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June 2011 p32 q7
2003
(a) The complex number u is defined by \(u = \frac{5}{a + 2i}\), where the constant a is real.
Express u in the form x + iy, where x and y are real.
Find the value of a for which \(\arg(u^*) = \frac{3}{4}\pi\), where u* denotes the complex conjugate of u.
(b) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z which satisfy both the inequalities \(|z| < 2\) and \(|z| < |z - 2 - 2i|\).
Solution
(a)(i) To express \(u\) in the form \(x + iy\), multiply the numerator and denominator by the conjugate of the denominator:
(a)(ii) The argument of \(u^*\) is \(-\arg(u)\). Given \(\arg(u^*) = \frac{3}{4}\pi\), we have \(\arg(u) = -\frac{3}{4}\pi\).
From \(u = \frac{5a}{a^2 + 4} + \frac{10i}{a^2 + 4}\), the argument is \(\arctan\left(\frac{10}{5a}\right) = -\frac{3}{4}\pi\).
Solving \(\tan\left(-\frac{3}{4}\pi\right) = \frac{10}{5a}\), we find \(a = -2\).
(b) The inequality \(|z| < 2\) represents a circle centered at the origin with radius 2. The inequality \(|z| < |z - 2 - 2i|\) represents the region closer to the origin than to the point \(2 + 2i\). This is the region below the perpendicular bisector of the line segment from the origin to \(2 + 2i\). Shade the region inside the circle and below this line.
