A-LevelMathematicsCoordinate geometry9709/12 · Oct/Nov 2024 · Q810 Marks

The equation of a circle is x² + y² +px+2y+q = 0, where p and q are constants.

(a)Express the equation in the form (x−a)² + (y−b)² = r², where a is to be given in terms of p and r² is to be given in terms of p and q.[2]
(b)(i)The line with equation x+2y = 10 is the tangent to the circle at the point A (4, 3). Find the equation of the normal to the circle at the point A.[3]
(b)(ii)Find the values of p and q.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

[B1*] (x-(-p/2))² + (y-(-1))² OE Allow a = -p/2 and b = -1, or centre is (-p/2, -1).
[DB1] (x-(-p/2))² + (y-(-1))² = -q+1+1/4 p² OE
[B1] [Gradient of tangent =] -1/2 OE SOI
[M1] [Gradient of normal =] 2 Use of m1m2= – 1with their numeric tangent gradient.
[A1] y-3 = 2(x-4) [y = 2x - 5] OE ISW Allow y = 2x +c, 3=2×4+c ⇒ c=-5.
[M1*] Using their stated centre or (±p/2, ±1) in their equation of the normal.
[A1] p=-4
[M1*] Using (4, 3) and their centre or (±their p/2, ±1) to find r² or r.
[DM1] Using their expression for r² (from (a)) equated to their 20.
[A1] q=-15

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Topic: Coordinate geometry, A-Level Mathematics (syllabus 9709). Structured question, 10 marks.
Source: A-Level Mathematics, Oct/Nov 2024 Paper 1 Variant 2, question 8.

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