Challenging +1.8 This is a Further Maths complex loci problem requiring geometric interpretation of two conditions, finding their intersection constraints, and algebraic manipulation to derive bounds on a parameter. It demands visualization of perpendicular bisector and circle loci, coordinate geometry to find extremal points, and careful algebraic work to express the answer in the required form. The multi-step reasoning and need to synthesize geometric and algebraic approaches place it well above average difficulty.
The complex number \(z\) satisfies the equations
$$|z^* - 1 - 2i| = |z - 3|$$
and
$$|z - a| = 3$$
where \(a\) is real.
Show that \(a\) must lie in the interval \([1 - s\sqrt{t}, 1 + s\sqrt{t}]\), where \(s\) and \(t\) are prime numbers.
[6 marks]
The complex number $z$ satisfies the equations
$$|z^* - 1 - 2i| = |z - 3|$$
and
$$|z - a| = 3$$
where $a$ is real.
Show that $a$ must lie in the interval $[1 - s\sqrt{t}, 1 + s\sqrt{t}]$, where $s$ and $t$ are prime numbers.
[6 marks]
\hfill \mbox{\textit{SPS SPS FM Pure 2022 Q8 [6]}}