Locus of Complex Numbers

Locus of Complex Numbers is obtained by letting ( z = x+yi ) and simplifying the expressions. Operations of modulus, conjugate pairs and arguments are to be used for determining the locus of complex numbers.

Question

Let $z = x + iy$ be a complex number such that $|z-1|^2 – |z-3|^2 = 1$. Find the locus of all complex numbers $z$.

\begin{aligned} \displaystyle \require{color} |x+yi-1|^2-|x+yi-3|^2 &= 1 \\ \Big[(x-1)^2 + y^2\Big] – \Big[(x-3)^2 + y^2\Big] &= 1 \\ x^2 – 2x + 1 + y^2 -x^2 + 6x – 9 – y^2 &= 1 \\ 4x &= 9 \\ \therefore x &= \frac{4}{9} \\ \end{aligned}