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 Complex Analysis Complex Analysis Math Forum

 July 17th, 2015, 01:56 AM #1 Member   Joined: Aug 2013 Posts: 31 Thanks: 1 Locus Hi guys, any help with this one would be great. Z is a complex number. Sketch the locus of z that satisfied by: arg(z-2/z-i)=pi/4. I break this down to: arg(z-2)-arg(z-i)=pi/4 My thought is to select three points and place them on the imaginary axis, then deduce the coordinate of z, three times, (ie start with z on the im axis, and we can show that coordinate of z is (0,2),then place z-2 on the im axis etc) Has anyone got an easier way to do this? I know the answer is a circle, but how can you tell? also, My the answer shows its just a major arc of a circle, but with no explanation as to why only part of the circle exists. if anyone can help deduce the equation and help with the above I would be appreciate it. Thanks in advance
 July 17th, 2015, 01:41 PM #2 Global Moderator   Joined: May 2007 Posts: 6,216 Thanks: 493 $\displaystyle \frac{z-2}{z-i}=r\frac{\sqrt{2}}{{2}}(1+i)$, where r > 0. Solve for r as a function of z and plot. Thanks from evaeva

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