The points $P$, $Q$ and $R$ representing the complex numbers $p$, $q$ and $r$ on an Arga

Question

The points $P$, $Q$ and $R$ representing the complex numbers $p$, $q$ and $r$ on an Argand diagram are such that $\arg(p) = \alpha$ and $\arg(q) = \beta$, where $0 < \alpha < \beta < \frac{\pi}{2}$, $\beta > 2\alpha$ and $r = p + q$. If $|p|=|q|$, describe the shape of the quadrilateral of $OPRQ$. Hence find $\arg(r)$ in terms of $\alpha$ and $\beta$.[3] The point $Q'$, representing the complex number $q'$, is the reflection of the point $Q$ in $OP$. State the angle $\angle POQ'$.[1] By leaving your answers in terms of $\alpha$, $\beta$ and $|q|$ where applicable, hence, or otherwise, find the argument of the complex number $q'$, [1] find the real and imaginary parts of $q'$ and write down $q'$ in $a+ib$ form. [3]

Accepted Answer

(a) $OPRQ$ is a rhombus, and $\arg(r)=\frac12(\alpha+\beta)$ (b) $\angle POQ'=\beta-\alpha$ (b)(i) $\arg(q')=2\alpha-\beta$ (b)(ii) $\Re(q')=|q|\cos(2\alpha-\beta)$, $\Im(q')=|q|\sin(2\alpha-\beta)$, and $q'=|q|\cos(2\alpha-\beta)+i|q|\sin(2\alpha-\beta)$

2025 NYJC P1 Q6

2025 NYJC P1 Q6

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