The complex number $z$ is given by $z = a + ib$, where $a$ and $b$ are real numbers. Its complex conjugate is written as $z^*$.
(a(i))[2]
Show that $|z|^2 = zz^*$ and also that $(z - ki)^* = z^* + ki$, where $k$ is real.
(a(ii))[5]
On an Argand diagram, the locus of complex numbers $z$ is given by $|z - 10i| = 2|z - 4i|$. By squaring both sides, show that $zz^* - 2iz^* + 2iz - 12 = 0$. Hence deduce that $|z - 2i| = 4$.
(a(iii))[1]
Give a geometric description of the set of points.
Worked solution & mark scheme
This 8-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Show that $(a+ib)(a-ib)=a^2+b^2$” …