| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | May |
| Marks | 6 |
| Topic | Complex Numbers Argand & Loci |
| Type | Optimization of argument on loci |
| Difficulty | Challenging +1.2 This is a Further Maths complex numbers question requiring geometric interpretation of modulus-argument form and optimization. Students must recognize that the point with least argument lies on the ray from the origin through the center, apply the tangency condition (perpendicular radius), and use coordinate geometry with exact arithmetic. While it requires multiple steps and geometric insight beyond routine manipulation, the setup is relatively standard for FM complex loci, and the 'prove that' format provides the target answer. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
A circle $C$ in the complex plane has equation $|z - 2 - 5i| = a$.
The point $z_1$ on $C$ has the least argument of any point on $C$, and $arg(z_1) = \frac{\pi}{4}$.
Prove that $a = \frac{3\sqrt{2}}{2}$. [6]
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q6 [6]}}