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The Anatomy of
the Hyperbola




                 12.3 Hyperbolas
What does a quot;sonic boomquot;
have to do with hyperbolas?




                   sonicboomplane_navy.jpg
Pre-Cal 40S May 25, 2009
Throw 2 stones in a pond. The resulting concentric ripples
meet in a hyperbola shape.
Hyperbolic
     Light




             Casting by flickr
             user harry harris
Hyperbolic
Cooling Towers




           Sheffield Cooling Towers (HDR)
           by flickr user Paul Denton Cocker
Generating Hyperbolas HOMEWORK
Pre-Cal 40S May 25, 2009
Pre-Cal 40S May 25, 2009
Constructing Hyperbolas



                 CONJUGATE
                   AXIS

                             VERTEX
        VERTEX
                     TRANSVERSE
                          AXIS



         ASYMPTOTES
The Anatomy of The Hyperbola               P

                              B1
                              c F
                   F2    a b      1
                      A 1 O c A2
                           B2


          O is the centre, it has coordinates (h, k).
The Anatomy of The Hyperbola                  P

                                 B1
                                  c F
                       F2    a b      1
                          A 1 O c A2
                               B2


             O is the centre, it has coordinates (h, k).
A1A2 is the length of the
transverse axis. It's length is 2a.
The Anatomy of The Hyperbola                  P

                                 B1
                                  c F
                       F2    a b      1
                          A 1 O c A2
                               B2


             O is the centre, it has coordinates (h, k).
A1A2 is the length of the              B1 B2 is the length of the
transverse axis. It's length is 2a.    conjugate axis. It's length is 2b.
The Anatomy of The Hyperbola                  P

                                  B1
                                  c F
                       F2    a b      1
                          A 1 O c A2
                               B2


             O is the centre, it has coordinates (h, k).
A1A2 is the length of the              B1 B2 is the length of the
transverse axis. It's length is 2a.    conjugate axis. It's length is 2b.
  F 1 & F 2 are called the foci. They
  are c units from the centre.
The Anatomy of The Hyperbola                  P

                                  B1
                                  c F
                       F2    a b      1
                          A 1 O c A2
                               B2


             O is the centre, it has coordinates (h, k).
A1A2 is the length of the              B1 B2 is the length of the
transverse axis. It's length is 2a.    conjugate axis. It's length is 2b.
  F 1 & F 2 are called the foci. They    PF1 & PF 2 are the focal
  are c units from the centre.           radii of the hyperbola.
The Anatomy of The Hyperbola                  P

                                  B1
                                  c F
                       F2    a b      1
                          A 1 O c A2
                               B2


             O is the centre, it has coordinates (h, k).
A1A2 is the length of the              B1 B2 is the length of the
transverse axis. It's length is 2a.    conjugate axis. It's length is 2b.
  F 1 & F 2 are called the foci. They    PF1 & PF 2 are the focal
  are c units from the centre.           radii of the hyperbola.
OA1 = OA2 is the length of the
semitransverse axis with length a.
The Anatomy of The Hyperbola                  P

                                  B1
                                  c F
                       F2    a b      1
                          A 1 O c A2
                               B2


             O is the centre, it has coordinates (h, k).
A1A2 is the length of the              B1 B2 is the length of the
transverse axis. It's length is 2a.    conjugate axis. It's length is 2b.
  F 1 & F 2 are called the foci. They    PF1 & PF 2 are the focal
  are c units from the centre.           radii of the hyperbola.
OA1 = OA2 is the length of the          A1 & A 2 are called the
semitransverse axis with length a.      vertices of the hyperbola.
The Anatomy of The Hyperbola                  P

                                  B1
                                  c F
                       F2    a b      1
                          A 1 O c A2
                               B2


             O is the centre, it has coordinates (h, k).
A1A2 is the length of the              B1 B2 is the length of the
transverse axis. It's length is 2a.    conjugate axis. It's length is 2b.
  F 1 & F 2 are called the foci. They    PF1 & PF 2 are the focal
  are c units from the centre.           radii of the hyperbola.
OA1 = OA2 is the length of the          A1 & A 2 are called the
semitransverse axis with length a.      vertices of the hyperbola.
  OB1 = OB2 is the length of the
  semiconjugate axis with length b.
The Anatomy of The Hyperbola                  P

                                  B1
                                  c F
                       F2    a b      1
                          A 1 O c A2
                               B2


             O is the centre, it has coordinates (h, k).
A1A2 is the length of the              B1 B2 is the length of the
transverse axis. It's length is 2a.    conjugate axis. It's length is 2b.
  F 1 & F 2 are called the foci. They    PF1 & PF 2 are the focal
  are c units from the centre.           radii of the hyperbola.
OA1 = OA2 is the length of the          A1 & A 2 are called the
semitransverse axis with length a.      vertices of the hyperbola.
  OB1 = OB2 is the length of the         B1 & B2 are called the endpoints
  semiconjugate axis with length b.      of the conjugate axis.
The Standard Form for the Equation of a Hyperbola

    Horizontal Orientation             Vertical Orientation




          Similarities                   Differences

       • Both equal one                  • a^2 with x^2 on left,
       • have (x-h)^2 and (y-k)^2        a^2 with y^2 on right.
       • have a^2 and b^2 in             • b^2 with y^2 on left,
       denominator.                      a^2 with x^2 on right
       • b^2 denominator in negative     • when y (+) is vertical,
       term.                             when y(-) is horizontal.
       • a^2 denominator in positive     • when x (-) is vertical,
       term.                             x(+) is horizontal.
Horizontal Hyperbola   Vertical Hyperbola




      a b                           a
        O
                                   O b
The Pythagorean Property


