A-LevelMathematicsCoordinate geometry9709/12 · Feb/Mar 2025 · Q66 Marks

The diagram shows a circle C of radius r, where x > 0 and y > 0 for all points on C. The least distance between any point on C and the x-axis is 8 units, and the least distance between any point on C and the y-axis is 5 units. [Figure X.X]

(a)State the coordinates of the centre of the circle in terms of r.[1]
(b)Given that the distance between the origin and the centre of the circle is 15 units, find the value of r.[3]
(c)The point on the circle furthest from the origin is denoted by P. Find the gradient of the tangent to the circle at P.[2]

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Mark scheme

Marking points from the official mark scheme (Feb/Mar 2025). The mark or mark type is shown in brackets.

[B1] (r+5, r+8) OE Allow x=r+5, y=r+8. If values are stated without reference to x and y, take the first value to be their x.
[B1] their(r+5)² +their(r+8)² = 15² FT Following their answers to (a), which must both contain r.
[M1] [r² +13r-68[=0]] (r+17)(r-4)[=0] Or other valid method of solution for their three-term quadratic.
[A1] [r =]4 CWO r=-4 ⇒ r= 4 scores A0. Special Case: After B1M0, r=4 scores SCB1, but after B1M0, r = -4⇒r=4 scores B0.
[M1] their (r+8) / their (r+5) from (a) with their r from (b), or (their r from(b))+8 / (their r from(b))+5 r>0 only.
[A1] 3/4 OE, i.e. (their (r+5)) / (their (r+8)) or (their r from(b))+8 / (their r from(b))+5

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