9709 P3 - Jun 2006 - Q7
1910
The complex number 2 + i is denoted by u. Its complex conjugate is denoted by u*.
(i) Show, on a sketch of an Argand diagram with origin O, the points A, B and C representing the complex numbers u, u* and u + u* respectively. Describe in geometrical terms the relationship between the four points O, A, B and C. [4]
(ii) Express \(\frac{u}{u^*}\) in the form \(x + iy\), where x and y are real. [3]
(iii) By considering the argument of \(\frac{u}{u^*}\), or otherwise, prove that \(\arctan\left(\frac{4}{3}\right) = 2 \arctan\left(\frac{1}{2}\right)\). [2]
