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June 2003 p3 q5
1916
The complex number 2i is denoted by u. The complex number with modulus 1 and argument \(\frac{2}{3} \pi\) is denoted by w.
(i) Find in the form x + iy, where x and y are real, the complex numbers w, uw and \(\frac{u}{w}\).
(ii) Sketch an Argand diagram showing the points U, A and B representing the complex numbers u, uw and \(\frac{u}{w}\) respectively.
(iii) Prove that triangle UAB is equilateral.
Solution
(i) The complex number w has modulus 1 and argument \(\frac{2}{3} \pi\), so \(w = \cos \frac{2}{3} \pi + i \sin \frac{2}{3} \pi = -\frac{1}{2} + \frac{\sqrt{3}}{2}i\).
To find uw, multiply u and w: \(uw = 2i \times \left(-\frac{1}{2} + \frac{\sqrt{3}}{2}i\right) = -\sqrt{3} - i\).
To find \(\frac{u}{w}\), multiply numerator and denominator by the conjugate of w: \(\frac{u}{w} = \frac{2i}{-\frac{1}{2} + \frac{\sqrt{3}}{2}i} \times \frac{-\frac{1}{2} - \frac{\sqrt{3}}{2}i}{-\frac{1}{2} - \frac{\sqrt{3}}{2}i} = \sqrt{3} - i\).
\((ii) On the Argand diagram, point U is at (0, 2), point A is at (-\sqrt{3}, -1), and point B is at (\sqrt{3}, -1).\)
(iii) To prove triangle UAB is equilateral, show that all sides are equal: \(AB = UA = UB\). Alternatively, show that angles are equal: \(\angle AUB = \angle ABU = \angle BAU = 60^\circ\).
