A-LevelMathematicsCoordinate geometry9709/12 · Oct/Nov 2021 · Q128 Marks

The diagram shows the circle with equation x² + y² – 6x + 4y – 27 = 0 and the tangent to the circle at the point P (5, 4). [Figure X.X]

(a)The tangent to the circle at P meets the x-axis at A and the y-axis at B. Find the area of triangle OAB, where O is the origin.[5]
(b)Points Q and R also lie on the circle, such that PQR is an equilateral triangle. Find the exact area of triangle PQR.[3]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

[B1] Centre is (3, -2)
[*M1] Finding gradient using their centre (not (0, 0)) and P (5,4).
[DM1] Using P and the negative reciprocal of their gradient to find the equation of AB.
[A1] Sight of [x =]17 and [y =] 17/3
[A1] [Area = 1/2 × 17 × 17/3 =] 289/6 Or 48 1/6 or AWRT 48.2.
[B1] Radius of circle = √40
[M1] Using 1/2 r²sinθ with their r and 120 or 60 [×3]
[A1] 30√3 AWRT 52[.0] implies B1M1A0.

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About this question

Topic

Coordinate geometry, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.

Source

A-Level Mathematics, Oct/Nov 2021 Paper 1 Variant 2, question 12.

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