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X2 T01 12 locus & complex numbers 3

N
Nigel Simmons

The document discusses finding the locus of complex numbers ω or z given some condition on ω or z. It provides examples of finding the locus when: 1) ω is a linear function of z and the condition is that ω is purely real or purely imaginary. The locus is an arc of a circle. 2) The condition is that the argument of a linear function of ω equals a constant θ. The locus is an arc of a circle that can be a minor arc, major arc, or semicircle depending on the value of θ. 3) An example finds the locus is a circle when ω is a rational function of z and the condition is that z equals a constant. 4) Another

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Locus and Complex Numbers
Locus and Complex Numbers   f  z  , find the locus of  or z given some condition for  or z
Locus and Complex Numbers   f  z  , find the locus of  or z given some condition for  or z (Make the condition the subject)
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 
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
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 
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   2
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   2  * major arc if   2
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   2  * major arc if   2  * semicircle if   2
z2 e.g .i  Find the locus of w if w  ,z 4 2
z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 w z
z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 w z zw  z  2
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
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
z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 2 w  4 z w  1 zw  z  2 z w  1  2 2 z w  1
z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 2 w  4 z w  1 zw  z  2 2 z w  1  2 4 w 1 2 z w  1
z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 2 w  4 z w  1 zw  z  2 2 z w  1  2 4 w 1 2 z w 1  1 w  1 2
z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 2 w  4 z w  1 zw  z  2 2 z w  1  2 4 w 1 2 z w 1  1 w  1 2 1  locus is a circle, centre 1,0  and radius 2 1 i.e.  x  1  y  2 2 4
z 1 ii  Find the locus of z if w  and w is purely real z 1
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
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
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
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
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
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
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 
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 
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 i.e. arg  z  1   0 or   x  12  y 2  z 1 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 
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 i.e. arg  z  1   0 or   x  12  y 2  z 1 If w is purely real then Imw  0 arg z  1  arg z  1  0 or  y i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0 x  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0 
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 i.e. arg  z  1   0 or   x  12  y 2  z 1 If w is purely real then Imw  0 arg z  1  arg z  1  0 or  y i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0 -1 1 x  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0 
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 i.e. arg  z  1   0 or   x  12  y 2  z 1 If w is purely real then Imw  0 arg z  1  arg z  1  0 or  y i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0 -1 1 x  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0 
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 i.e. arg  z  1   0 or   x  12  y 2  z 1 If w is purely real then Imw  0 arg z  1  arg z  1  0 or  y i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0 -1 1 x  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0 
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 i.e. arg  z  1   0 or   x  12  y 2  z 1 If w is purely real then Imw  0 arg z  1  arg z  1  0 or  y i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0 -1 1 x  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0  locus is y  0, excluding  1,0 
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 i.e. arg  z  1   0 or   x  12  y 2  z 1 If w is purely real then Imw  0 arg z  1  arg z  1  0 or  y i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0 -1 1 x  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0  locus is y  0, excluding  1,0  Note : locus is y  0, excluding 1,0  only i.e. answer the original question
 z  iii  Find the locus of z if arg   z  4 6
 z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6
 z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6
 z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6  arg z  arg z  4   6 y x
 z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6  arg z  arg z  4   6 y 4x  6
 z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6  arg z  arg z  4   6 y 4x  6 NOTE: arg z  arg z-4   below axis
 z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6  arg z  arg z  4   6 y 2 4x  6 NOTE: arg z  arg z-4   below axis
 z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6  arg z  arg z  4   6 y 2 r 4x (2,y)  6 NOTE: arg z  arg z-4   below axis
 z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6  arg z  arg z  4   6 y 2 r 4x (2,y) 30  6 NOTE: arg z  arg z-4   below axis
 z  iii  Find the locus of z if arg   z  4 6 arg  z  y  tan 60   z  4 6 2  arg z  arg z  4   6 y 2 r 4x (2,y) 30  6 NOTE: arg z  arg z-4   below axis
 z  iii  Find the locus of z if arg   z  4 6 arg  z  y  tan 60   z  4 6 2  y  2 tan 60 arg z  arg z  4   6 y 2 3 2 r 4x (2,y) 30  6 NOTE: arg z  arg z-4   below axis
 z  iii  Find the locus of z if arg   z  4 6 arg  z  y  tan 60   z  4 6 2  y  2 tan 60 arg z  arg z  4   6 y 2 3  centre is 2,2 3  2 r 4x (2,y) 30  6 NOTE: arg z  arg z-4   below axis
 z  iii  Find the locus of z if arg   z  4 6 arg  z  y  tan 60 r 2  2 2  2 3  2   z  4 6 2  y  2 tan 60 arg z  arg z  4   6 y 2 3  centre is 2,2 3  2 r 4x (2,y) 30  6 NOTE: arg z  arg z-4   below axis
 z  iii  Find the locus of z if arg   z  4 6 arg  z  y  tan 60 r 2  2 2  2 3  2   z  4 6 2  r 2  16 arg z  arg z  4   y  2 tan 60 6 r4 y 2 3  centre is 2,2 3  2 r 4x (2,y) 30  6 NOTE: arg z  arg z-4   below axis
 z  iii  Find the locus of z if arg   z  4 6 arg  z  y  tan 60 r 2  2 2  2 3  2   z  4 6 2  r 2  16 arg z  arg z  4   y  2 tan 60 6 r4 y 2 3  centre is 2,2 3   locus is the major arc of the circle 2  x  2   y  2 3   16 formed by the 2 2 r 4x chord joining 0,0  and 4,0  but not (2,y) 30 including these points.  6 NOTE: arg z  arg z-4   below axis
 z  iii  Find the locus of z if arg   z  4 6 arg  z  y  tan 60 r 2  2 2  2 3  2   z  4 6 2  r 2  16 arg z  arg z  4   y  2 tan 60 6 r4 y 2 3  centre is 2,2 3   locus is the major arc of the circle 2  x  2   y  2 3   16 formed by the 2 2 x r4 chord joining 0,0  and 4,0  but not (2,y) 30 including these points.  6 Exercise 4N; 5, 6 NOTE: arg z  arg z-4  Exercise 4O; 3 to 10, 12, 13a, 14a  below axis

