1. Ellipse
Section 7.3

2. Definitions
The ellipse with foci F1 and F2 is the set of all point M for which the sum of the distances to the foci is constant.
There are two cases according to the major axis containing the foci.

3. What does it look like? B2(0,b) A2(0,a) M M F2(c,0) F1(-c,0) A2(a,0)

4. Observations For any point M on the ellipse we have:
d(M,F1) + d(M,F2) = 2a
The four vertices of the ellipse are A1, A2, B1 and B2
The major axis is the line segment A1A2 with length 2a
The foci F1 and F2 are located on the major axis and the focal distance between them is 2c

5. Observations
The minor axis is the line segment B1B2 with length 2b
The axes of the ellipse are axes of symmetry
The center O of the ellipse is the center of symmetry
The parameters a, b and c form a Pythagoras Triangle
a2 = b2 + c2 B2 a b
NOTICE a is the hypotenuse!! c F2

6. How to Draw an Ellipse B2(0,3) F2(4,0) F1(-4,0) A2(5,0)
Find a (bigger number)
a = √25 = 5
Find b
b = √9 = 3
Find c
a2 = b2 + c2  a2 – b2 = c2
c2 = 25-9
c2 = 16
c = √16 = 4
B1(0,-3)
Major Axis?
x-axis
Find vertices & Foci
A1(-5,0), A2(0,-5)
B1(0,-3), B2(0,3)
F1(-4,0), F2(4,0)

7. How to Find the Rule 8 Major Axis? y-axis What do we have? a=8 and b=4
Find a2 and b2
a2=82=64
a2=42=16
Write Rule
(we do not need c to write the rule!) 4

8. Example Major Axis? x-axis 2 What do we have? c=3 and b=2
Find a2 and b2
a2 = b2 + c2
a2 =
a2 = 13
b2 = 4
Write Rule 2
F2(3,0)

9. Homework
Workbook
p.332 #1,3,4
p. 333 #5,6,7