Need conics help (1 Viewer)

[globetrotter]

Hi everyone,

I am having trouble with this conics question:

Prove that the sum of the distances from any point P on the ellipse x^2/a^2 + y^2/b^2 = 1 to the two foci is equal to 2a. Do not use the paramatric form - use cartesian coordinates.

Could someone please help me?

Thank you!

 

[Trebla]

Consider the point P(x₁, y₁)
Let:
- S be the positive focus
- S' be the negative focus
- M be the point (a/e, y₁) [i.e. horizontal line passing through P and passes through M at the positive directrix)
- M' be the point (- a/e, y₁)

From the fundamental conic definition:
PS/PM = e
=> PS = e.PM (1)
Similarly,
PS' = e.PM' (2)

(1) + (2):
PS + PS' = e(PM + PM')

When you draw the diagram it should be clear that from PM + PM' = MM' since P is the common point of PM and PM'.
=> PS + PS' = e.MM'
But MM' is the distance between the two directrices which is 2a/e
=> PS + PS' = e.(2a/e)
.: PS + PS' = 2a

 

[nottellingu]

I dont get it

 

[kony]

Go ask your teacher at school.

Trebla's explanation is about as good as it's going to get without using a diagram.

The following might help though:

MM' is the distance between M and M', which is the same as the distance from M' to O added to the distance from O to M'. From the anatomy of the ellipse, we know that MM' is 2a/e.

Also, PS = ePM => This is your basic ellipse formula. "The distance from a point P to a focus S is a constant ratio, e, to the distance from the point P to a straight line (the directrix)."

It's basically an algebraic manipulation.

 

[globetrotter]

Consider the point P(x<SUB>1</SUB>, y<SUB>1</SUB>)
Let:
- S be the positive focus
- S' be the negative focus
- M be the point (a/e, y<SUB>1</SUB>) [i.e. horizontal line passing through P and passes through M at the positive directrix)
- M be the point (- a/e, y<SUB>1</SUB>)

From the fundamental conic definition:
PS/PM = e
=> PS = e.PM (1)
Similarly,
PS' = e.PM' (2)

(1) + (2):
PS + PS' = e(PM + PM')

When you draw the diagram it should be clear that from PM + PM' = MM' since P is the common point of PM and PM'.
=> PS + PS' = e.MM'
But MM' is the distance between the two directrices which is 2a/e
=> PS + PS' = e.(2a/e)
.: PS + PS' = 2a

Ohh I get it now. Thanks very much, Trebla!

 

[nottellingu]

Yepp got it ! I wouldnt hav thought of using the definition.