3 Two loci, \(L _ { 1 }\) and \(L _ { 2 }\), in an Argand diagram are given by $$\begin{aligned} & L _ { 1 } : | z + 1 + 3 \mathrm { i } | = | z - 5 - 7 \mathrm { i } |
& L _ { 2 } : \arg z = \frac { \pi } { 4 } \end{aligned}$$

1. Verify that the point represented by the complex number \(2 + 2 \mathrm { i }\) is a point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).
2. Sketch \(L _ { 1 }\) and \(L _ { 2 }\) on one Argand diagram.
3. Shade on your Argand diagram the region satisfying
   both $$| z + 1 + 3 i | \leqslant | z - 5 - 7 i |$$ and $$\frac { \pi } { 4 } \leqslant \arg z \leqslant \frac { \pi } { 2 }$$