9709 P33 - Jun 2022 - Q5
1927
The complex number 3 - i is denoted by u.
(a) Show, on an Argand diagram with origin O, the points A, B and C representing the complex numbers u, u^* and u^* - u respectively. State the type of quadrilateral formed by the points O, A, B and C.
(b) Express \(\frac{u^*}{u}\) in the form \(x + iy\), where \(x\) and \(y\) are real.
(c) By considering the argument of \(\frac{u^*}{u}\), or otherwise, prove that \(\arctan\left(\frac{3}{4}\right) = 2 \arctan\left(\frac{1}{3}\right)\).
