The complex number \(w\) is such that
$$\arg ( w + 2 \mathrm { i } ) = \tan ^ { - 1 } \frac { 1 } { 2 }$$
It is given that \(w = x + \mathrm { i } y\), where \(x\) and \(y\) are real and \(x > 0\)
Find an equation for \(y\) in terms of \(x\)
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The complex number \(z\) satisfies both
$$- \frac { \pi } { 2 } \leq \arg ( z + 2 \mathrm { i } ) \leq \tan ^ { - 1 } \frac { 1 } { 2 } \quad \text { and } \quad | z - 2 + 3 \mathrm { i } | \leq 2$$
The region \(R\) is the locus of \(z\)
Sketch the region \(R\) on the Argand diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-10_1015_1020_1683_511}
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\(\quad z _ { 1 }\) is the point in \(R\) at which \(| z |\) is minimum.
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Calculate the exact value of \(\left| z _ { 1 } \right|\)
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(ii) Express \(z _ { 1 }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real.