Notes 8.2 Notes 8.2

Conics Sections –Conics Sections – The Ellipse The Ellipse

I. IntroductionI. Introduction

A.) A.) Def: The set of all points in a plane Def: The set of all points in a plane whose distances from two fixed points whose distances from two fixed points in the plane have a constant sum.in the plane have a constant sum.

1.) The fixed points are the 1.) The fixed points are the FOCIFOCI..2.) The line through the foci is the 2.) The line through the foci is the

FOCAL AXIS.FOCAL AXIS.3.) The 3.) The CENTER CENTER is ½ way is ½ way

between the foci and/or the vertices.between the foci and/or the vertices.

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

P(x, y)

FocusFocus

d1 d2

(x, y)

(F1, 0) (F2, 0)

C.) C.) Pictures – By Definition - Pictures – By Definition -

Minor Axis

Focus

Focus

Major Axis

a is the SEMI-

MAJOR axis

Center

(-a, 0) (a, 0)

(0, -b)

(0, b)

(-c, 0) (c, 0)(0, 0)

b is the SEMI-

MINOR axis

Pictures -Expanded- Pictures -Expanded-

VertexVertex

2 2 2 2

2 2 2 21 or 1

x y y x

a b a b

Where b2 + c2 = a2.

D.) D.) Standard Form Equation - Standard Form Equation -

a

0, cy axisx axis

a

2 2

2 21

y x

a b

b

,0c

2 2

2 21

x y

a b St. Fm.

Focal axis

Foci

Semi-Major

Semi-Minor

Pyth. Rel. 2 2 2a b c

b2 2 2a b c

E.) ELLIPSES - E.) ELLIPSES - Center at (0,0) Center at (0,0)

a

,h k cx hy k

a

2 2

2 21

y k x h

a b

b

,h c k

2 2

2 21

x h y k

a b

St. fm.

Focal axis

Foci

Semi-Major

Semi-Minor

Pyth. Rel. 2 2 2a b c

b2 2 2a b c

F.) ELLIPSES - F.) ELLIPSES - Center at (Center at (hh, , kk) )

A. ) Ex. 1- Find the vertices and foci of the following ellipse:

II. ExamplesII. Examples

2c

27 5 c

2 2 2a b c

2 2

17 5

x y

2 25 7 35x y

Foci =

Vertices = 7,0 and 7,0

2,0 and 2,0

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

2 2

125 9

x y

5a

2 2 23 4a

4, 3c b

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

foci : 3 7,5

center : 3,5

7c

216 9 c

2 23 5

116 9

x y

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).

216 4b

4

2

a

c

center 2,3 foci 2,1 & 2,5

vertices. 2, 1 & 2,7

2 23 2

116 12

y x

III. EccentricityIII. Eccentricity

A.)A.)

B.) What it tells us – B.) What it tells us –

1.) 1.) ee close to 0 close to 0 foci close to center foci close to center

2.) e2.) e close to 1 close to 1 foci close to vertices foci close to vertices

0 1e c

ea

IV. Ellipsoids of RevolutionIV. 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.