An Approach to Mathematic Functions Basics

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

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Section XVI - The Asypmtotes for the Hyperbola - Part II

Summary for the Section :

• In this Section , we shall do the geometric construction of the Asymptotes ,
  • using the [ straight-edge and compass ] method .

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Geometric Construction of the Asymptotes :

• Let us now do a step-by-step construction of the Asymptotes .

  STEP ONE :

  • We first identify the Vertex of the Hyperbola as the point [ Point Q ] and then :
    • we draw a straight-line parallel to the V-Axis passing thru [ Point Q ] ,
      • as per the [ blue-color line ] in the diagram on-the-left below .

  

  STEP TWO :

  • We then draw a circle using [ Point O ] as the center-of-the-circle ,
    • and the radius of the circle is then equal to the length [ L-sub-F ] ,
      • as per the [ cyan-color circle ] as shown in the diagram on-the-right above .

    We then mark-off the 2 points-of-intersection between the [ Straight-Line ] in Step One and the [ Circle ] here

    • as the points [ Point T1 ] and [ Point T2 ] respectively .

  STEP THREE :

  • We then :
    • draw a straight-line passing thru [ Point O ] and [ Point T1 ] to serve as the first Asmptote ,

      and

    • draw another straight-line passing thru [ Point O ] and [ Point T2 ] to serve as the second Asymptote ;

      as per the 2 [ magenta-color dotted lines ] in the diagram on-the-left below .

  

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Double-checking the Asymptotes :

• Let us quickly double-check our construction procedure here .

  We see here that :

  • for the right-angle triangle [ Triangle O-Q-T1 ] in the diagram on-the-right above :
    • 

      same as the radius of the cyan-color circle ;

    • 

  Therefore :

  • 

• Let us now calculate the slope of the first Asymptote , and we have :
  • 

  Consequently :

  • 

• Let us recall , at this point , the equations for the Asymptotes from the last Section :
  • For the first Asymptote :
    • 

  • For the second Asymptote :
    • 

    where :

    • 

      for both equations .

  This then confirms that we have constructed the Asymptotes properly :

  • both Aymptotes do pass thru the { Point-Of-Origin } , and

  • the slopes of both Asymptotes are in-line with the presumed values .

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go to the next section : Section XVII - A Statement of Purpose for Part IV

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

return to the MATH Functions Basics HomePage

Original dated 2014-12-07