Deduccion de la ecuación de la Elipse

Deducción de d e l a ecuac ecuacii ón de la elipse eli pse con cent cent r o en en el ori or i gen gen y ej e mayor sobre obr e l as abscisas:

 PF   2a   L a suma suma de l as dis di st anci anci as ent r e l os focos focos y un u n punt pu nt o

PF

de l a el i pse es i gual a una cant cant i dad cons constt ant e  2a



d PF d PF  x

x

 c 2  y 2 

  x  c 2  y2  

 c 2  y2 

 x c 

2

y

 x   c  2

x 2

x

 2

2

 c  2  y 2  2a  



 2a 

x c

2



2

y   2

 y 2

 4a 2  4a  x  c  2  y 2   x  c  2  y 2  

 2cx  c 2

 4a 2  4a  x  c  2  y 2  x 2    2cx  c 2

4cx  4a2  4a  x  c  2  y 2  



cx  a 2

c2x 2



2 2

a b 





 a x  c2  y2  

2

 2a 2cx   a 4  a 2x 2  2a 2cx   a 2c2  a 2y 2  a 4  a 2c 2  a 2x 2  c 2x 2  a 2y 2  a2 a 2

a2b 2



2



x 2 b 2 2

2

a b 

 c2   x 2 a 2  c2   a 2y 2   

a 2 y 2

x2 1 2 2 2 a a  b 

y2  2 b



x2  2 a

y2    2 1 b  

Deducción de la ecuación de la elipse con centro en el origen y eje mayor en las ordenadas:

 PF   2a  

d PF

d PF

 x 2  y  c 2 

d PF  x2

 x 2  y  c 2 

 y  c 2  x 2  y  c2  2a  



x

2

 y  c 

2





2a  x

2



 y  c    2

2

 y  c 2  4a2  4a x 2  y  c 2  x 2   y  c 2

x 2 y 2

 2

 4a2  4a x 2  y  c 2  y 2   2yc  c 2

 2yc  c 2

4yc  4a2  4a x 2  y  c 2 

yc  a 

2 2





a x

2

 y  c    2

 2

 2a2yc  a 4  a2 x 2  y 2  2yc  c2   y 2c2  2a 2yc   a 4  a 2x 2  a 2y 2  2a 2yc    a 2c 2 4 2 2 2 2 2 2 2 2 a a c  a x a y y c   y 2c2



a2 a 2 a 2b 2 2 2

a b 



a 2 x 2 2

2

a  b 

 c2   a2x 2  y 2 a 2  c2  



y 2 b 2 2

2

a b 

 1

x2 b2

y2  2 a



x2 b2

y2    2 1 a  

Deducción de d e l a ecuac ecuacii ón de l a el el i pse con con ce cent nt r o en en ((h, h, k) k ) y eje mayor paral p arale el o a l as abscisas:

d PF

 d PF   2a  

d PF

 x  h  c   y  k 2  2

d PF 

 x  h  c   y  k 2  2

x  h  c   y  k  x  h   c   y  k 



2

2

 x  h  c   y  k 2  2a  

2

2

 x  h   c   y  k 2  2a  

x  h   c 

2

2

2

 y  k 2

  2

x  h   c   y  k    2

2



2a 

 c2 x  h 

2

c2 x  h 

2



 y  k 2 

 2

2





a

2

x  h   c   y  k   x  h     2c x  h   c   x  h   c   y  k    2

4a2  4a

4c x  h   4a2  4a 2

2

 4a2  4a x  h   c   y  k 2  x  h   c   y  k 2 

x  h 2  2c x  h   c2   c x  h   a2

x  h   c 

2

x  h   c 2  y  k 2 



 2a2c x  h   a 4  a 2 x  h   c   y  k 2  2

2

2

2

2

 2



 2a2c x  h    a4  a2 x  h 2  2a2c x  h    a2 y  k 2 a 4  a2c2

 a2 x  h 2  c2 x  h 2 a 2 y k 2 



 x  h 2 a2  c2  a 2 y  k 2 



a2 a2  c2 b2 = a2 - c2 

a 2b2 a2b 2 a2b 2

1

 x  h 2 b2 a 2 y k  2 2 2 x  h  b 2 a 2 y  k     2 2 2 2



a b  x  h 2 a2

a  b 

2 2 2 y  k x  h  y  k         1 2 2 2

b

a

b

 

Deducción de d e l a ecuac ecuacii ón de l a el el i pse con con ce cent nt r o en en ((h, h, k) k ) y eje mayor paral p arale el o a l as ordenadas:

d PF

 d PF   2a  

d PF

 x  h 2  y  k  c  

d PF 

2

 x  h 2  y  k  c  

2

x  h 2  y  k  c 2  x  h 2  y  k  c 2  2a

 

x  h 2  y  k   c 2  x  h 2  y  k   c 2  2a

 



x  h 2  y  k   c 2

  2



2 x  h 2  y  k   c  

y  k 2  2c y  k   c 2  

2a 



x  h 2  y  k   c 2 

2

4a 2  4a x  h   y  k   c   x  h     y  k   c  2

2

2

2

4a2  4a x  h   y  k   c   y  k     2c y  k   c 2   2

2

2

4c y  k   4a 2  4a x  h 2  y  k   c    2

c y  k   a 

2 2

c2 y  k 

2



 a x  h   y  k   c    2

2

 2

 2a 2c y  k    a 4  a2 x  h 2  a2 y  k 2  2a 2c y  k    a2c 2 a4



 a2c2  a2 x  h 2  a2 y  k 2  c2 y  k 2 

a2 a2  c2



a 2b2 a 2b 2 a 2b 2

1

 a2 x  h 2  y  k 2 a 2  c2    a2 x  h 2  y  k 2 b2   a 2 x  h 

2



a 2 b 2

2 y  k  b 2  

a2 b 2

2 2 x h y k     2 2

b

a

2 2 x  h  y  k       1 2 2

b

a