Determine values of $\theta$ for which $\arg(z-4+2i)=\theta$ and $|z+6+6i|=4$ have no common solutions

Question

So there is this question that's asking for a "range of values for theta from $-\pi$ to $\pi$, for which $\arg(z-4+2i)=\theta$ and $|z+6+6i|=4$ have no common solutions."

I'm not really sure how to do it as my teacher didn't explain these sort of questions at all to us. I just don't know where to start or what exactly to equate.

Any help would be appreciated. Thanks in advance!

Answer 1

Hint: Let $a=-6-6i,\,b=4-2i$ be points on the complex plane. $|z-a|=4$ is a circle of radius $4$ and center in $a$. What is $\arg(z-b)$?
If you start with all the values if $z$ on the circle and determine somehow the range of $\arg(z-b)$ for all these $z$ (say we have set $B$ being the range), what will the answer be?