2022HSCShort answer

Q16 (a)

4 marks

Question

A square in the Argand plane has vertices $5+5i$ , $5-5i$ , $-5-5i$ and $-5+5i$ . The complex numbers $z_A=5+i$ , $z_B$ and $z_C$ lie on the square and form the vertices of an equilateral triangle, as shown in the diagram.

Find the exact value of the complex number $z_B$ .

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Solution

Criteria	Marks
Provides correct solution	4
Equates real or imaginary parts from $z_C-z_A=e^{i\pi/3}(z_B-z_A)$ OR obtains a quadratic in $x$ from $	z_B-z_A
Attempts to use either $z_C-z_A=e^{i\pi/3}(z_B-z_A)$ OR $	z_B-z_A
Observes that $z_B=x+5i$ , for some real number $x$ , or equivalent merit	1

Sample answer:
Let $z_B=b+5i$ and $z_C=c-5i$ where $b$ and $c$ are real numbers.
$e^{i\pi/3}(z_B-z_A)=z_C-z_A$

\left(\frac12+i\frac{\sqrt3}{2}\right)((b-5)+4i)=(c-5)-6i

Equating the imaginary parts, we get
$2+\frac{\sqrt3}{2}(b-5)=-6$
$\frac{\sqrt3}{2}(b-5)=-8$
$b=5-\frac{16}{\sqrt3}$
$\therefore z_B=\left(5-\frac{16}{\sqrt3}\right)+5i$

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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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