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

(F , 0) 1 (F2, 0) Pictures -Expanded- Minor Axis

Center Focus (0, b) Focus Major Axis

Vertex (-a, 0)(-c, 0)(0, 0) (c, 0) (a, 0) Vertex

(0, -b) a is the SEMI- b is the SEMI- MAJOR axis MINOR axis D.) Standard Form Equation -

xy22 yx 22 1 or  1 ab22 ab 22

Where b2 + c2 = a2. E.) ELLIPSES - Center at (0,0)

xy22 yx22  1 St. Fm. 22 1 22 ab ab Focal axis x  axis yaxis Foci c,0 0, c Semi-Major a a

Semi-Minor b b Pyth. Rel. abc222  abc222  F.) ELLIPSES - Center at (h, k)

22 22 xh yk yk xh St. fm.  1 22 1 ab22 ab Focal axis y  k xh Foci hck ,  hk,  c Semi-Major a a

Semi-Minor b b Pyth. Rel. abc222  abc222  II. Examples

A. ) Ex. 1- Find the vertices and foci of the following ellipse: 75  c2 57xy22 35 c  2

22 xy Vertices = 7,0 and  7,0 1 75 Foci =  2,0 and  2,0 abc222 B.) Ex. 2- Find a equation of the ellipse with foci (4,0) and (-4,0) whose minor axis has a length of 6. cb4, 3

222 xy22 a 34  1 25 9 a  5 C.) Ex. 3- Find the center, foci, and vertices of the following ellipse:

22 xy35 1 16 9 center : 3,5

2 16 9 c foci : 3 7,5 c  7 vertices : 7,5 & 1,5 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). vertices.  2,  1 &  2,7 foci  2,1 &  2,5 center 2,3 a  4 22 yx32 c  2 1 16b2 4 16 12 III. Eccentricity c A.) e  01 e  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.