In an Argand diagram, the loci
$\arg(z-2i)=\frac{\pi}{6} $ and $ |z-3|=|z-3i|$ intersect at the point $P$. Express the complex number represented by $P$ in the form $re^{i\theta}$
I tried to sketch the lock in argand(sorry for poor image)
Is the point $P$ intersect as in the image, and how to get the argument of $P$ as I'm getting the wrong answer of $\frac{\pi}{3}$ but the answer is $\frac{\pi}{4}$
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.