If $A=\{z|arg(z)=\frac{\pi}{4}\}$ and $B=\{z|arg(z-3-3i)=\frac{2\pi}{3}\}$, $z\in C$, then $n(A\cap B)$ is equal to?

Question

So the locus of $z$ will be the ray $x=y$ on the argand plane.

In the case of B, the Ray originates from $(3,3)$, which lies on $x=y$ and makes an angle $2\pi/3$ with the +ve $x$

So there is one point of intersection, so the answer should be 1. The given answer is 0. Can someone please Explain this?

Answer 1

Visually there is an intersection at $(3,3)$ but $(3,3)$ is an element of $A$ only, not $B$.

Hint: check $arg(z-3-i3)$ at $z=3+i3$

Answer 2

Note that the first locus is the ray $y=x$, in the first quadrant having a point $(3,3)$ fb it but the second locus is a ray $y-3=-\sqrt{3}(x-3)$, with $x<3$ and $y>3$ this locus leaves out the said point so there is no common point in both the loci.