A circle \(C\) in the Argand diagram has equation
$$| z + 5 - \mathrm { i } | = \sqrt { 2 }$$
Write down its radius and the complex number representing its centre.
A half-line \(L\) in the Argand diagram has equation
$$\arg ( z + 2 \mathrm { i } ) = \frac { 3 \pi } { 4 }$$
Show that \(z _ { 1 } = - 4 + 2 \mathrm { i }\) lies on \(L\).
Show that \(z _ { 1 } = - 4 + 2 \mathrm { i }\) also lies on \(C\).
Hence show that \(L\) touches \(C\).
Sketch \(L\) and \(C\) on one Argand diagram.
The complex number \(z _ { 2 }\) lies on \(C\) and is such that \(\arg \left( z _ { 2 } + 2 \mathrm { i } \right)\) has as great a value as possible.
Indicate the position of \(z _ { 2 }\) on your sketch.