(a)[2]
If $z = 1 + yi$ and $y$ is a real number, write $\frac{1}{z}$ in the form $a + bi$, with $a$ and $b$ expressed as functions of $y$.
(b)[3]
Show that $\left(a - \frac{1}{2}\right)^2 + b^2 = \frac{1}{4}$, where $a$ and $b$ are the functions of $y$ found in part (a).
(c)[3]
On one Argand diagram, sketch the loci defined by the equations $\operatorname{Re}(z) = 1$ and $\left|z - \frac{1}{2}\right| = \frac{1}{2}$, where $z$ is a complex number.
(d)[1]
The complex number $z$ satisfies $\operatorname{Re}(z) = 1$. Use your answer to part (b) to describe geometrically the locus of $\frac{1}{z}$.