Hyperbola - Focal-Point based Co-ordinates ( I )

An Approach to Mathematic Functions Basics

Section XIII - The Hyperbola via [ Focal-Point based Co-ordinates ] - Part I

We shall now mark-off the distance between [ Point F1 ] and the [ Vertex ] as the distance [ R-sub-zero ] ,
- i.e. :
  as per the diagram on-the-right above .
And we can see from this diagram that :
Consequently , we have the relations :
- and
Let us recall , at this point , that we have already established this relation for the value of [ D ] :
- towards the end of Section VIII .
Thus , we can now re-express [ D ] in terms of [ R-sub-zero ] in this manner :
- based on the expression we have for [ RHO-sub-zero ] immediately above .
And with this expression in-hand , we are now ready to proceed with the derivation process ,
- to follow immediately below .

Let us now bring-in this set of 2 diagrams below :

And we see here that , for the point [ Point P ] on the Hyperbola :
And for right-angle-triangle [ Triangle F2-Q-P ] on the right-hand-side diagram , we have :
Then , for the original [ Constant-Difference Equation ] for the hyperbola , i.e. this equation :
we can now write :
And on re-arranging terms , we have :
Let us now square both sides of the equation to arrive at this :
Further expansion on the right-hand-side then yields us this :
And consequently :
Further cancelling terms then yields us this equation :
And on re-arranging terms , we have :
- yielding :
Let us now take a closer look at the terms within the 3 brackets in the above equation , and :
- we bring-in , from above , this expression for [ D ] in terms of [ L-sub-F ] and [ R-sub-zero ] :
Let us now substitute this into each of the 3 brackets , and :
- For the terms in the 1st bracket , we now have :
- For the terms in the 2nd bracket , we now have :
- For the terms in the 3rd bracket , we now have :
Thus , we can now re-write the original equation above in this manner :
And on re-arranging terms , we have :
- yielding :
Let us now re-write the above the above equation in this special format :
Therefore :
- yielding :
And consequently , we can now express [ R ] in terms of [ R-sub-zero ] via the following :

Let us now introduce the value [ Lamda ] such that :
We can then write the expression above in this easier-to-read format :

Let us recall , from above , that :
- for the [ R-sub-zero ] we have defined in this Section , we did establish this particular relation :
Thus , for the value of [ Lamda ] here being defined by :
- we can now re-write this as :
Let us also recall that :
- we did place a very special condition on the Fixed-Difference [ D ] back in Section VIII ,
  
  namely that :
And in that same Section VIII :
- we did establish this relation :
  consequently yielding :
Thus , with [ Lamda ] now being re-written as :
- the relevent range for our [ Lamda ] here is therefore :
Consequently , our equation expressing [ R ] in terms of [ R-sub-zero ] can now be re-written in this final format :
And this is an important equation that we shall be coming back to in later Sections .