proof for an Ellipse (1 Viewer)

[sincred91]

I need to prove that the distance from any point on an ellipse, P, to the two foci, is equal to 2a? i know the distance to the directrix is constant but, im not sure how to do the rest..
help thanks?

 

[lychnobity]

let M & M' be the end of the perpendicular lines from P to the directrix

S & S' are the foci

PS/PM = e ie PS = ePM

Similarly, PS' = ePM'

PS + PS' = e(PM + PM')

= e(MM')
= e (2a/e) (ie distance from the 2 directrices, a/e + a/e)
= 2a

 

[alakazimmy]

Let P be an arbitrary point on the ellipse called (a cos @, b sin @)

Use the eccentricity rule, that: ^PS/_PM = e, where S is your positive focus, and M is the perpendicular from your positive directrix

Then, you have that PS = PM*e.

Now, you repeat for for S', which is the negative focus, and M' which is the perpendicular from the negative directrix.

So PS' = PM' * e

Then, just add PS with PS' = e(PM + PM') = 2a somehow

 

[lychnobity]

look at the drawing

 

[sincred91]

e(MM') That step how did you derive that from PS + PS' = e(PM + PM')?

 

[azureus88]

PS=ePM, thats just the definition of an ellipse, so you dont need to derive it.

 

[lychnobity]

e(MM') That step how did you derive that from PS + PS' = e(PM + PM')?

The distances PM & PM' form the straight line MM' (look at the diagram)