X2 t01 08 locus & complex nos 2 (2013)

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X2 t01 08 locus & complex nos 2 (2013)

  1. 1. Locus and Complex Numbers
  2. 2. Locus and Complex Numbers   f  z  , find the locus of  or z given some condition for  or z
  3. 3. Locus and Complex Numbers   f  z  , find the locus of  or z given some condition for  or z (Make the condition the subject)
  4. 4. Locus and Complex Numbers   f  z  , find the locus of  or z given some condition for  or z (Make the condition the subject)  is purely real  Im   0, arg   0 or 
  5. 5. Locus and Complex Numbers   f  z  , find the locus of  or z given some condition for  or z (Make the condition the subject)  is purely real  Im   0, arg   0 or   is purely imaginary  Re   0, arg     2
  6. 6. Locus and Complex Numbers   f  z  , find the locus of  or z given some condition for  or z (Make the condition the subject)  is purely real  Im   0, arg   0 or   is purely imaginary  Re   0, arg     2  linear function  arg      locus is an arc of a circle  linear function 
  7. 7. Locus and Complex Numbers   f  z  , find the locus of  or z given some condition for  or z (Make the condition the subject)  is purely real  Im   0, arg   0 or   is purely imaginary  Re   0, arg     2  linear function  arg      locus is an arc of a circle  linear function   * minor arc if   * major arc if   2  * semicircle if   2  2
  8. 8. z2 e.g .i  Find the locus of w if w  ,z 4 2
  9. 9. z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 w z zw  z  2
  10. 10. z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 w z zw  z  2 z w  1  2
  11. 11. z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 w z zw  z  2 z w  1  2 2 z w  1
  12. 12. z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 w z zw  z  2 z w  1  2 2 z w  1 2  4 w  1
  13. 13. z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 w z zw  z  2 z w  1  2 2 z w  1 2  4 w  1 2 4 w 1
  14. 14. z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 w z zw  z  2 z w  1  2 2 z w  1 2  4 w  1 2 4 w 1 w 1  1 2
  15. 15. z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 w z zw  z  2 z w  1  2 2 z w  1 2  4 w  1 2 4 w 1 w 1  1 2 1  locus is a circle, centre 1,0  and radius 2 1 2 2 i.e.  x  1  y  4
  16. 16. z 1 ii  Find the locus of z if w  and w is purely real z 1
  17. 17. z 1 ii  Find the locus of z if w  and w is purely real z 1  x  1  iy  x  1  iy w   x  1  iy  x  1  iy
  18. 18. z 1 ii  Find the locus of z if w  and w is purely real z 1  x  1  iy  x  1  iy w   x  1  iy  x  1  iy x  2  1  i x  1 y  i x  1 y  y 2  x  12  y 2
  19. 19. z 1 ii  Find the locus of z if w  and w is purely real z 1  x  1  iy  x  1  iy w   x  1  iy  x  1  iy x  2  1  i x  1 y  i x  1 y  y 2  x  12  y 2 If w is purely real then Imw  0
  20. 20. z 1 ii  Find the locus of z if w  and w is purely real z 1  x  1  iy  x  1  iy w   x  1  iy  x  1  iy x  2  1  i x  1 y  i x  1 y  y 2  x  12  y 2 If w is purely real then Imw  0 i.e.   x  1 y   x  1 y  0
  21. 21. z 1 ii  Find the locus of z if w  and w is purely real z 1  x  1  iy  x  1  iy w   x  1  iy  x  1  iy x  2  1  i x  1 y  i x  1 y  y 2  x  12  y 2 If w is purely real then Imw  0 i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0
