Notes 8.2 Conics Sections – The Ellipse

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.

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.

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

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

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

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

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

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

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

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

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).

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

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.