17 The circle \(C\) represents the locus of points satisfying the equation
$$| z - a \mathrm { i } | = b$$
where \(a\) and \(b\) are real constants.
The circle \(C\) intersects the imaginary axis at 2 i and 8 i
The circle \(C\) is shown on the Argand diagram in Figure 2
\begin{figure}[h]
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\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-24_764_770_778_699}
\end{figure}
17
- Write down the value of \(a\)
17
- (ii) Write down the value of \(b\)
17 - The half-line \(L\) represents the locus of points satisfying the equation
$$\arg ( z ) = \tan ^ { - 1 } ( k )$$
where \(k\) is a positive constant.
The point \(P\) is the only point which lies on both \(C\) and \(L\), as shown in Figure 3
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-25_766_770_685_699}
\end{figure}
17 - The point \(O\) represents the number \(0 + 0 \mathrm { i }\)
Calculate the length \(O P\)
17
- (ii) Calculate the exact value of \(k\)
17 - (iii) Find the complex number represented by point \(P\)
Give your answer in the form \(x + y i\) where \(x\) and \(y\) are real.