Higher June 2019 Paper 3 Q22

EdexcelCurrent spec4 marksBasic Trigonometry Circle Theorems

22 The diagram shows a circle, centre \(O\).

Circle centre O at the origin of x and y axes; P(3p, p) lies on the circle; B(16, 0) is on the x-axis outside the circle; the tangent from B touches the circle at A below the x-axis, and angle OBA = 30 degrees

\(AB\) is the tangent to the circle at the point \(A\).
Angle \(OBA = 30^\circ\)

Point \(B\) has coordinates \((16, 0)\)
Point \(P\) has coordinates \((3p, p)\)

Find the value of \(p\).
Give your answer correct to 1 decimal place.
You must show all your working. (4)

Mark scheme

Answer	Mark	Mark scheme
2.5	P1	use of \(\sin 30 = \tfrac{1}{2}\) to find \(OA\ (= 8)\) or \(OAB = 90^\circ\) eg \(OA = 16\sin 30^\circ\) or right angle marked on diagram
	P1	recognition that equation of circle is \(x^2 + y^2 = r^2\)
	P1	Correct substitution of \(p\), \(3p\) and \(r\) in \(x^2 + y^2 = r^2\) eg \(9p^2 + p^2 = OA^2\) or \((3p)^2 + p^2 = \text{``}8^2\text{''}\)
	A1	for answer in the range 2.5 to 2.53

Additional guidance

Accept \(3p^2 + p^2 = r^2\) for the award of this mark

Do not accept \(3p^2 + p^2 = 8^2\) for the award of this mark

Accept \(\sqrt{6.4}\) or \(\dfrac{4\sqrt{10}}{5}\)
If an answer within the given range is seen in working and rounded incorrectly award full marks.
Award 0 marks for the answer without supportive working.