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X2 T01 12 locus & complex numbers 3

  • 2. Locus and Complex Numbers   f  z  , find the locus of  or z given some condition for  or z
  • 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. 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. 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. 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. 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   2
  • 8. 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   2  * major arc if   2
  • 9. 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   2  * major arc if   2  * semicircle if   2
  • 10. z2 e.g .i  Find the locus of w if w  ,z 4 2
  • 11. z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 w z
  • 12. z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 w z zw  z  2
  • 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
  • 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
  • 15. z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 2 w  4 z w  1 zw  z  2 z w  1  2 2 z w  1
  • 16. z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 2 w  4 z w  1 zw  z  2 2 z w  1  2 4 w 1 2 z w  1
  • 17. z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 2 w  4 z w  1 zw  z  2 2 z w  1  2 4 w 1 2 z w 1  1 w  1 2
  • 18. z2 e.g .i  Find the locus of w if w  ,z 4 2 z2 2 w  4 z w  1 zw  z  2 2 z w  1  2 4 w 1 2 z w 1  1 w  1 2 1  locus is a circle, centre 1,0  and radius 2 1 i.e.  x  1  y  2 2 4
  • 19. z 1 ii  Find the locus of z if w  and w is purely real z 1
  • 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
  • 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
  • 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
  • 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
  • 24. 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
  • 25. 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
  • 26. 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 
  • 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  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 
  • 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  x 2  1  i x  1 y  i x  1 y  y 2 i.e. arg  z  1   0 or   x  12  y 2  z 1 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 
  • 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  x 2  1  i x  1 y  i x  1 y  y 2 i.e. arg  z  1   0 or   x  12  y 2  z 1 If w is purely real then Imw  0 arg z  1  arg z  1  0 or  y i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0 x  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0 
  • 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  x 2  1  i x  1 y  i x  1 y  y 2 i.e. arg  z  1   0 or   x  12  y 2  z 1 If w is purely real then Imw  0 arg z  1  arg z  1  0 or  y i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0 -1 1 x  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0 
  • 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  x 2  1  i x  1 y  i x  1 y  y 2 i.e. arg  z  1   0 or   x  12  y 2  z 1 If w is purely real then Imw  0 arg z  1  arg z  1  0 or  y i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0 -1 1 x  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0 
  • 32. 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 i.e. arg  z  1   0 or   x  12  y 2  z 1 If w is purely real then Imw  0 arg z  1  arg z  1  0 or  y i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0 -1 1 x  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0 
