An Approach to Mathematic Functions Basics

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

Top Of Page

Section XV - The Asymptotes for the Hyperbola - Part I

Summary for the Section :

• In this Section , we shall derive the Asymptotes for the Hyperbola ,
  • using the conventional approach .

go to Top Of Page

Preparing for the Derivation Process :

• Let us bring-in , from Section IX , the equation of the Hyperbola in [ U-V Co-ordinates ] ,
  • i.e. :
    • 

      where :

      • 

  

• And we take special note here that the official value for [ Beta ] is then :
  • 

  But for our discussion purposes here in this Section , we shall use instead the [ absolute value ] , i.e. :

  • 

  so that :

  • the range for the value of [ Beta ] is now :
    • 

    based here again on this special restriction on [ RHO-sub-zero ] :

    • 

      same as in previous Sections .

go to Top Of Page

Deriving the Asymtotes :

• Let us bring-in again the equation for the Hyperbola in [ U-V Co-ordinates ] :
  • 

  Re-arranging terms then yields :

  • 

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

  Dividing throughout by [ RHO-sub-zero square ] then yields us this equation :

  • 

  We then have :

  • 

    yielding :

  • 

• Let us now take a look at the ratio [ V-over-U ] and we have :
  • 

  Consequently :

  • 

    yielding :

  • 

• Let us take a quick look at what would happen if we were to let [ U ] approach [ infinity ] .

  We then have :

  • 

  And we take special note here that :

  • 

  Therefore :

  • 

• As such , the equations for the 2 Asymptotes are then :
  • 

    and

  • 

    as per the diagram on-the-left below .

  

  And the meaning of the Asymptotes is simply this :

  • IF :
    • we were to start-off at the Vertex of the Hyperbola and move along the Hyperbola in either direction ,

    THEN :

    • points on the Hyperbola will get progressively closer-and-closer to an asymptote ,
      • as we move further-and-further away from the Vertex of the Hyperbola .

• The Asymptotes , to a certain extent , then represent a boundary for the Hyperbola .

  And that-is-to-say , the Hyperbolic Curve itself must stay within boundaries as set our by the Asymptotes ,

  • i.e. , within the region as marked-off in light-green color in the diagram on-the-right , above .

• Let us take a quick look , next , as to how we would go about constructing the Asymptotes .

go to Top Of Page

go to the next section : Section XVI - The Asymptotes for the Hyperbola - Part II

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

return to the MATH Functions Basics HomePage

Original dated 2014-12-07