The point $P$ represents a variable point $z = x + iy$ in an Argand diagram. The point $Q$ represents a variable point $w = u + iv$ in a second Argand diagram and $x$, $y$, $u$ and $v$ are real variables.
Given that $w = \frac{z}{iz+1}$, find an equation of the locus of $Q$ as $P$ moves along the line with equation $y=1$
There is the question. I have been trying this for some time but to no avail; I am not sure whether to rearrange the equation in $w$ to one in $z$, or to sub in $x+i$ for $z$. Having tried both, neither seems to be correct.
Let's sub in $z = x + i$ in $w$ $$ w = \frac{z}{iz + 1} = \frac{x +i}{i(x+i)+1} = \frac{x+i}{ix} = \frac{ix -1}{-x} = \frac{1}{x} - i = u + iv.$$ So $u = \frac{1}{x}$ and $v=-1$