$ z = x + \text{j} \cdot y $ and $ \arg(z) = \frac{\pi}{4} $. Find the relationship between $x$ any $y$, then find $z$ for
$|z - 3 + 2 \cdot \text{j}| = |z +3 \cdot \text{j}|$, where $\text{j}^2= -1$,
In the above question do I have to compare like this:
$Z= x+\text{j}\cdot y$, $Z=\tan^{-1}\left(\dfrac{\text{j} \cdot y}{x}\right)=arg(z)= \dfrac{\pi}{4}$? to find $x$, $y$ and $z$?
Well, when we know that the argument of a complex number is $\frac{\pi}{4}$ we can write:
$$\arg\left(\text{a}+\text{b}i\right)=\arctan\left(\frac{\text{b}}{\text{a}}\right)=\frac{\pi}{4}\space\Longleftrightarrow\space\frac{\text{b}}{\text{a}}=\tan\left(\frac{\pi}{4}\right)=1\space\Longleftrightarrow\space\text{a}=\text{b}>0\tag1$$
So, we get:
$$\left|\text{z}-3+2i\right|=\left|\text{z}+3i\right|\space\Longleftrightarrow\space\sqrt{\left(\text{a}-3\right)^2+\left(\text{a}+2\right)^2}=\sqrt{\text{a}^2+\left(\text{a}+3\right)^2}\tag2$$
Which give:
$$\text{a}=\frac{1}{2}\tag3$$