1. Notes 8.2 Conics Sections – The Ellipse

2. I. Introduction
A.) Def: The set of all points in a plane whose distances from two fixed points in the plane have a constant sum.
1.) The fixed points are the FOCI.
2.) The line through the foci is the FOCAL AXIS.
3.) The CENTER is ½ way between the foci and/or the vertices.

3. B.) Forming an Ellipse - When a plane intersects a double-napped cone and is neither parallel nor perpendicular to the base of the cone, an ellipse is formed.

4. C.) Pictures – By Definition -
P(x, y)
Focus
Focus
(x, y) d1 d2
(F1, 0)
(F2, 0)

5. Pictures -Expanded- Minor Axis Center Focus Focus Major Axis Vertex
(0, b)
Focus
Major Axis
Vertex
Vertex
(-a, 0)
(-c, 0)
(0, 0)
(c, 0)
(a, 0)
(0, -b)
b is the SEMI-
MINOR axis
a is the SEMI-
MAJOR axis

6. D.) Standard Form Equation -
Where b2 + c2 = a2.

7. E.) ELLIPSES - Center at (0,0)
St. Fm.
Focal axis
Foci
Semi-Major
Semi-Minor
Pyth. Rel.

8. F.) ELLIPSES - Center at (h, k)
St. fm.
Focal axis
Foci
Semi-Major
Semi-Minor
Pyth. Rel.

9. II. Examples
A. ) Ex. 1- Find the vertices and foci of the following ellipse:
Vertices =
Foci =

10. B.) Ex. 2- Find a equation of the ellipse with foci (4,0) and (-4,0) whose minor axis has a length of 6.

11. C.) Ex. 3- Find the center, foci, and vertices of the following ellipse:

12. D.) Ex. 2- Find the equation of an ellipse with foci (-2, 1) and (-2, 5) and major-axis endpoints (-2, -1) and (-2,7).

13. III. Eccentricity A.) B.) What it tells us –
1.) e close to 0  foci close to center
2.) e close to 1  foci close to vertices

14. IV. Ellipsoids of Revolution
A.) Rotate ellipse about its focal axis to get an ellipsoid of revolution
B.) Examples of these include whispering galleries and a lithotripter, a device which uses shockwaves to destroy kidney stones.