                       B1
                       b a c
                              F1
                F2 A2 O c A 1
                           B2
                                             2       2   2
                                            c = a +b


         Horizontal Hyperbola   Vertical Hyperbola
Conics Animations Source
                           http://tinyurl.com/conicAni
Conics Animations Source

                           http://tinyurl.com/conicAni
Conics Animations Source


                           http://tinyurl.com/conicAni
Conics Animations Source


                           http://tinyurl.com/conicAni
For the hyperbola whose equation is given below.
     (i) Write the equation in standard form
     (ii) Determine the lengths of the transverse and conjugate axes,
the coordinates of the verticies and foci, and the equations of the
asymptotes.                                  HOMEWORK
     (iii) Sketch a graph of the hyperbola.

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Pre-Cal 40S May 25, 2009

  • 1. The Anatomy of the Hyperbola 12.3 Hyperbolas
  • 2. What does a quot;sonic boomquot; have to do with hyperbolas? sonicboomplane_navy.jpg
  • 4. Throw 2 stones in a pond. The resulting concentric ripples meet in a hyperbola shape.
  • 5. Hyperbolic Light Casting by flickr user harry harris
  • 6. Hyperbolic Cooling Towers Sheffield Cooling Towers (HDR) by flickr user Paul Denton Cocker
  • 10. Constructing Hyperbolas CONJUGATE AXIS VERTEX VERTEX TRANSVERSE AXIS ASYMPTOTES
  • 11. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k).
  • 12. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the transverse axis. It's length is 2a.
  • 13. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the B1 B2 is the length of the transverse axis. It's length is 2a. conjugate axis. It's length is 2b.
  • 14. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the B1 B2 is the length of the transverse axis. It's length is 2a. conjugate axis. It's length is 2b. F 1 & F 2 are called the foci. They are c units from the centre.
  • 15. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the B1 B2 is the length of the transverse axis. It's length is 2a. conjugate axis. It's length is 2b. F 1 & F 2 are called the foci. They PF1 & PF 2 are the focal are c units from the centre. radii of the hyperbola.
  • 16. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the B1 B2 is the length of the transverse axis. It's length is 2a. conjugate axis. It's length is 2b. F 1 & F 2 are called the foci. They PF1 & PF 2 are the focal are c units from the centre. radii of the hyperbola. OA1 = OA2 is the length of the semitransverse axis with length a.
  • 17. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the B1 B2 is the length of the transverse axis. It's length is 2a. conjugate axis. It's length is 2b. F 1 & F 2 are called the foci. They PF1 & PF 2 are the focal are c units from the centre. radii of the hyperbola. OA1 = OA2 is the length of the A1 & A 2 are called the semitransverse axis with length a. vertices of the hyperbola.
  • 18. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the B1 B2 is the length of the transverse axis. It's length is 2a. conjugate axis. It's length is 2b. F 1 & F 2 are called the foci. They PF1 & PF 2 are the focal are c units from the centre. radii of the hyperbola. OA1 = OA2 is the length of the A1 & A 2 are called the semitransverse axis with length a. vertices of the hyperbola. OB1 = OB2 is the length of the semiconjugate axis with length b.
  • 19. The Anatomy of The Hyperbola P B1 c F F2 a b 1 A 1 O c A2 B2 O is the centre, it has coordinates (h, k). A1A2 is the length of the B1 B2 is the length of the transverse axis. It's length is 2a. conjugate axis. It's length is 2b. F 1 & F 2 are called the foci. They PF1 & PF 2 are the focal are c units from the centre. radii of the hyperbola. OA1 = OA2 is the length of the A1 & A 2 are called the semitransverse axis with length a. vertices of the hyperbola. OB1 = OB2 is the length of the B1 & B2 are called the endpoints semiconjugate axis with length b. of the conjugate axis.
  • 20. The Standard Form for the Equation of a Hyperbola Horizontal Orientation Vertical Orientation Similarities Differences • Both equal one • a^2 with x^2 on left, • have (x-h)^2 and (y-k)^2 a^2 with y^2 on right. • have a^2 and b^2 in • b^2 with y^2 on left, denominator. a^2 with x^2 on right • b^2 denominator in negative • when y (+) is vertical, term. when y(-) is horizontal. • a^2 denominator in positive • when x (-) is vertical, term. x(+) is horizontal.
  • 21. Horizontal Hyperbola Vertical Hyperbola a b a O O b
  • 22. The Pythagorean Property B1 b a c F1 F2 A2 O c A 1 B2 2 2 2 c = a +b Horizontal Hyperbola Vertical Hyperbola
  • 23. Conics Animations Source http://tinyurl.com/conicAni
  • 24. Conics Animations Source http://tinyurl.com/conicAni
  • 25. Conics Animations Source http://tinyurl.com/conicAni
  • 26. Conics Animations Source http://tinyurl.com/conicAni
  • 27. For the hyperbola whose equation is given below. (i) Write the equation in standard form (ii) Determine the lengths of the transverse and conjugate axes, the coordinates of the verticies and foci, and the equations of the asymptotes. HOMEWORK (iii) Sketch a graph of the hyperbola.