3 A circle \(C\) and a half-line \(L\) have equations
$$| z - 2 \sqrt { 3 } - \mathrm { i } | = 4$$
and
$$\arg ( z + i ) = \frac { \pi } { 6 }$$
respectively.
- Show that:
- the circle \(C\) passes through the point where \(z = - \mathrm { i }\);
- the half-line \(L\) passes through the centre of \(C\).
- On one Argand diagram, sketch \(C\) and \(L\).
- Shade on your sketch the set of points satisfying both
$$| z - 2 \sqrt { 3 } - \mathrm { i } | \leqslant 4$$
and
$$0 \leqslant \arg ( z + i ) \leqslant \frac { \pi } { 6 }$$