2023TrialShort answer

Q15 (c) (ii)

3 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.

Given that $z_2-z_1$ and $z_4-z_3$ are perpendicular, prove that

k=\frac{4|z_2|^2-7|z_1||z_2|\cos\theta+3|z_1|^2}{|z_2|^2-|z_1||z_2|\cos\theta},

where $\theta$ is the angle between $z_1$ and $z_2$ .

Try the question before revealing the answer.

Solution

\begin{aligned} z_4-z_3&\perp z_2-z_1\\ z_4-z_3&=ip(z_2-z_1)\quad\text{for positive real }p\\ z_4&=z_3+ip(z_2-z_1)\\ kz_2&=4z_2-3z_1+ip(z_2-z_1)\\ k&=4-\frac{3z_1}{z_2}+ip\left(1-\frac{z_1}{z_2}\right) \end{aligned}

Let $z_1=ae^{i\alpha}$ and $z_2=be^{i\beta}$ and $\theta=\alpha-\beta$ .

\begin{aligned} k&=4-\frac{3a}{b}e^{i(\alpha-\beta)}+ip\left(1-\frac{a}{b}e^{i(\alpha-\beta)}\right)\\ &=4-\frac{3a}{b}e^{i\theta}+ip\left(1-\frac{a}{b}e^{i\theta}\right)\\ &=4-\frac{3a}{b}(\cos\theta+i\sin\theta)+ip\left(1-\frac{a}{b}(\cos\theta+i\sin\theta)\right)\\ &=4-\frac{3a}{b}\cos\theta+\frac{pa}{b}\sin\theta+i\left(p-\frac{pa}{b}\cos\theta-\frac{3a}{b}\sin\theta\right) \end{aligned}

Since $k$ is real,

p-\frac{pa}{b}\cos\theta-\frac{3a}{b}\sin\theta=0

p=\frac{3a\sin\theta}{b-a\cos\theta}

Now,

\begin{aligned} k&=4-\frac{3a}{b}\cos\theta+\frac{3a\sin\theta}{b-a\cos\theta}\cdot\frac{a}{b}\sin\theta\\ &=4-\frac{3a}{b}\cos\theta+\frac{3a^2\sin^2\theta}{b(b-a\cos\theta)}\\ &=\frac{4(b^2-ab\cos\theta)-3ab\cos\theta(b-a\cos\theta)+3a^2\sin^2\theta}{b(b-a\cos\theta)}\\ &=\frac{4b^2-4ab\cos\theta-3ab^2\cos\theta+3a^2b\cos^2\theta+3a^2\sin^2\theta}{b^2-ab\cos\theta}\\ &=\frac{4b^2-7ab\cos\theta+3a^2}{b^2-ab\cos\theta}\\ &=\frac{4|z_2|^2-7|z_1||z_2|\cos\theta+3|z_1|^2}{|z_2|^2-|z_1||z_2|\cos\theta} \end{aligned}

Supplied marking guidance · human reviewed

Compare this with your working, not only your final sentence.

Syllabus alignment

Complex Numbers

Geometrical implications of complex numbersInterpret complex-number operations geometrically and solve problems involving roots and loci in the complex plane.
Geometric representation of a complex numberRepresent complex numbers in the complex plane and use modulus, argument and polar form with their identities.

Spot a problem? Send feedback