2023TrialShort answer

Q15 (c) (i)

2 marks

Question

$z_1$ and $z_2$ are two complex numbers representing the two points $A$ and $B$ in the Argand diagram. $z_3$ is a complex number representing the point $C$ such that $|AB|:|AC|=1:4$ . $z_4$ is a complex number representing the point $D$ , such that $|OB|:|OD|=1:k$ , for some constant $k$ and $O$ is the origin. The points $A$ , $B$ and $C$ are collinear.

Find $z_3$ in terms of $z_1$ and $z_2$ .

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Solution

\begin{aligned} \overrightarrow{AC}&=4\overrightarrow{AB}\\ z_3-z_1&=4(z_2-z_1)\\ z_3&=z_1+4z_2-4z_1\\ z_3&=4z_2-3z_1 \end{aligned}

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Syllabus alignment

Complex Numbers

Geometrical implications of complex numbersInterpret complex-number operations geometrically and solve problems involving roots and loci in the complex plane.

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