Optimization of argument on loci

7 The complex number \(u\) is given by \(u = \frac { 7 + 4 \mathrm { i } } { 3 - 2 \mathrm { i } }\).

1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Sketch an Argand diagram showing the point representing the complex number \(u\). Show on the same diagram the locus of the complex number \(z\) such that \(| z - u | = 2\).
3. Find the greatest value of \(\arg z\) for points on this locus.

2

1. Draw on an Argand diagram the locus \(L\) of points satisfying the equation \(\arg z = \frac { \pi } { 6 }\).
   (1 mark)
2. 1. A circle \(C\), of radius 6, has its centre lying on \(L\) and touches the line \(\operatorname { Re } ( z ) = 0\). Draw \(C\) on your Argand diagram from part (a).
   2. Find the equation of \(C\), giving your answer in the form \(\left| z - z _ { 0 } \right| = k\).
   3. The complex number \(z _ { 1 }\) lies on \(C\) and is such that \(\arg z _ { 1 }\) has its least possible value. Find \(\arg z _ { 1 }\), giving your answer in the form \(p \pi\), where \(- 1 < p \leqslant 1\).

A circle \(C\) in the complex plane has equation \(|z - 2 - 5\mathrm{i}| = a\) The point \(z_1\) on \(C\) has the least argument of any point on \(C\), and \(\arg(z_1) = \frac{\pi}{4}\) Prove that \(a = \frac{3\sqrt{2}}{2}\) [6 marks]

The Argand diagram below shows a circle \(C\) \includegraphics{figure_17}

1. Write down the equation of the locus of \(C\) in the form $$|z - w| = a$$ where \(w\) is a complex number whose real and imaginary parts are integers, and \(a\) is an integer. [2 marks]
2. It is given that \(z_1\) is a complex number representing a point on \(C\). Of all the complex numbers which represent points on \(C\), \(z_1\) has the least argument.
   1. Find \(|z_1|\) Give your answer in an exact form. [3 marks]
   2. Show that \(\arg z_1 = \arcsin\left(\frac{6\sqrt{3} - 2}{13}\right)\) [4 marks]

Sketch on an Argand diagram the locus of all points that satisfy \(|z + 4 - 4i| = 2\sqrt{2}\) and hence find \(\theta, \phi \in (-\pi, \pi]\) such that \(\theta \leq \arg z \leq \phi\). [5]

A circle \(C\) in the complex plane has equation \(|z - 2 - 5i| = a\). The point \(z_1\) on \(C\) has the least argument of any point on \(C\), and \(arg(z_1) = \frac{\pi}{4}\). Prove that \(a = \frac{3\sqrt{2}}{2}\). [6]