AS June 2024 Paper 1 Q17

AQACurrent spec9 marksComplex Numbers

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 \(2\mathrm{i}\) and \(8\mathrm{i}\)

The circle \(C\) is shown on the Argand diagram in Figure 2

Figure 2: Argand diagram showing the circle C, centred on the imaginary axis, crossing it at 2 and 8
Figure 2
(a)
(i) Write down the value of \(a\) [1 mark]
(ii) Write down the value of \(b\) [1 mark]
(b) 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

Figure 3: the circle C with the half-line L from O touching the circle at the point P in the first quadrant
Figure 3
(i) The point \(O\) represents the number \(0 + 0\mathrm{i}\)

Calculate the length \(OP\) [2 marks]

(ii) Calculate the exact value of \(k\) [2 marks]
(iii) Find the complex number represented by point \(P\)

Give your answer in the form \(x + y\mathrm{i}\) where \(x\) and \(y\) are real. [3 marks]

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