I need to find the locus of points (on an Argand diagram) such that:(i) arg ⁡ ( z − ( − 1 − 4 i ) ) + arg ⁡ ( z − ( 5 + 8 i ) ) = 0(ii) arg ⁡ ( z − ( − 1 − 4 i ) ) + arg ⁡ ( z − ( 5 + 8 i ) ) = π 2 I could not see a way to solve these problems other than plotting arbitrary points and trying to observe a general pattern.I am aware that arg ⁡ ( z − ( − 1 − 4 i ) ) − arg ⁡ ( z − ( 5 + 8 i ) ) = π 2 is a semicircle, and for other angles, say π 3 or π 4 , part of the arc of a circle, but this is only because I knew that this equation represents the locus of points that made a certain angle between the two complex numbers. I was unable to find a similar representation, however, for (i) and (ii).I am also interested in whether problems (i) and (ii) can be generalised to any angle between 0 to π. Any help here would be greatly appreciated.

Question

I need to find the locus of points (on an Argand diagram) such that:(i)   arg  ⁡  (  z  −  (  −  1  −  4  i  )  )  +  arg  ⁡  (  z  −  (  5  +  8  i  )  )  =  0(ii)   arg  ⁡  (  z  −  (  −  1  −  4  i  )  )  +  arg  ⁡  (  z  −  (  5  +  8  i  )  )  =      π    2  I could not see a way to solve these problems other than plotting arbitrary points and trying to observe a general pattern.I am aware that   arg  ⁡  (  z  −  (  −  1  −  4  i  )  )  −  arg  ⁡  (  z  −  (  5  +  8  i  )  )  =      π    2   is a semicircle, and for other angles, say       π    3   or       π    4  , part of the arc of a circle, but this is only because I knew that this equation represents the locus of points that made a certain angle between the two complex numbers. I was unable to find a similar representation, however, for (i) and (ii).I am also interested in whether problems (i) and (ii) can be generalised to any angle between 0 to   π. Any help here would be greatly appreciated.

yen1291kp6 · Accepted Answer

Denote the points   A  (  −  1  −  4  i  )  ,    B  (  5  +  8  i  )  ,    M  (  z  )  .(i) is equivalent to  arg  ⁡  (  z  −  (  −  1  −  4  i  )  )  =  −  arg  ⁡  (  z  −  (  5  +  8  i  )  )This signifies that the direction from   A towards   M is opposite to the one from   B towards   M  . The locus of points   M  (  z  ) is the segment   A  B except   A  ,  B  .

I need to find the locus of points (on an Argand diagram) such that: (i) arg(z−(−1−4i))+arg(z−(5+8i))=0 (ii) arg(z−(−1−4i))+arg(z−(5+8i))=pi/2

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