has a length of 2a.
• The value for a is the semimajor axis.
• The chord through the center perpendicular to the
  focal axis is the minor axis of the ellipse and
  has a length of 2b.
• The value of b is the semiminor axis

5

• An ellipse centered at the origin with the y-axis
  as its focal axis has the form
• So when looking at the equation of an ellipse the
  variable with the larger denominator will be the
  focal axis.

6
Ellipses with center (0,0)
Standard Equation
Focal axis x-axis y-axis
Foci (c, 0) (0, c)
Vertices (a, 0) (0, a)
Semimajor axis a a
Semiminor axis b b
Pythagorean relation c2 a2 b2 c2 a2 b2
7
Example 1

• Find the vertices and foci of the ellipse 4x2
  8y2 64
• Vertices (-4, 0), (4, 0)
• c2 16 8 8
• c 2.8
• Foci (-2.8, 0), (2.8,0)

8
Example 2

• Find an equation of the ellipse with foci (0, -3)
  and (0,3) whose minor axis has length 4.
• From the foci c 3 and focal axis is y-axis
• 2b 4
• b 2
• 32 a2 22
• 9 a2 4
• 13 a2

9
Sketching an ellipse

• Vertices (0, -3.6), (0, 3.6) Points on minor
  axis (-2, 0), (2,0)

10
Graphing an ellipse

• Just as with the parabolas, must solve the
  equation for y

11
Homework

• Page 653 1, 2, 5, 6, 12, 14, 18, 22-30even