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By definition, an ellipse is the set of points the sum of the distances to two fixed points, the foci, is a constant.
In the complex plane if u & v are points then the distance from u to v is |u-v|.
Because $\displaystyle \pm 2i$ are the foci we have:
$\displaystyle \begin{array}{l}
\left| {z - 2i} \right| + \left| {z + 2i} \right| = \left| {3 + 2i - 2i} \right| + \left| {3 + 2i + 2i} \right| \\
\left| {z - 2i} \right| + \left| {z + 2i} \right| = 3 + \sqrt {25} \\
\left| {z - 2i} \right| + \left| {z + 2i} \right| = 8 \\
\end{array}$.
Now you finish.