18 The locus of points \(L _ { 1 }\) satisfies the equation \(| z | = 2\)
The locus of points \(L _ { 2 }\) satisfies the equation \(\arg ( z + 4 ) = \frac { \pi } { 4 }\)
18
- Sketch \(L _ { 1 }\) on the Argand diagram below.
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\includegraphics[max width=\textwidth, alt={}, center]{86aa9e6f-261c-40d4-8271-a0dc560d8a72-26_1152_1195_644_427}
18 - Sketch \(L _ { 2 }\) on the Argand diagram above.
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[1 mark]
18 - The complex number \(a + \mathrm { i } b\), where \(a\) and \(b\) are real, lies on \(L _ { 1 }\)
The complex number \(c + \mathrm { i } d\), where \(c\) and \(d\) are real, lies on \(L _ { 2 }\)
Calculate the least possible value of the expression
$$( c - a ) ^ { 2 } + ( d - b ) ^ { 2 }$$
\includegraphics[max width=\textwidth, alt={}, center]{86aa9e6f-261c-40d4-8271-a0dc560d8a72-28_2492_1721_217_150}