Five points from one polar vector · source reference

Question

$z$ has modulus $r$ and argument $\theta$, with $0<r<1$ and $\pi/2<\theta<\pi$. $w$ satisfies $|w|=|z|$ and $\arg w+\arg z=\pi$.

A, B, C, D and E represent $z,w,z-w,z^*,-iz$ respectively, where $z^*$ is the conjugate of $z$. On a single Argand diagram illustrate all five points. [5]

[modified 2019/NJC/Prelim/I/1]

Source working

Printed working and teacher notes

$\arg z=\theta$, $\arg w=\pi-\theta$; OA and OB are equally inclined to the real axis.

$z-w$ is represented by $\overrightarrow{BA}$ or $\overrightarrow{OC}$, where $\overrightarrow{OC}=\overrightarrow{BA}$. Since $|\overrightarrow{OA}|=|\overrightarrow{OB}|$, BA is parallel to the real axis. C therefore lies on the real axis.

E is A rotated clockwise through $90^\circ$ about the origin.

All diagram labels

Re, Im, O, A, B, C, D, E; radius $r$ on OA, OB, OD and OE; angles $\theta$, $\pi-\theta$, $-\theta$; right angle between OA and OE; $C=(2r\cos\theta,0)$.

Native coordinates clarify the diagram without changing its content:

$$A=(r\cos\theta,r\sin\theta),\quad B=(-r\cos\theta,r\sin\theta),$$ $$C=(2r\cos\theta,0),\quad D=(r\cos\theta,-r\sin\theta),\quad E=(r\sin\theta,-r\cos\theta).$$

Printed answer key

No separate short-key entry is printed; see the complete working diagram.

Native diagrams

Provenance

2025 TJC H2 Math (9758) JC2 MSM: COMPLEX NUMBERS. Question pages 3; solution pages 18. These are original document page numbers, retained as text metadata.