  22. 22. z 1 ii  Find the locus of z if w  and w is purely real z 1  x  1  iy  x  1  iy w   x  1  iy  x  1  iy x  2  1  i x  1 y  i x  1 y  y 2  x  12  y 2 If w is purely real then Imw  0 i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0
  23. 23. z 1 ii  Find the locus of z if w  and w is purely real z 1  x  1  iy  x  1  iy w   x  1  iy  x  1  iy x  2  1  i x  1 y  i x  1 y  y 2  x  12  y 2 If w is purely real then Imw  0 i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0 
  24. 24. z 1 ii  Find the locus of z if w  and w is purely real z 1  x  1  iy  x  1  iy OR If w is purely real then arg w  0 or  w   x  1  iy  x  1  iy x  2  1  i x  1 y  i x  1 y  y 2  x  12  y 2 If w is purely real then Imw  0 i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0 
  25. 25. z 1 ii  Find the locus of z if w  and w is purely real z 1  x  1  iy  x  1  iy OR If w is purely real then arg w  0 or  w   x  1  iy  x  1  iy  z  1   0 or  x 2  1  i x  1 y  i x  1 y  y 2 i.e. arg   2  z 1  x  1  y 2 If w is purely real then Imw  0 i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0 
  26. 26. z 1 ii  Find the locus of z if w  and w is purely real z 1  x  1  iy  x  1  iy OR If w is purely real then arg w  0 or  w   x  1  iy  x  1  iy  z  1   0 or  x 2  1  i x  1 y  i x  1 y  y 2 i.e. arg   2  z 1  x  1  y 2 If w is purely real then Imw  0 i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0  arg z  1  arg z  1  0 or  y x
  27. 27. z 1 ii  Find the locus of z if w  and w is purely real z 1  x  1  iy  x  1  iy OR If w is purely real then arg w  0 or  w   x  1  iy  x  1  iy  z  1   0 or  x 2  1  i x  1 y  i x  1 y  y 2 i.e. arg   2  z 1  x  1  y 2 If w is purely real then Imw  0 i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0  arg z  1  arg z  1  0 or  y -1 1 x
  28. 28. z 1 ii  Find the locus of z if w  and w is purely real z 1  x  1  iy  x  1  iy OR If w is purely real then arg w  0 or  w   x  1  iy  x  1  iy  z  1   0 or  x 2  1  i x  1 y  i x  1 y  y 2 i.e. arg   2  z 1  x  1  y 2 If w is purely real then Imw  0 i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0  arg z  1  arg z  1  0 or  y -1 1 x
  29. 29. z 1 ii  Find the locus of z if w  and w is purely real z 1  x  1  iy  x  1  iy OR If w is purely real then arg w  0 or  w   x  1  iy  x  1  iy  z  1   0 or  x 2  1  i x  1 y  i x  1 y  y 2 i.e. arg   2  z 1  x  1  y 2 If w is purely real then Imw  0 i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0  arg z  1  arg z  1  0 or  y -1 1 x
  30. 30. z 1 ii  Find the locus of z if w  and w is purely real z 1  x  1  iy  x  1  iy OR If w is purely real then arg w  0 or  w   x  1  iy  x  1  iy  z  1   0 or  x 2  1  i x  1 y  i x  1 y  y 2 i.e. arg   2  z 1  x  1  y 2 If w is purely real then Imw  0 i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0  arg z  1  arg z  1  0 or  y -1 1 x locus is y  0, excluding  1,0 
