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X2 T01 11 locus & complex numbers 2

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Nigel Simmons
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Locus and Complex Numbers
Locus and Complex Numbers Lines
Locus and Complex Numbers Lines y x
Locus and Complex Numbers Lines y c x
Locus and Complex Numbers Lines y c x Im z   c
Locus and Complex Numbers Lines y y c x x Im z   c
Locus and Complex Numbers Lines y y c x k x Im z   c
Locus and Complex Numbers Lines y y c x k x Im z   c Re z   k
Locus and Complex Numbers Lines y y c x k x Im z   c Re z   k y x
Locus and Complex Numbers Lines y y c x k x Im z   c Re z   k y x
Locus and Complex Numbers Lines y y c x k x Im z   c Re z   k y 1 x 2
Locus and Complex Numbers Lines y y c x k x Im z   c Re z   k y 1 z  1  z  2 x 2
e.g . z  1  i  z  2  i
e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12
e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1 y2  2 y 1  x2  4x  4  y2  2 y 1
e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1 y2  2 y 1  x2  4x  4  y2  2 y 1 6x  4 y  3  0
e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1 y2  2 y 1  x2  4x  4  y2  2 y 1 6x  4 y  3  0 OR  bisector of 1,1 and  2,1
e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1 y2  2 y 1  x2  4x  4  y2  2 y 1 6x  4 y  3  0 OR  bisector of 1,1 and  2,1 M 1  2 ,1  1   2 2   1    ,0   2 
e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1 y2  2 y 1  x2  4x  4  y2  2 y 1 6x  4 y  3  0 OR  bisector of 1,1 and  2,1 M 1  2 ,1  1 11  m  2 2  1 2  1   2   ,0   2  3
e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1 y2  2 y 1  x2  4x  4  y2  2 y 1 6x  4 y  3  0 OR  bisector of 1,1 and  2,1 M 1  2 ,1  1 11  m  2 2  1 2  1   2 3   ,0   required slope is   2  3 2
e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1 y2  2 y 1  x2  4x  4  y2  2 y 1 6x  4 y  3  0 OR  bisector of 1,1 and  2,1 M 1  2 ,1  1 11  m  2 2  1 2  1   2 3   ,0   required slope is   2  3 2 3 1 y0   x  2 2
e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1  y2  2 y 1  x2  4x  4  y 2  2 y 1 6x  4 y  3  0 OR  bisector of 1,1 and  2,1 M 1  2 ,1  1 11  m  2 2  1 2  1   2 3   ,0   required slope is   2  3 2 3 1 y0   x  2 2 3 2 y  3 x  2 6x  4 y  3  0
ii  Sketch z  2i  z  4i
ii  Sketch z  2i  z  4i y x
ii  Sketch z  2i  z  4i y 4 x -2
ii  Sketch z  2i  z  4i y 4 1 x -2
ii  Sketch z  2i  z  4i y 4 1 x -2
ii  Sketch z  2i  z  4i y 4 1 x -2 Rays
ii  Sketch z  2i  z  4i y 4 1 x -2 Rays y x
ii  Sketch z  2i  z  4i y 4 1 x -2 Rays y  x
ii  Sketch z  2i  z  4i y 4 1 x -2 Rays y  x arg z  
ii  Sketch z  2i  z  4i y 4 1 x -2 Rays y y  x x arg z  
ii  Sketch z  2i  z  4i y 4 1 x -2 Rays y y    x x arg z  
ii  Sketch z  2i  z  4i y 4 1 x -2 Rays y y    x x arg z   arg z     
 e.g. z  1 and 0  arg z  4
 e.g. z  1 and 0  arg z  4 y x
 e.g. z  1 and 0  arg z  4 y 1 z 1 -1 1 x -1
 e.g. z  1 and 0  arg z  4 y  1 arg z  z 1 4  4 -1 1 x -1
 e.g. z  1 and 0  arg z  4 y  1 arg z  z 1 4  4 -1 1 x arg z  0 -1
 e.g. z  1 and 0  arg z  4 y  1 arg z  z 1 4  4 -1 1 x arg z  0 -1
 e.g. z  1 and 0  arg z  4 y  1 arg z  z 1 4  4 -1 1 x arg z  0 -1
 e.g. z  1 and 0  arg z  4 y  1 arg z  z 1 4  4 -1 1 x arg z  0 -1 Exercise 4N; 1a to j, 2ace, 3ace etc, 4ace

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X2 T01 11 locus & complex numbers 2

