The circle C has equation x² + y² - 6x + 8y - 24 = 0. (a) Find the coordinates of the centre of C and the exact radius of C. (b) The point P(8, -10) lies on the circle. Find an equation of the tangent to C at the point P. Give your answer in the form ax + by + c = 0, where a, b and c are integers.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. (a) M1 for attempting to complete the square for x or y terms (e.g. (x-3)² or (y+4)²)
2. (a) A1 for centre $(3, -4)$
3. (a) M1 for using r² = g² + f² - c or (x-a)² + (y-b)² = r² to find r² or r
4. (a) A1 for radius r=7 (accept √49)
5. (b) M1 for finding the gradient of the radius CP using their centre $(3, -4)$ and P(8, -10)
6. (b) A1 for gradient of radius CP = (-10 - (-4))/(8 - 3) = (-6)/5
7. (b) M1 for using the perpendicular condition to find the gradient of the tangent, m_(tan) = -1/(m_(radius))
8. (b) A1 for gradient of tangent = 5/6
9. (b) M1 for forming the equation of the tangent using y - y₁ = m(x - x₁) with their gradient and point P(8, -10)
10. (b) A1 for 5x - 6y - 110 = 0 (or any integer multiple)