An Approach to Mathematic Functions Basics

by Frank C. Fung ( 1st published in December, 2014. )

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Section XIV - The Hyperbola via [ Focal-Point based Co-ordinates ] - Part II

Summary for the Section :

• In this Section ,
  • we shall again be deriving the equations for the Hyperbola using [ Focal-Point based Co-ordinates ] ,

  but here in this Section :

  • we shall be using [ Focal Point #2 ] as the base-point for our [ Focal-Point based Co-ordinate System ] .

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The [ Focal-Point based Co-ordinate System ] here in this Section :

• Here in this Section , we shall select [ Focal-Point #2 ] , i.e. :
  • the [ focal-point ] farthest away from the Vertex of the Hyperbola ,
    • as the [ base-point ] of our [ Focal-Point based Co-ordinate System ] .

  Then , for any point [ Point P ] on the Hyperbola :

  • its position can therefore be uniquely defined by the Position Vector [ Vector F2-P ] ;
    • as per the diagram on-the-left below .

  

  And the position-vector [ Vector F2-P ] can now be defined via the two (2) variables :

  • the distance [ R ] :
    • 

  • the angle [ PSI ] :
    • 

      as shown in the diagram on-the-left above .

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Preparing for the Derivation Process :

• We shall now mark-off the distance between [ Point F2 ] 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 :

  • 

  Therefore :

  • 

    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-write the expression for [ 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 .

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Developing the equation for the Hyperbola :

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

  

  And we see here that , for the point [ Point P ] on the Hyperbola :

  • 

    based on the right-angle-triangle [ Triangle F1-Q-P ] on the right-hand-side diagram above .

  And for [ L2 ] , we simply 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 equation in this special format :
  • 

  We then have :

  • 

    yielding :

  • 

  Therefore :

  • 

• Let us now re-write the above equation in this special format :
  • 

  And we bring-in , at this point , these 2 relations from above :

  • 

    and

  • 

  Consequently , we can now re-write the equation immediately preceding in this new format :

  • 

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Introducing the value [ Lamda ] :

• Let us now introduce the value [ Lamda ] such that :
  • 

  We can then re-write the expression above in this easier-to-read format :

  • 

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A Brief Comment of the value of [ Lamda ] here :

• Let us recall again 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 also establish this relation :
    • 

    consequently yielding :

    • 

  Thus , with [ Lamda ] here being :

  • 

    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 .

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go to the next section : Section XV - The Asymptotes for the Hyperbola - Part I

go to the last section : Section XIII - The Hyperbola via [ Focal-Point based Co-ordinates ] - Part I

return to the MATH Functions Basics HomePage

Original dated 2014-12-07