Ellipses

Elliptical billiards.

Reflective properties of ellipse.

Ellipse: The set of all points in a plane, such that the sum of the distances from two fixed points, called the foci, to a point on the ellipse, is constant.

Ellipse construction:http://tube.geogebra.org/material/show/id/1225391

Demonstration with geogebra

Major axis

min

or

axis

“Watermelon”

Center (h,k) (h,k)

Foci

Length of major axis

2a 2a

Length of minor axis

2b 2b

f f

f

f

a

c

b

a

2 2

2 2

( ) ( )1

x h y k

a b

2 2 2 2

2 2 2 2

( ) ( ) ( ) ( )1 or 1

x h y k y k x h

b a a b

“Larry”

( , )h c k ( , )h k c

2 2( 2) ( 3)1

25 16

x y

GraphLabel “corners”

Identify:CenterFociLength of major axisLength of minor axis

GraphLabel “corners”

Identify:Center (2,-3)Foci (5,-3)(-1,-3)Length of major axis 10 unitsLength of minor axis 8 units

Write the equation for the ellipse in standard form [(h,k) form.]

2 22 8 12 8 0x y x y

2 2 2 2( 2) ( 6) ( 6) ( 2)1 or 1

18 36 36 18

x y y x

2 2

2 2

2 2

1 2 2

18

36 364 4

2 8 12 8

( 4 ) 12 8

2( 2) ( 6) 36

2 ( 2) ( 6) 36

36

2

3

2

36 6

x x y y

x x y y

x y

x y

LarryCenter: ( 2, 6), M =12 units, m = 6 2units, foci ( 2,6 3 2)