N2011 P1 Q10

Do not use a graphing calculator in answering this question.

1. The roots of the equation \({{z}^{2}}=-8\mathbf{i}\) are \({{z}_{1}}\) and \({{z}_{2}}\). Find \({{z}_{1}}\) and \({{z}_{2}}\) in cartesian form \(x+\mathbf{i}y\), showing your working.[4]
2. Hence, or otherwise, find in cartesian form the roots \({{w}_{1}}\) and \({{w}_{2}}\) of the equation
   \({{w}^{2}}+4w+\left( 4+2\mathbf{i} \right)=0\)[3]

3. Using a single Argand diagram, sketch the loci
   1. \(|z-z_1|=|z-z_2|\),[1]
   2. \(|z-w_1|=|z-w_2|\).[1]
4. Give a reason why there are no points which lie on both of these loci.[1]