Challenging +1.8 This is a Further Maths loci problem requiring students to find the geometric interpretation of two conditions (perpendicular bisector and circle), determine their intersection constraints, and express the result in a specific form with prime numbers. It demands visualization, algebraic manipulation of modulus equations, and optimization thinking, but follows a relatively standard Further Maths approach to loci intersection problems.
8 The complex number \(z\) satisfies the equations
$$\left| z ^ { * } - 1 - 2 i \right| = | z - 3 |$$
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. [0pt]
[6 marks]
8 The complex number $z$ satisfies the equations
$$\left| z ^ { * } - 1 - 2 i \right| = | z - 3 |$$
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.\\[0pt]
[6 marks]\\
\hfill \mbox{\textit{AQA Further Paper 2 2021 Q8 [6]}}