Given $Z=\frac{4-z}{4+z}$, find the locus of $Z$ if $|z|=4$

Question

Given $Z=\frac{4-z}{4+z}$, find the locus of $Z$ if $|z|=4$

I tried letting $z=x+iy$ and subbing into $Z=\frac{4-z}{4+z}$, rationalising the denominator but I always end up with $\frac{3-2yi}{5+2x}$ and I don't know how to find the locus from that. Am I doing something wrong?

Thanks so much in advance

Answer 1

Write $Z=-1+\frac{8}{4+z}$. The points satisfying $|z|=4$ is a circle with center at the origin and radius $4$. That means that those points applying $z+4$ is a circle of the same radius and center at $4$. This is a circle that passes through the origin. If you apply to them the function $1/z$ they become a perpendicular line passing through $1/8$. If you multiply the result by $8$ it becomes a perpendicular line passing through $1$. Finally, if you subtract $1$ it becomes the $Y$ axis.

Answer 2

You can rewrite $w=Z$, then $$z={4(1-w)\over 1+w}\;\;\; \Longrightarrow \;\;\; |z|=4\Big|{1-w\over 1+w}\Big|$$ so $|w-(-1)|=|w-1|$ so $w$ is equally distance from $-1$ and $1$, thus $w$ describes $Y$-axis.