  • 33. 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 i.e. arg  z  1   0 or   x  12  y 2  z 1 If w is purely real then Imw  0 arg z  1  arg z  1  0 or  y i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0 -1 1 x  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0  locus is y  0, excluding  1,0 
  • 34. 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 i.e. arg  z  1   0 or   x  12  y 2  z 1 If w is purely real then Imw  0 arg z  1  arg z  1  0 or  y i.e.   x  1 y   x  1 y  0  xy  y  xy  y  0  2y  0 y0 -1 1 x  locus is y  0, excluding 1,0   z  1  0, bottom of fraction  0  locus is y  0, excluding  1,0  Note : locus is y  0, excluding 1,0  only i.e. answer the original question
  • 35.  z  iii  Find the locus of z if arg   z  4 6
  • 36.  z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6
  • 37.  z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6
  • 38.  z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6  arg z  arg z  4   6 y x
  • 39.  z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6  arg z  arg z  4   6 y 4x  6
  • 40.  z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6  arg z  arg z  4   6 y 4x  6 NOTE: arg z  arg z-4   below axis
  • 41.  z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6  arg z  arg z  4   6 y 2 4x  6 NOTE: arg z  arg z-4   below axis
  • 42.  z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6  arg z  arg z  4   6 y 2 r 4x (2,y)  6 NOTE: arg z  arg z-4   below axis
  • 43.  z  iii  Find the locus of z if arg   z  4 6 arg  z    z  4 6  arg z  arg z  4   6 y 2 r 4x (2,y) 30  6 NOTE: arg z  arg z-4   below axis
  • 44.  z  iii  Find the locus of z if arg   z  4 6 arg  z  y  tan 60   z  4 6 2  arg z  arg z  4   6 y 2 r 4x (2,y) 30  6 NOTE: arg z  arg z-4   below axis
  • 45.  z  iii  Find the locus of z if arg   z  4 6 arg  z  y  tan 60   z  4 6 2  y  2 tan 60 arg z  arg z  4   6 y 2 3 2 r 4x (2,y) 30  6 NOTE: arg z  arg z-4   below axis
  • 46.  z  iii  Find the locus of z if arg   z  4 6 arg  z  y  tan 60   z  4 6 2  y  2 tan 60 arg z  arg z  4   6 y 2 3  centre is 2,2 3  2 r 4x (2,y) 30  6 NOTE: arg z  arg z-4   below axis
  • 47.  z  iii  Find the locus of z if arg   z  4 6 arg  z  y  tan 60 r 2  2 2  2 3  2   z  4 6 2  y  2 tan 60 arg z  arg z  4   6 y 2 3  centre is 2,2 3  2 r 4x (2,y) 30  6 NOTE: arg z  arg z-4   below axis
  • 48.  z  iii  Find the locus of z if arg   z  4 6 arg  z  y  tan 60 r 2  2 2  2 3  2   z  4 6 2  r 2  16 arg z  arg z  4   y  2 tan 60 6 r4 y 2 3  centre is 2,2 3  2 r 4x (2,y) 30  6 NOTE: arg z  arg z-4   below axis
  • 49.  z  iii  Find the locus of z if arg   z  4 6 arg  z  y  tan 60 r 2  2 2  2 3  2   z  4 6 2  r 2  16 arg z  arg z  4   y  2 tan 60 6 r4 y 2 3  centre is 2,2 3   locus is the major arc of the circle 2  x  2   y  2 3   16 formed by the 2 2 r 4x chord joining 0,0  and 4,0  but not (2,y) 30 including these points.  6 NOTE: arg z  arg z-4   below axis
  • 50.  z  iii  Find the locus of z if arg   z  4 6 arg  z  y  tan 60 r 2  2 2  2 3  2   z  4 6 2  r 2  16 arg z  arg z  4   y  2 tan 60 6 r4 y 2 3  centre is 2,2 3   locus is the major arc of the circle 2  x  2   y  2 3   16 formed by the 2 2 x r4 chord joining 0,0  and 4,0  but not (2,y) 30 including these points.  6 Exercise 4N; 5, 6 NOTE: arg z  arg z-4  Exercise 4O; 3 to 10, 12, 13a, 14a  below axis