LastBivin
Updated:
April 11, 2006
Jeff
-- LZHS
Ellipse
The set of all co-planar points whose sum
of the distances from two fixed points (foci)
are constant.
Jeff Bivin -- LZHS
Ellipse
a2 = b2 + c2
Jeff Bivin -- LZHS
Ellipse
The set of all co-planar points whose sum
of the distances from two fixed points (foci)
are constant.
Jeff Bivin -- LZHS
Ellipse
The set of all co-planar points whose sum
of the distances from two fixed points (foci)
are constant.
Jeff Bivin -- LZHS
Ellipse
The set of all co-planar points whose sum
of the distances from two fixed points (foci)
are constant.
Jeff Bivin -- LZHS
Ellipse
The set of all co-planar points whose sum
of the distances from two fixed points (foci)
are constant.
Jeff Bivin -- LZHS
Ellipse
The set of all co-planar points whose sum
of the distances from two fixed points (foci)
are constant.
Jeff Bivin -- LZHS
Ellipse
The set of all co-planar points whose sum
of the distances from two fixed points (foci)
are constant.
Jeff Bivin -- LZHS
Ellipse
The set of all co-planar points whose sum
of the distances from two fixed points (foci)
are constant.
Sum of the distances from any point on an
ellipse to the two foci is 2a
Jeff Bivin -- LZHS
Ellipse
The set of all co-planar points whose sum
of the distances from two fixed points (foci)
are constant.
Jeff Bivin -- LZHS
Ellipse
The set of all co-planar points whose sum
of the distances from two fixed points (foci)
are constant.
Jeff Bivin -- LZHS
Ellipse
The set of all co-planar points whose sum
of the distances from two fixed points (foci)
are constant.
Jeff Bivin -- LZHS
Ellipse
P
F1 (-c, 0)
(0, 0)
(x, y)
(c, 0) F2
PF1 + PF2 = 2a
[( x  (c)]2  ( y  o) 2 
( x  c) 2  y 2 
Jeff Bivin -- LZHS
( x  c ) 2  ( y  o) 2  2a
( x  c ) 2  y 2  2a
( x  c) 2  y 2 
( x  c)  y
2
 ( x  c)
2
 y
2
2
( x  c ) 2  y 2  2a
 2a  ( x  c )  y
2
  2a 
2
2
( x  c)  y
2
2

2
( x  c ) 2  y 2  4a 2  4a ( x  c ) 2  y 2  ( x  c ) 2  y 2
x  2 xc  c  4a  4a ( x  c)  y  x  2 xc  c
2
2
2
2
2
 4a  4 xc   4a ( x  c)  y
4
2
a 2  xc 
Jeff Bivin -- LZHS
2
a ( x  c) 2  y 2
2
2
2
a 2  xc 
a
2

 xc
2
a ( x  c) 2  y 2

 a ( x  c)  y
2
2

2

a 4  2a 2 xc  x 2c 2  a 2 ( x  c) 2  y 2


a 4  2a 2 xc  x 2c 2  a 2 x 2  2 xc  c 2  y 2

a 4  2a 2 xc  x 2c 2  a 2 x 2  2a 2 xc  a 2c 2  a 2 y 2
a 4  x 2c 2  a 2 x 2  a 2c 2  a 2 y 2
Jeff Bivin -- LZHS
a 4  x 2c 2  a 2 x 2  a 2c 2  a 2 y 2
remember that a 2  b 2  c 2  c 2  a 2  b 2




a 4  x2 a 2  b2  a 2 x 2  a 2 a 2  b2  a 2 y 2
a 4  a 2 x 2  b 2 x 2  a 2 x 2  a 4  a 2b 2  a 2 y 2
 b 2 x 2   a 2b 2  a 2 y 2
 b 2 x 2  a 2 y 2   a 2b 2
 a 2b 2
b x
a y
a b


2 2
2 2
a b a b
 a 2b 2
x2
y2
 2  1
2
a
b
2
Jeff Bivin -- LZHS
2
2
2
2 2
2
2
x
y
 2  1
2
a
b
this has a center at (0, 0)
if the center is at (h, k )
x  h 2
a
Jeff Bivin -- LZHS
2

 y  k 2
b
2
 1
Ellipse
The set of all co-planar points whose sum
of the distances from two fixed points (foci)
are constant.
Jeff Bivin -- LZHS
Ellipse
The set of all co-planar points whose sum
of the distances from two fixed points (foci)
are constant.
Jeff Bivin -- LZHS
Ellipse
a2 = b2 + c2
Jeff Bivin -- LZHS