  31. 31. z 1 ii  Find the locus of z if w  and w is purely real z 1  x  1  iy  x  1  iy OR If w is purely real then arg w  0 or  w   x  1  iy  x  1  iy  z  1   0 or  x 2  1  i x  1 y  i x  1 y  y 2 i.e. arg   2  z 1  x  1  y 2 If w is purely real then Imw  0 i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0  arg z  1  arg z  1  0 or  y -1 1 x locus is y  0, excluding  1,0  Note : locus is y  0, excluding 1,0  only i.e. answer the original question
  32. 32.  z  iii  Find the locus of z if arg   z  4 6
  33. 33.  z  iii  Find the locus of z if arg   z  4 6  z  arg   z  4 6
  34. 34.  z  iii  Find the locus of z if arg   z  4 6  z  arg   z  4 6
  35. 35.  z  iii  Find the locus of z if arg   z  4 6  z  arg   z  4 6  arg z  arg z  4   6 y x
  36. 36.  z  iii  Find the locus of z if arg   z  4 6  z  arg   z  4 6  arg z  arg z  4   6 y 4x  6
  37. 37.  z  iii  Find the locus of z if arg   z  4 6  z  arg   z  4 6  arg z  arg z  4   6 y 4x  6 NOTE: arg z  arg z-4   below axis
  38. 38.  z  iii  Find the locus of z if arg   z  4 6  z  arg   z  4 6  arg z  arg z  4   6 y 2 4x  6 NOTE: arg z  arg z-4   below axis
  39. 39.  z  iii  Find the locus of z if arg   z  4 6  z  arg   z  4 6  arg z  arg z  4   6 y 2 r 4x (2,y)  6 NOTE: arg z  arg z-4   below axis
  40. 40.  z  iii  Find the locus of z if arg   z  4 6  z  arg   z  4 6  arg z  arg z  4   6 y 2 r 4x (2,y)  6 NOTE: arg z  arg z-4   below axis 30
  41. 41.  z  iii  Find the locus of z if arg   z  4 6 y  z   tan 60 arg  2  z  4 6 arg z  arg z  4   y  6 2 r 4x (2,y)  6 NOTE: arg z  arg z-4   below axis 30
  42. 42.  z  iii  Find the locus of z if arg   z  4 6 y  z   tan 60 arg  2  z  4 6  y  2 tan 60 arg z  arg z  4   6 2 3 y 2 r 4x (2,y)  6 NOTE: arg z  arg z-4   below axis 30
  43. 43.  z  iii  Find the locus of z if arg   z  4 6 y  z   tan 60 arg  2  z  4 6  y  2 tan 60 arg z  arg z  4   6 2 3 y  centre is 2,2 3  2 r 4x (2,y)  6 NOTE: arg z  arg z-4   below axis 30
  44. 44.  z  iii  Find the locus of z if arg   z  4 6 y  z   tan 60 arg  2  z  4 6  y  2 tan 60 arg z  arg z  4   6 2 3 y  centre is 2,2 3  2 r 4x (2,y)  6 NOTE: arg z  arg z-4   below axis 30 r 2  2 2  2 3  2
  45. 45.  z  iii  Find the locus of z if arg   z  4 6 y  z   tan 60 arg  2  z  4 6  y  2 tan 60 arg z  arg z  4   6 2 3 y  centre is 2,2 3  2 r 4x (2,y)  6 NOTE: arg z  arg z-4   below axis 30 r 2  2 2  2 3  2 r 2  16 r4
  46. 46.  z  iii  Find the locus of z if arg   z  4 6 y  z   tan 60 arg  2  z  4 6  y  2 tan 60 arg z  arg z  4   6 2 3 y r 2  2 2  2 3  2 r 2  16 r4  centre is 2,2 3   locus is the major arc of the circle 2 r 4x (2,y)  6 NOTE: arg z  arg z-4   below axis  x  2   y  2 3   16 formed by the chord joining 0,0  and 4,0  but not 2 2 30 including these points.
  47. 47.  z  iii  Find the locus of z if arg   z  4 6 y  z   tan 60 arg  2  z  4 6  y  2 tan 60 arg z  arg z  4   6 2 3 y r 2  2 2  2 3  2 r 2  16 r4  centre is 2,2 3   locus is the major arc of the circle 2 r 4x (2,y)  6 NOTE: arg z  arg z-4   below axis  x  2   y  2 3   16 formed by the chord joining 0,0  and 4,0  but not 2 2 30 including these points. Patel: Exercise 4N; 5, 6 Cambridge: Exercise 1F; 10 to 20 HSC Geometrical Complex Numbers Questions

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