1. Lesson 9.2 Ellipses

2. Ellipse
Set of all points where the sum of the distances to two fixed points (foci) is constant.
8cm
7cm
5cm
4cm
9cm
3cm
10cm
2cm
Focus
Focus

3. Other parts of an ellipse:
Major axis
Center
Minor axis
Vertices
Major & Minor axis can run in opposite direction:

4. a → distance from center to vertex on major axis
Deriving the Equation for an Ellipse
Equation comes from distances: b a c
a → distance from center to vertex on major axis
b → distance from center to vertex on minor axis
c → distance from center to a focus

5. Equation of an ellipse derives from distances:
Relation among a, b, & c:
Equation of an ellipse derives from distances: or

6. USING THE EQUATION
1) Center is at h and k
2) a is the larger denominator/ b is smaller (major/minor)
Use a, b, and/or c to find missing values using the equations
Use a, b, and c distances to find foci and vertices from center
If a is under x, the ellipse “goes” Left/Right
If a is under y, the ellipse “goes” Up/Down

7. Example Name the foci, center, length of major and minor axes.
Then sketch the ellipse.

8. e close to 1 → more elliptical
Eccentricity
where 0 < e < 1
small e → circular
e close to 1 → more elliptical
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
orbital eccentricity
0.2056
0.0068
0.0167
0.0934
0.0483
0.0560
0.0461
0.0097

9. Example Find the center, foci, vertices and eccentricity.
9x2 + 4y2 + 36x – 24y + 36 = 0

10. Example Write the equation for the ellipse with foci (0,0), (0,8); and major axis length of 16.