7. A complex number \(z\) is given by
$$z = a + 2 i$$
where \(a\) is a non-zero real number.
- Find \(z ^ { 2 } + 2 z\) in the form \(x +\) iy where \(x\) and \(y\) are real expressions in terms of \(a\).
Given that \(z ^ { 2 } + 2 z\) is real,
- find the value of \(a\).
Using this value for \(a\),
- find the values of the modulus and argument of \(z\), giving the argument in radians, and giving your answers to 3 significant figures.
- Show the points \(P , Q\) and \(R\), representing the complex numbers \(z , z ^ { 2 }\) and \(z ^ { 2 } + 2 z\) respectively, on a single Argand diagram with origin \(O\).
- Describe fully the geometrical relationship between the line segments \(O P\) and \(Q R\).