  • 2. Locus and Complex Numbers Lines
  • 3. Locus and Complex Numbers Lines y x
  • 4. Locus and Complex Numbers Lines y c x
  • 5. Locus and Complex Numbers Lines y c x Im z   c
  • 6. Locus and Complex Numbers Lines y y c x x Im z   c
  • 7. Locus and Complex Numbers Lines y y c x k x Im z   c
  • 8. Locus and Complex Numbers Lines y y c x k x Im z   c Re z   k
  • 9. Locus and Complex Numbers Lines y y c x k x Im z   c Re z   k y x
  • 10. Locus and Complex Numbers Lines y y c x k x Im z   c Re z   k y x
  • 11. Locus and Complex Numbers Lines y y c x k x Im z   c Re z   k y 1 x 2
  • 12. Locus and Complex Numbers Lines y y c x k x Im z   c Re z   k y 1 z  1  z  2 x 2
  • 13. e.g . z  1  i  z  2  i
  • 14. e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12
  • 15. e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1 y2  2 y 1  x2  4x  4  y2  2 y 1
  • 16. e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1 y2  2 y 1  x2  4x  4  y2  2 y 1 6x  4 y  3  0
  • 17. e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1 y2  2 y 1  x2  4x  4  y2  2 y 1 6x  4 y  3  0 OR  bisector of 1,1 and  2,1
  • 18. e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1 y2  2 y 1  x2  4x  4  y2  2 y 1 6x  4 y  3  0 OR  bisector of 1,1 and  2,1 M 1  2 ,1  1   2 2   1    ,0   2 
  • 19. e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1 y2  2 y 1  x2  4x  4  y2  2 y 1 6x  4 y  3  0 OR  bisector of 1,1 and  2,1 M 1  2 ,1  1 11  m  2 2  1 2  1   2   ,0   2  3
  • 20. e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1 y2  2 y 1  x2  4x  4  y2  2 y 1 6x  4 y  3  0 OR  bisector of 1,1 and  2,1 M 1  2 ,1  1 11  m  2 2  1 2  1   2 3   ,0   required slope is   2  3 2
  • 21. e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1 y2  2 y 1  x2  4x  4  y2  2 y 1 6x  4 y  3  0 OR  bisector of 1,1 and  2,1 M 1  2 ,1  1 11  m  2 2  1 2  1   2 3   ,0   required slope is   2  3 2 3 1 y0   x  2 2
  • 22. e.g . z  1  i  z  2  i  x  12   y  12   x  22   y  12 x2  2x 1  y2  2 y 1  x2  4x  4  y 2  2 y 1 6x  4 y  3  0 OR  bisector of 1,1 and  2,1 M 1  2 ,1  1 11  m  2 2  1 2  1   2 3   ,0   required slope is   2  3 2 3 1 y0   x  2 2 3 2 y  3 x  2 6x  4 y  3  0
  • 23. ii  Sketch z  2i  z  4i
  • 24. ii  Sketch z  2i  z  4i y x
  • 25. ii  Sketch z  2i  z  4i y 4 x -2
  • 26. ii  Sketch z  2i  z  4i y 4 1 x -2
  • 27. ii  Sketch z  2i  z  4i y 4 1 x -2
  • 28. ii  Sketch z  2i  z  4i y 4 1 x -2 Rays
  • 29. ii  Sketch z  2i  z  4i y 4 1 x -2 Rays y x
  • 30. ii  Sketch z  2i  z  4i y 4 1 x -2 Rays y  x
  • 31. ii  Sketch z  2i  z  4i y 4 1 x -2 Rays y  x arg z  
  • 32. ii  Sketch z  2i  z  4i y 4 1 x -2 Rays y y  x x arg z  
  • 33. ii  Sketch z  2i  z  4i y 4 1 x -2 Rays y y    x x arg z  
  • 34. ii  Sketch z  2i  z  4i y 4 1 x -2 Rays y y    x x arg z   arg z     
  • 35.  e.g. z  1 and 0  arg z  4
  • 36.  e.g. z  1 and 0  arg z  4 y x
  • 37.  e.g. z  1 and 0  arg z  4 y 1 z 1 -1 1 x -1
  • 38.  e.g. z  1 and 0  arg z  4 y  1 arg z  z 1 4  4 -1 1 x -1
  • 39.  e.g. z  1 and 0  arg z  4 y  1 arg z  z 1 4  4 -1 1 x arg z  0 -1
  • 40.  e.g. z  1 and 0  arg z  4 y  1 arg z  z 1 4  4 -1 1 x arg z  0 -1
  • 41.  e.g. z  1 and 0  arg z  4 y  1 arg z  z 1 4  4 -1 1 x arg z  0 -1
  • 42.  e.g. z  1 and 0  arg z  4 y  1 arg z  z 1 4  4 -1 1 x arg z  0 -1 Exercise 4N; 1a to j, 2ace, 3ace etc, 4ace