An Approach to Mathematic Functions Basics

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

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Section XI - Equation of the Hyperbola in [ U-V Axis Co-ordinates ]

Summary for the Section :

• In this Section , we shall derive the equation for the Hyperbola in [ U-V Axis Co-ordinates ] ,
  • i.e. , this equation :
    • 

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Preparing for the Derivation Process :

• We take special note at this point that :
  • for the Hyperbola we have developed in the previous Section ,
    • the relevent range of [ u ] is :
      • 

  We shall now split this { relevent range of [ u ] } into two (2) sub-regions :

  • [ Region One ] being the region where :
    • 

  • [ Region Two ] being the region where :
    • 

    as per this set of 2 diagrams below .

  

  This is a necessary procedure here , because :

  • the set-up for [ L1 ] and [ L2 ] are slightly different for the 2 sub-regions .

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Calculating [ L1 ] and [ L2 ] in [ Region One ] :

• Let us now take a quick look at the set-up for [ L1 ] and [ L2 ] in [ Region One ] ,
  • and we identify the point [ Point P ] as any point on the Hyperbola within this sub-region ,
    • as per the diagram on-the-left below .

  

  We then have , for the diagram on-the-right above :

  • for [ Right-Angle Triangle F1-Q-P ] :
    • 

  • for [ Right-Angle Triangle F2-Q-P ] :
    • 

• Let us quickly double-check the validity of the 2 equations here for both the [ Vertex ] and for [ u = L-sub-F ] .
  • For the [ Vertex ] , whose [ U-V co-ordinates ] are :
    • 

    the above 2 equations then become :

    • 

      and

    • 

    And these check-out O.K. , as per the diagram on-the-left below .

  

• Let us also quickly double-check the validity of the 2 equations at the [ cross-over point ] from [ Region I ] into [ Region II ] :
  • For [ u = L-sub-F ] , i.e. :
    • 

    the above 2 equations then become :

    • 

      and

    • 

    And these also check-out O.K. , as per the diagram on-the-right above .

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Calculating [ L1 ] and [ L2 ] in [ Region Two ] :

• Let us now take a quick look at the set-up for [ L1 ] and [ L2 ] in [ Region Two ] ,
  • and we identify the point [ Point P ] as any point on the Hyperbola within this sub-region .

  

  We then have :

  • for [ Right-Angle Triangle F1-Q-P ] in the diagram on-the-left above :
    • 

  • for [ Right-Angle Triangle F2-Q-P ] in the diagram on-the-right above :
    • 

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Consolidating [ Region One ] and [ Region Two ] :

• We see here that the equations for [ L2 ] are the same for both regions here ,
  • i.e. , this equation :
    • 

      is in fact common to both .

  But the equations for [ L1 ] are slightly different ;

  • for [ Region One ] :
    • 

  • for [ Region Two ] :
    • 

  However , these 2 equations immediately above are different in format only ,

  • and are in-essence the same equation .

• Thus , we can use either of these equations in our development process to follow ,
  • and we shall now elect to use the first equation here from-here-on-in .

  As such we can now write :

  • 

    yielding :

  • 

  And this shall be the core equation we shall be using for our development process , next .

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Developing the equation for the Hyperbola :

• We now bring back the core equation from immediately above :
  • 

  Re-arranging terms then yields :

  • 

• Let us now square both sides of the equation to arrive at this :
  • 

  And on re-arranging and cancelling terms , we have :

  • 

  We take note , at this point , that on algebriac expansion , the right-hand-side would become :

  • 

  Therefore , we can now write :

  • 

  And on re-arranging terms , we have :

  • 

  Dividing throughout by [ 2*D ] then yields us this :

  • 

• Let us now square both sides of the equation to arrive at this :
  • 

  And on further expansion , we have :

  • 

  Cancelling terms then yields us this :

  • 

  And on re-arranging , we have :

  • 

• Let us recall , at this point , that we have already established this relation :
  • 

    towards the end of the last section , Section VIII .

  Therefore :

  • 

  Substituting this into the equation immediately preceding then yields us this :

  • 

  And on consolidation , we have :

  • 

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

  And on further munipulations , we have :

  • 

    yielding :

  • 

  Therefore :

  • 

    yielding :

  • 

  Dividing throughout by { [ RHO-sub-zero ] square } then yield us this :

  • 

• Let us now introduce the quantity [ BETA ] such that :
  • 

  or ,

  • 

  We can then re-write the equation immediately preceding in this format :

  • 

  Consequently :

  • 

  yielding :

  • 

• And the final equation for the Hyperbola in [ U-V Axis Co-ordinates ] is then :
  • 

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go to the next section : Section XII - Equation of the Hyperbola in [ Polar Co-ordinates ]

go to the last section : Section X - The Geometric Definition of the Hyperbola

return to the MATH Functions Basics HomePage

Original dated 2014-12-07