The circle C has equation x^2 + y^2 - 6x + 8y - 24 = 0. (a) Find the coordinates of the centre of C and the radius of C. (b) The line L has equation 3x + 4y - k = 0, where k is a constant. Given that L is tangent to C, find the two possible values of k.

Mark scheme

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

1. (a) M1: Completes the square for x and y terms: (x^2 - 6x) + (y^2 + 8y) = 24.
2. (a) A1: (x - 3)^2 - 9 + (y + 4)^2 - 16 = 24 => (x - 3)^2 + (y + 4)^2 = 49 (correct equation of circle).
3. (a) A1: Centre (3, -4) and Radius = sqrt(49) = 7 (both correct).
4. (b) M1: Uses the formula for the perpendicular distance from the centre (their (h, k) from part a) to the line 3x + 4y - k = 0.
5. (b) A1: Perpendicular distance = |3(3) + 4(-4) - k| / sqrt(3^2 + 4^2) = |9 - 16 - k| / sqrt(25) = |-7 - k| / 5.
6. (b) M1: Sets the perpendicular distance equal to the radius of the circle (their radius from part a).
7. (b) A1: |-7 - k| / 5 = 7 => |-7 - k| = 35.
8. (b) M1: Solves the absolute value equation correctly: -7 - k = 35 OR -7 - k = -35.
9. (b) A1: k = -42 or k = 28. (Both values required for the mark).