A-LevelMathematicsCoordinate geometry9709/12 · Feb/Mar 2020 · Q1211 Marks

A diameter of a circle C₁ has end-points at (−3, −5) and (7, 3).

(a)Find an equation of the circle C₁.[3]
(b)Find an equation of the circle C₂.[2]
(c)Show that the equation of the line RS is y = −2x + 13.[4]
(d)Hence show that the x-coordinates of R and S satisfy the equation 5x² – 60x + 159 = 0.[2]

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Mark scheme

Marking points from the official mark scheme (Feb/Mar 2020). The mark or mark type is shown in brackets.

[B1] (a) Centre = (2, -1)
[M1] (a) r² =[2-(-3)]² + [-1-(-5)]² or [2-7]² + [-1-3]² OE OR 1/2[(-3-7)²+(-5-3)²] OE
[A1] (a) (x−2)² + (y+1)² = 41 Must not involve surd form SCB3 (x+3)(x−7)+(y+5)(y-3) = 0
[B1FT] (b) Centre = their (2, -1) + (8, 4) = (10, 3) SOI FT on their (2, -1)
[B1FT] (b) (x-10)²+(y-3)² = their 41 FT on their 41 even if in surd form SCB2 (x-5)(x-15)+(y+1)(y−7) = 0
[M1] (c) (x-2)² + (y+1)²-41=(x-10)² + (y-3)² -41 OE
[A1] (c) x² - 4x + 4 + y² + 2y +1 = x² - 20x +100+ y² −6y+9 OE Condone 1 error or errors caused by 1 error in the first line
[A1] (c) 16x+8y=104
[A1] (c) y = -2x+13 AG
[M1] (d) (x-10)²+(-2x+13-3)² = 41 Or eliminate y between C1 and C2
[A1] (d) x² - 20x +100+ 4x² − 40x +100 = 41 → 5x² − 60x+159=0 AG

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