A circle `C` has equation `x^2 + y^2 - 4x + 6y - 12 = 0`. (a) Find the coordinates of the centre of `C` and the radius of `C`. (b) The point `P(5, 1)` lies on `C`. (i) Find an equation of the tangent to `C` at `P`. (ii) This tangent intersects the y-axis at `A` and the x-axis at `B`. Find the exact area of triangle `OAB`, where `O` is the origin.

Mark scheme

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

1. M1 for completing the square for `x` or `y` terms OR using `(-g, -f)` for centre.
2. A1 for centre `(2, -3)`.
3. A1 for radius `r = √(2² + (-3)² - (-12)) = √(4+9+12) = √25 = 5`.
4. M1 for finding gradient of radius `CP`. (`(1 - (-3)) / (5 - 2) = 4/3`).
5. M1 for finding gradient of tangent. (`-1 / (gradient of radius) = -3/4`).
6. M1 for using `y - y_1 = m(x - x_1)` with point `P(5, 1)` and their tangent gradient.
7. A1 for `y - 1 = -3/4(x - 5)` OR `4y + 3x - 19 = 0` (or equivalent integer form).
8. M1 for setting `x=0` to find y-intercept (A) or `y=0` to find x-intercept (B) for their tangent equation.
9. A1 for `A = (0, 19/4)` and `B = (19/3, 0)`.
10. M1 for using area formula `1/2 * base * height`.
11. A1 for `1/2 * (19/3) * (19/4) = 361/24`.