A real conjugate sum has two directions · source reference

Question

$w=-1-i\sqrt3$ and $z^*+z\ne0$. P and Q represent $w$ and $z^*+z$ respectively. Find $\arg w$ exactly. Hence state possible values of $\angle POQ$, where O is the origin. [3]

[2020/VJC/Prelim/II/3a]

Source working

Printed working

$$\tan^{-1}(\sqrt3/1)=\pi/3,\qquad\arg w=-\pi+\pi/3=-2\pi/3.$$

Since $z^*+z\in\mathbb R$, Q lies on the real axis. Thus $\angle POQ=\pi/3$ or $2\pi/3$; in degrees, $60^\circ$ or $120^\circ$.

Teacher notes

P representing w is in the third quadrant. Let $z=x+iy$, $x,y\in\mathbb R$:

$$z+z^*=(x+iy)+(x-iy)=2x\in\mathbb R.$$

Diagram: Re, Im, O, $P=(-1,-\sqrt3)$, a dashed OP ray and its clockwise argument arc. Q cannot be O because $z^*+z\ne0$.

Printed answer key

$\arg w=-2\pi/3$; $\angle POQ=\pi/3$ or $2\pi/3$.

Native diagrams

Provenance

2025 TJC H2 Math (9758) JC2 MSM: COMPLEX NUMBERS. Question pages 4; solution pages 18. These are original document page numbers, retained as text metadata.