A circle C has equation x² + y² - 6x + 8y - 24 = 0. (a) Find the coordinates of the centre of C and the radius of C. (b) The point P(7, 3) lies on the circumference of the circle C. Find the equation of the tangent to C at P. Give your answer in the form ax+by+c=0.

Mark scheme

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

1. (a)
2. M1 for attempting to complete the square for x and y terms OR using formula for centre (-g, -f) and radius √(g²+f²-c).
3. A1 for centre $(3, -4)$. (Both coordinates correct)
4. A1 for radius r = √((-3)² + (4)² - (-24)) = √(9+16+24) = √49 = 7. (cao)
5. (b)
6. M1 for finding the gradient of the radius CP. Centre $(3, -4)$, Point P(7, 3). m_(CP) = (3 - (-4))/(7 - 3) = 7/4.
7. M1 for using the perpendicular condition to find the gradient of the tangent, m_(tangent) = -1/(m_(CP)).
8. A1 for m_(tangent) = -4/7.
9. M1 for using the equation of a straight line y - y₁ = m(x - x₁) with point P(7, 3) and their gradient of the tangent.
10. A1 for y - 3 = -4/7(x - 7).
11. A1 for 4x + 7y - 49 = 0 (or equivalent integer form ax+by+c=0). (cao)