Conics- PS + PS' =2a ...The proof? (1 Viewer)

[Sparcod]

Can someone proof for me that the sum of the distances between a variable and both foci is a constant?

 

[Mill]

I'll assume you are talking about for an ellipse.

|PS| + |PS'| = e|PD| + e|PD'| = e(a/e - x) + e(a/e + x) = 2a

And you're done.

We can say that |PS| + |PS'| = e|PD| + e|PD'| due to the focus-directrix definition of an ellipse, where D is the point on the directrix.


It can also be proved algebraically but it really kind of blows since you have to wade through a lot of algebra (use of distance formula).

 

[exquisite]

PS + PS' = e(PM + PM')
= e(MM')
= e(2a/e)
= 2a

nb; where M is a point on the directrix

 

[Trebla]

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

This statement comes from the definition:
PS/PM = e
.: PS = ePM
Similarly
PS' = ePM'
.: PS + PS' = e(PM + PM')
then work your way through.....

 

[c0okies]

just in case.. M is a point on the directrix and PM is the line perpendicular to the directrix

 

[Sparcod]

Can I possibly use two distance formulae (PS + PS') and add them up?

 

[Mountain.Dew]

Can I possibly use two distance formulae (PS + PS') and add them up?

you could, but its gonna be a lot A LOT OF WORK. plus, i think you need to use parameters to do this as well.

i strongly advise using the distance formula, and go with the easy PS + PS' = ePD + ePD'

 

[Sparcod]

you could, but its gonna be a lot A LOT OF WORK. plus, i think you need to use parameters to do this as well.

i strongly advise using the distance formula, and go with the easy PS + PS' = ePD + ePD'

Which parameters?

 

[Mill]

It can also be proved algebraically but it really kind of blows since you have to wade through a lot of algebra (use of distance formula).

Can I possibly use two distance formulae (PS + PS') and add them up?

:jaw: :jaw: :jaw: :jaw: :jaw: :jaw: :jaw: :jaw: :jaw: :jaw: :jaw: :jaw: :jaw: :jaw: :jaw: :jaw: :jaw: :jaw: :jaw: :jaw:

 

[Rax]

Ok I have the proof.
Its not in my textbook
We did it in class......and I can tell you It took over 1 and a half A4 pages of pure working to get the answer down to 2a
its a fucking bitch.......you really want it?

 

[Mountain.Dew]

Which parameters?

the parameters for an ellipse is:

x = acos@, y =bsin@, @ denotes angle.

 

[Sparcod]

PS + PS' = e(PM + PM')
= e(MM')
= e(2a/e)
= 2a

nb; where M is a point on the directrix

Woh! I thought that PS/PM=e
That means: PS + PS'/ PM + PM' =2e
PS +PS' =2e (PM + PM')


Please explain?

 

[_ShiFTy_]

Woh! I thought that PS/PM=e
That means: PS + PS'/ PM + PM' =2e
PS +PS' =2e (PM + PM')


Please explain?

Umm...you are wrong
PS/PM = e
PS = ePM

PS'/PM' = e
PS' = ePM'

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

 

[hyparzero]

I'll assume you are talking about for an ellipse.

|PS| + |PS'| = e|PD| + e|PD'| = e(a/e - x) + e(a/e + x) = 2a

And you're done.

We can say that |PS| + |PS'| = e|PD| + e|PD'| due to the focus-directrix definition of an ellipse, where D is the point on the directrix.


It can also be proved algebraically but it really kind of blows since you have to wade through a lot of algebra (use of distance formula).

Proving the sum of the lengths from a variable point P to the foci equals 2a is the classic definition of an ellipse. The modern definition is where the eccentricity comes into play, which makes everything nice and easy.

But to prove PS + PS' = 2a, you cannot utilise the modern definition of an ellipse (you cannot use eccentricity), only pure algebra or otherwise.

Btw, the proof isn't that long, its in one of my exercise books, i'll type it up if needed.

 

[Sparcod]

Ok I have the proof.
Its not in my textbook
We did it in class......and I can tell you It took over 1 and a half A4 pages of pure working to get the answer down to 2a
its a fucking bitch.......you really want it?

We've shown you an easier way.

 

[Rax]

LOL I know
We just did it in class, our teacher thought it would be funny
there were a few bits where the lines of working were like 1 and a half A4 style
and then it all cancels down
Quite humerous really
GG

 

[Sparcod]

LOL I know
We just did it in class, our teacher thought it would be funny
there were a few bits where the lines of working were like 1 and a half A4 style
and then it all cancels down
Quite humerous really
GG

It is funny.
I'm not too sure how that ended up to be 1 and a half pages.