My mathematics teacher taught us that if we draw the locus of $|Z-Z_0| = \theta $ ($Z$ is a variable complex number and $Z_0$ is a constant complex number), it will be a ray originating from the point $P$ denoting $Z_0$ and make an angle of $\theta$ with the horizontal. He also told us that $P$ itself may or may not be included in the ray.
He gave 2 examples:
- $ arg(Z - (i + 1)) = π/4 $
Here the ray includes $(1,1)$
- $ arg(Z - (i + \sqrt{3})) = π/4 $
Here the ray doesn't include $( \sqrt{3} , 1)$
If I have not misunderstood his explanation (which I am almost sure I have not), he meant that the argument of $(1 + i)$ itself is $π/4$ so it's corresponding point was included in the ray but in the other example, the argument was something else, so it's corresponding point got excluded from the ray.
But I find it very wierd if we put $Z^0$ in $arg(Z - Z_0) = \theta$
We will get $arg(Z_0 - Z_0)$ and $Z_0 - Z_0$ is obviously $0$ and $arg(0)$ is undefined. So according to me, $Z_0$ should never be included.
Can you please tell me whether I am correct or not? If I am wrong, then what did I miss?
Thank you
