I want to draw the point cloud represented by the following term.
$$M_{4}=\left\{z \in \mathbb{C} : | z-1|=\frac{1}{2}| z-j|\right\}$$
$j$ equals $i$, the imaginary square root of $-1$.
I have made several attempts to get a solution for the equation. This is the one that looks the most promising. I don't really have an approach on how to continue. I think I have to generate a $j$, but I haven't seen the right way to do so. Thank you in advance.
$ |x+j y-1|=\frac{1}{2}|x+j y-j| $
$ |(x-1)+jy|=\frac{1}{2}|x+(j y-j)| $
$ \sqrt{(x-1)^2+jy^2}=\frac{1}{4}\sqrt{x^2+(j y-j)^2} $
$ (x-1)^2+jy^2=\frac{1}{4}(x^2+(j y-j)^2) $
$ x^2 -2x +1 -y^2=\frac{1}{4}(x^2+(jy^2 -2jy^2 +j^2) $
$ x^2 -2x +1 -y^2=\frac{1}{4}(x^2-y^2 +2y^2 -1) $
$ \frac{3} {4}x^2 - 2x + \frac{5}{4}y^2 - \frac{1}{2}y = 0$
I am sorry if I am giving you a bum steer. I believe by point cloud we are looking for the locus of points described by the above equation. I don't know TEX. Six steps to solution. First, multiply both sides by of (z+i). Simplify. Second, add (iz)^2 to both sides. Simplify. Third, add the imaginary unit to both sides and simplify. Fourth, bring all terms involving z over to the left hand side of the equation and factor out z. Fifth, divide both sides by the constant coefficient of z. Sixth, and lastly, rationalize the denominator on the right hand side by multiplying top and bottom by (1-i). The answer I get is z equals modulus of [(3+i)/4]. I am not certain this is the right answer. Sorry if I am wrong.