Conic help, please (1 Viewer)

[wantingtoknow]

Help me with hyperbolas, please.

This question is from Terry Lee's book, page 118. Question is: For a point P on the hyperbola x^2/a^2 - y^2/b^2 = 1, using the definiation of the hyperbola, show that |SP - S'P| = 2a, where S and S' are the foci.

I looked in the worked solutions and they went from |ePM - ePM'| to eMM'. Why? How?

Conics* ._.'

 

[leoyh]

you use the definition of a hyperbola, which is PS/PM = e. Rearrange that and you get PS = ePM

Similarly, PS'/PM' = e, PS' = ePM'

So PS - PS' = ePM - ePM'
= e (PM - PM')
= e (MM')
= e (2a/e)
= 2a

Drawing a diagram helps

 

[wantingtoknow]

Lol, that's exactly what the worked solution said but how do you get from |ePM - ePM'| to eMM'?

 

[leoyh]

from e (PM - PM'), it becomes e (MM'), look at your diagram or the diagram in the book if theres one and you should see that PM - PM' = MM'

 

[wantingtoknow]

The book doesn't have a diagram for hyperbola. I tried drawing it just then but still don't understand. Is there any working out you can show or is drawing a diagram the only way?

 

[wantingtoknow]

Never mind, I got it. Thanks ;D

 

[kevda1st]

how come MM' turns into 2a/e ???

 

[shaon0]

how come MM' turns into 2a/e ???

distance between directrices