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X2 t01 08  locus & complex nos 2 (2013)
 

X2 t01 08 locus & complex nos 2 (2013)

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  • Major arc of a circle standing on a chord joining (1,-2) and (0,-3), but not including those points, the arc would be on the left side of the chord
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    X2 t01 08  locus & complex nos 2 (2013) X2 t01 08 locus & complex nos 2 (2013) Presentation Transcript

    • 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   * major arc if   2  * semicircle if   2  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 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 w z zw  z  2 z w  1  2 2 z w  1 2  4 w  1
    • 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
    • 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
    • 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
    • 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  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 
    • 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
    • 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
    • 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
    • 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
    • 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 
    • 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
    •  z  iii  Find the locus of z if arg   z  4 6
    •  z  iii  Find the locus of z if arg   z  4 6  z  arg   z  4 6
    •  z  iii  Find the locus of z if arg   z  4 6  z  arg   z  4 6
    •  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
    •  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
    •  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
    •  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
    •  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
    •  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
    •  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
    •  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
    •  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
    •  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
    •  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
    •  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.
    •  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