| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2024 |
| Session | January |
| Marks | 14 |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Standard +0.3 This is a standard Further Maths loci question requiring algebraic manipulation to find a circle equation, sketching two loci (circle and half-line), and finding their intersection. Part (a) uses routine techniques (expanding |z-a|=k|z-b| and completing the square), parts (b) and (c) are straightforward applications. Slightly easier than average due to the mechanical nature of the algebra and standard intersection method. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
The point $P$ represents a complex number $z$ on an Argand diagram such that
$$|z - 6i| = 2|z - 3|$$
\begin{enumerate}[label=(\alph*)]
\item Show that, as $z$ varies, the locus of $P$ is a circle, stating the radius and the coordinates of the centre of this circle. [6]
\end{enumerate}
The point $Q$ represents a complex number $z$ on an Argand diagram such that
$$\arg(z - 6) = -\frac{3\pi}{4}$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Sketch, on the same Argand diagram, the locus of $P$ and the locus of $Q$ as $z$ varies. [4]
\item Find the complex number for which both $|z - 6i| = 2|z - 3|$ and $\arg(z - 6) = -\frac{3\pi}{4}$. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q7 [14]}}