The coordinates of points A, B and C are (6, 4), (p, 7) and (14, 18) respectively, where p is a constant. The line AB is perpendicular to the line BC.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. [*M1] Their gradients must both come from (Difference in the ys / Difference in the xs) OR Scalar product leading to (14-p)(6-p)-33=0.
2. [A1] Clearing of fractions and collecting terms to arrive at the three-term quadratic. Allow integer multiples.
3. [DM1] OE Solving their three-term quadratic.
4. [A1] If M1A1DM0 scored then SC B1 is available for final answer.
5. [B1] SOI by final answer.
6. [M1] Finding half of the length of AC or using their centre, which cannot be A, B or C, to find r² or r. Note: r=65 is M0.
7. [A1] (x-6)(x-14)+(y-4)(y-18)=0 scores 3/3.
8. [*M1] Gradient of their centre, which cannot be A, B or C, from part (b), to A or C or the gradient of AC but working needed if incorrect centre. OR by clearly differentiating and substitution of (14,18).
9. [DM1] OE Using (14,18) and -1/their 7/4 to form the equation of a straight line.
10. [A1] All terms on one side in any order. Allow multiples of this format by an integer only.