Q. For the real constant a find the loci of all points z = x + yi in the complex plane that satisfy:(adsbygoogle = window.adsbygoogle || []).push({});

a) [tex]{\mathop{\rm Re}\nolimits} \left\{ {\log \left( {\frac{{z - ia}}{{z + ia}}} \right)} \right\} = c,c > 0[/tex]

b) [tex]{\mathop{\rm Im}\nolimits} \left\{ {\log \left( {\frac{{z - ia}}{{z + ia}}} \right)} \right\} = k,0 \le k \le \frac{\pi }{2}[/tex]

I have very little idea as to how to do these questions.

For each of them I've thought about first 'ignoring' the Im and Re to see where I could get. I thought, maybe exponentiate both sides but then I'm still left with a quotient of involving z, with the quotient being equal to the exponential of a positive number or an angle(depending on if I'm working on part a or b). There doesn't seem to be an easy way to do this question.

I get the feeling that perhaps these two require some sort of geometric interpretation but I can't really see anyway to interpret the equation. Can someone please help me get started on deducing what the locus of points for each question is?

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# Homework Help: Locus complex plane

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