3 In this question you must show detailed reasoning.
The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by \(z _ { 1 } = 2 - 3 i\) and \(z _ { 2 } = a + 4 i\) where \(a\) is a real number.
- Express \(z _ { 1 }\) in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures.
- Find \(z _ { 1 } z _ { 2 }\) in terms of \(a\), writing your answer in the form \(c + \mathrm { id }\).
- The real and imaginary parts of a complex number on an Argand diagram are \(x\) and \(y\) respectively. Given that the point representing \(z _ { 1 } z _ { 2 }\) lies on the line \(y = x\), find the value of \(a\).
- Given instead that \(z _ { 1 } z _ { 2 } = \left( z _ { 1 } z _ { 2 } \right) ^ { * }\) find the value of \(a\).