- The points representing the complex numbers \(z _ { 1 } = 35 - 25 i\) and \(z _ { 2 } = - 29 + 39 i\) are opposite vertices of a regular hexagon, \(H\), in the complex plane.
The centre of \(H\) represents the complex number \(\alpha\)
- Show that \(\alpha = 3 + 7 \mathrm { i }\)
Given that \(\beta = \frac { 1 + \mathrm { i } } { 64 }\)
- show that
$$\beta \left( z _ { 1 } - \alpha \right) = 1$$
The vertices of \(H\) are given by the roots of the equation
$$( \beta ( z - \alpha ) ) ^ { 6 } = 1$$
- Write down the roots of the equation \(w ^ { 6 } = 1\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\)
- Hence, or otherwise, determine the position of the other four vertices of \(H\), giving your answers as complex numbers in Cartesian form.