June 2024 Paper 3 Q9

AQACurrent spec9 marksCo-ordinate GeometryTrigonometry

9 Figure 1 below shows a circle.

A circle with centre P, to the left of the y-axis and above the x-axis; the circle crosses the y-axis, with the upper intersection marked Q
Figure 1

The centre of the circle is \(P\) and the circle intersects the \(y\)-axis at \(Q\) as shown in Figure 1.

The equation of the circle is

\[x^2 + y^2 = 12y - 8x - 27\]
(a) Express the equation of the circle in the form\[(x - a)^2 + (y - b)^2 = k\]

where \(a\), \(b\) and \(k\) are constants to be found. [3 marks]

(b) State the coordinates of \(P\) [1 mark]
(c) Find the \(y\)-coordinate of \(Q\) [2 marks]
(d) The line segment \(QR\) is a tangent to the circle as shown in Figure 2 below.
The circle with centre P; Q on the y-axis; the tangent QR from Q to the point R below the x-axis to the right; the line segments PQ and PR are drawn
Figure 2

The point \(R\) has coordinates (9, −3).

Find the angle \(QPR\)

Give your answer in radians to three significant figures. [3 marks]

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