In an Argand diagram, the loci

$Arg(z-2i)=\pi/6 $ and $ |z-3|=|z-3i|$

intersect at the point P. Express the complex number represented by P in the form re^iQ

I try to sketch the Argand(sorry for poor image) enter image description here

Is the point P intersect as in the image, and how to get the angle of P as am getting the wrong answer of $\pi/3$ where the answer is $\pi/4$


The argument of the complex number corresponding to $P$ is by definition the angle of $OP$ counter-clockwise from the real axis.

But the line OP is parallel to (in fact, it is part of) the locus of $|z - 3| = |z - 3i|$.

Hence, the required argument is exactly the angle of the locus from the real axis. Now note that the locus is exactly the line $y = x$. This line lies $\frac{\pi}{4}$ above the real axis.

  • $\begingroup$ How do you calculate to get $\pi/4$ i try $\pi - 2\pi/3$ but the answer is $\pi/3$ $\endgroup$ – Arodi007 Oct 23 '14 at 9:23
  • $\begingroup$ By definition (en.wikipedia.org/wiki/…) the argument of a complex number is the angle of $\vec{OP}$ from the real axis. "The angle of $P$" refers to this argument, not the angle of the vertex $P$. $\endgroup$ – Yiyuan Lee Oct 23 '14 at 9:25

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